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monocypher.c (110141B)

---

 // Monocypher version 3.1.2
 //
 // This file is dual-licensed.  Choose whichever licence you want from
 // the two licences listed below.
 //
 // The first licence is a regular 2-clause BSD licence.  The second licence
 // is the CC-0 from Creative Commons. It is intended to release Monocypher
 // to the public domain.  The BSD licence serves as a fallback option.
 //
 // SPDX-License-Identifier: BSD-2-Clause OR CC0-1.0
 //
 // ------------------------------------------------------------------------
 //
 // Copyright (c) 2017-2020, Loup Vaillant
 // All rights reserved.
 //
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are
 // met:
 //
 // 1. Redistributions of source code must retain the above copyright
 //    notice, this list of conditions and the following disclaimer.
 //
 // 2. Redistributions in binary form must reproduce the above copyright
 //    notice, this list of conditions and the following disclaimer in the
 //    documentation and/or other materials provided with the
 //    distribution.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
 // HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 //
 // ------------------------------------------------------------------------
 //
 // Written in 2017-2020 by Loup Vaillant
 //
 // To the extent possible under law, the author(s) have dedicated all copyright
 // and related neighboring rights to this software to the public domain
 // worldwide.  This software is distributed without any warranty.
 //
 // You should have received a copy of the CC0 Public Domain Dedication along
 // with this software.  If not, see
 // <https://creativecommons.org/publicdomain/zero/1.0/>
 
 #include "monocypher.h"
 
 /////////////////
 /// Utilities ///
 /////////////////
 #define FOR_T(type, i, start, end) for (type i = (start); i < (end); i++)
 #define FOR(i, start, end)         FOR_T(size_t, i, start, end)
 #define COPY(dst, src, size)       FOR(i, 0, size) (dst)[i] = (src)[i]
 #define ZERO(buf, size)            FOR(i, 0, size) (buf)[i] = 0
 #define WIPE_CTX(ctx)              crypto_wipe(ctx   , sizeof(*(ctx)))
 #define WIPE_BUFFER(buffer)        crypto_wipe(buffer, sizeof(buffer))
 #define MIN(a, b)                  ((a) <= (b) ? (a) : (b))
 #define MAX(a, b)                  ((a) >= (b) ? (a) : (b))
 
 typedef int8_t   i8;
 typedef uint8_t  u8;
 typedef int16_t  i16;
 typedef uint32_t u32;
 typedef int32_t  i32;
 typedef int64_t  i64;
 typedef uint64_t u64;
 
 static const u8 zero[128] = {0};
 
 // returns the smallest positive integer y such that
 // (x + y) % pow_2  == 0
 // Basically, it's how many bytes we need to add to "align" x.
 // Only works when pow_2 is a power of 2.
 // Note: we use ~x+1 instead of -x to avoid compiler warnings
 static size_t align(size_t x, size_t pow_2)
 {
     return (~x + 1) & (pow_2 - 1);
 }
 
 static u32 load24_le(const u8 s[3])
 {
     return (u32)s[0]
         | ((u32)s[1] <<  8)
         | ((u32)s[2] << 16);
 }
 
 static u32 load32_le(const u8 s[4])
 {
     return (u32)s[0]
         | ((u32)s[1] <<  8)
         | ((u32)s[2] << 16)
         | ((u32)s[3] << 24);
 }
 
 static u64 load64_le(const u8 s[8])
 {
     return load32_le(s) | ((u64)load32_le(s+4) << 32);
 }
 
 static void store32_le(u8 out[4], u32 in)
 {
     out[0] =  in        & 0xff;
     out[1] = (in >>  8) & 0xff;
     out[2] = (in >> 16) & 0xff;
     out[3] = (in >> 24) & 0xff;
 }
 
 static void store64_le(u8 out[8], u64 in)
 {
     store32_le(out    , (u32)in );
     store32_le(out + 4, in >> 32);
 }
 
 static void load32_le_buf (u32 *dst, const u8 *src, size_t size) {
     FOR(i, 0, size) { dst[i] = load32_le(src + i*4); }
 }
 static void load64_le_buf (u64 *dst, const u8 *src, size_t size) {
     FOR(i, 0, size) { dst[i] = load64_le(src + i*8); }
 }
 static void store32_le_buf(u8 *dst, const u32 *src, size_t size) {
     FOR(i, 0, size) { store32_le(dst + i*4, src[i]); }
 }
 static void store64_le_buf(u8 *dst, const u64 *src, size_t size) {
     FOR(i, 0, size) { store64_le(dst + i*8, src[i]); }
 }
 
 static u64 rotr64(u64 x, u64 n) { return (x >> n) ^ (x << (64 - n)); }
 static u32 rotl32(u32 x, u32 n) { return (x << n) ^ (x >> (32 - n)); }
 
 static int neq0(u64 diff)
 {   // constant time comparison to zero
     // return diff != 0 ? -1 : 0
     u64 half = (diff >> 32) | ((u32)diff);
     return (1 & ((half - 1) >> 32)) - 1;
 }
 
 static u64 x16(const u8 a[16], const u8 b[16])
 {
     return (load64_le(a + 0) ^ load64_le(b + 0))
         |  (load64_le(a + 8) ^ load64_le(b + 8));
 }
 static u64 x32(const u8 a[32],const u8 b[32]){return x16(a,b)| x16(a+16, b+16);}
 static u64 x64(const u8 a[64],const u8 b[64]){return x32(a,b)| x32(a+32, b+32);}
 int crypto_verify16(const u8 a[16], const u8 b[16]){ return neq0(x16(a, b)); }
 int crypto_verify32(const u8 a[32], const u8 b[32]){ return neq0(x32(a, b)); }
 int crypto_verify64(const u8 a[64], const u8 b[64]){ return neq0(x64(a, b)); }
 
 void crypto_wipe(void *secret, size_t size)
 {
     volatile u8 *v_secret = (u8*)secret;
     ZERO(v_secret, size);
 }
 
 /////////////////
 /// Chacha 20 ///
 /////////////////
 #define QUARTERROUND(a, b, c, d)     \
     a += b;  d = rotl32(d ^ a, 16);  \
     c += d;  b = rotl32(b ^ c, 12);  \
     a += b;  d = rotl32(d ^ a,  8);  \
     c += d;  b = rotl32(b ^ c,  7)
 
 static void chacha20_rounds(u32 out[16], const u32 in[16])
 {
     // The temporary variables make Chacha20 10% faster.
     u32 t0  = in[ 0];  u32 t1  = in[ 1];  u32 t2  = in[ 2];  u32 t3  = in[ 3];
     u32 t4  = in[ 4];  u32 t5  = in[ 5];  u32 t6  = in[ 6];  u32 t7  = in[ 7];
     u32 t8  = in[ 8];  u32 t9  = in[ 9];  u32 t10 = in[10];  u32 t11 = in[11];
     u32 t12 = in[12];  u32 t13 = in[13];  u32 t14 = in[14];  u32 t15 = in[15];
 
     FOR (i, 0, 10) { // 20 rounds, 2 rounds per loop.
         QUARTERROUND(t0, t4, t8 , t12); // column 0
         QUARTERROUND(t1, t5, t9 , t13); // column 1
         QUARTERROUND(t2, t6, t10, t14); // column 2
         QUARTERROUND(t3, t7, t11, t15); // column 3
         QUARTERROUND(t0, t5, t10, t15); // diagonal 0
         QUARTERROUND(t1, t6, t11, t12); // diagonal 1
         QUARTERROUND(t2, t7, t8 , t13); // diagonal 2
         QUARTERROUND(t3, t4, t9 , t14); // diagonal 3
     }
     out[ 0] = t0;   out[ 1] = t1;   out[ 2] = t2;   out[ 3] = t3;
     out[ 4] = t4;   out[ 5] = t5;   out[ 6] = t6;   out[ 7] = t7;
     out[ 8] = t8;   out[ 9] = t9;   out[10] = t10;  out[11] = t11;
     out[12] = t12;  out[13] = t13;  out[14] = t14;  out[15] = t15;
 }
 
 static void chacha20_init_key(u32 block[16], const u8 key[32])
 {
     load32_le_buf(block  , (const u8*)"expand 32-byte k", 4); // constant
     load32_le_buf(block+4, key                          , 8); // key
 }
 
 void crypto_hchacha20(u8 out[32], const u8 key[32], const u8 in [16])
 {
     u32 block[16];
     chacha20_init_key(block, key);
     // input
     load32_le_buf(block + 12, in, 4);
     chacha20_rounds(block, block);
     // prevent reversal of the rounds by revealing only half of the buffer.
     store32_le_buf(out   , block   , 4); // constant
     store32_le_buf(out+16, block+12, 4); // counter and nonce
     WIPE_BUFFER(block);
 }
 
 u64 crypto_chacha20_ctr(u8 *cipher_text, const u8 *plain_text,
                         size_t text_size, const u8 key[32], const u8 nonce[8],
                         u64 ctr)
 {
     u32 input[16];
     chacha20_init_key(input, key);
     input[12] = (u32) ctr;
     input[13] = (u32)(ctr >> 32);
     load32_le_buf(input+14, nonce, 2);
 
     // Whole blocks
     u32    pool[16];
     size_t nb_blocks = text_size >> 6;
     FOR (i, 0, nb_blocks) {
         chacha20_rounds(pool, input);
         if (plain_text != 0) {
             FOR (j, 0, 16) {
                 u32 p = pool[j] + input[j];
                 store32_le(cipher_text, p ^ load32_le(plain_text));
                 cipher_text += 4;
                 plain_text  += 4;
             }
         } else {
             FOR (j, 0, 16) {
                 u32 p = pool[j] + input[j];
                 store32_le(cipher_text, p);
                 cipher_text += 4;
             }
         }
         input[12]++;
         if (input[12] == 0) {
             input[13]++;
         }
     }
     text_size &= 63;
 
     // Last (incomplete) block
     if (text_size > 0) {
         if (plain_text == 0) {
             plain_text = zero;
         }
         chacha20_rounds(pool, input);
         u8 tmp[64];
         FOR (i, 0, 16) {
             store32_le(tmp + i*4, pool[i] + input[i]);
         }
         FOR (i, 0, text_size) {
             cipher_text[i] = tmp[i] ^ plain_text[i];
         }
         WIPE_BUFFER(tmp);
     }
     ctr = input[12] + ((u64)input[13] << 32) + (text_size > 0);
 
     WIPE_BUFFER(pool);
     WIPE_BUFFER(input);
     return ctr;
 }
 
 u32 crypto_ietf_chacha20_ctr(u8 *cipher_text, const u8 *plain_text,
                              size_t text_size,
                              const u8 key[32], const u8 nonce[12], u32 ctr)
 {
     u64 big_ctr = ctr + ((u64)load32_le(nonce) << 32);
     return (u32)crypto_chacha20_ctr(cipher_text, plain_text, text_size,
                                     key, nonce + 4, big_ctr);
 }
 
 u64 crypto_xchacha20_ctr(u8 *cipher_text, const u8 *plain_text,
                          size_t text_size,
                          const u8 key[32], const u8 nonce[24], u64 ctr)
 {
     u8 sub_key[32];
     crypto_hchacha20(sub_key, key, nonce);
     ctr = crypto_chacha20_ctr(cipher_text, plain_text, text_size,
                               sub_key, nonce+16, ctr);
     WIPE_BUFFER(sub_key);
     return ctr;
 }
 
 void crypto_chacha20(u8 *cipher_text, const u8 *plain_text, size_t text_size,
                      const u8 key[32], const u8 nonce[8])
 {
     crypto_chacha20_ctr(cipher_text, plain_text, text_size, key, nonce, 0);
 
 }
 void crypto_ietf_chacha20(u8 *cipher_text, const u8 *plain_text,
                           size_t text_size,
                           const u8 key[32], const u8 nonce[12])
 {
     crypto_ietf_chacha20_ctr(cipher_text, plain_text, text_size, key, nonce, 0);
 }
 
 void crypto_xchacha20(u8 *cipher_text, const u8 *plain_text, size_t text_size,
                       const u8 key[32], const u8 nonce[24])
 {
     crypto_xchacha20_ctr(cipher_text, plain_text, text_size, key, nonce, 0);
 }
 
 /////////////////
 /// Poly 1305 ///
 /////////////////
 
 // h = (h + c) * r
 // preconditions:
 //   ctx->h <= 4_ffffffff_ffffffff_ffffffff_ffffffff
 //   ctx->c <= 1_ffffffff_ffffffff_ffffffff_ffffffff
 //   ctx->r <=   0ffffffc_0ffffffc_0ffffffc_0fffffff
 // Postcondition:
 //   ctx->h <= 4_ffffffff_ffffffff_ffffffff_ffffffff
 static void poly_block(crypto_poly1305_ctx *ctx)
 {
     // s = h + c, without carry propagation
     const u64 s0 = ctx->h[0] + (u64)ctx->c[0]; // s0 <= 1_fffffffe
     const u64 s1 = ctx->h[1] + (u64)ctx->c[1]; // s1 <= 1_fffffffe
     const u64 s2 = ctx->h[2] + (u64)ctx->c[2]; // s2 <= 1_fffffffe
     const u64 s3 = ctx->h[3] + (u64)ctx->c[3]; // s3 <= 1_fffffffe
     const u32 s4 = ctx->h[4] +      ctx->c[4]; // s4 <=          5
 
     // Local all the things!
     const u32 r0 = ctx->r[0];       // r0  <= 0fffffff
     const u32 r1 = ctx->r[1];       // r1  <= 0ffffffc
     const u32 r2 = ctx->r[2];       // r2  <= 0ffffffc
     const u32 r3 = ctx->r[3];       // r3  <= 0ffffffc
     const u32 rr0 = (r0 >> 2) * 5;  // rr0 <= 13fffffb // lose 2 bits...
     const u32 rr1 = (r1 >> 2) + r1; // rr1 <= 13fffffb // rr1 == (r1 >> 2) * 5
     const u32 rr2 = (r2 >> 2) + r2; // rr2 <= 13fffffb // rr1 == (r2 >> 2) * 5
     const u32 rr3 = (r3 >> 2) + r3; // rr3 <= 13fffffb // rr1 == (r3 >> 2) * 5
 
     // (h + c) * r, without carry propagation
     const u64 x0 = s0*r0+ s1*rr3+ s2*rr2+ s3*rr1+ s4*rr0; // <= 97ffffe007fffff8
     const u64 x1 = s0*r1+ s1*r0 + s2*rr3+ s3*rr2+ s4*rr1; // <= 8fffffe20ffffff6
     const u64 x2 = s0*r2+ s1*r1 + s2*r0 + s3*rr3+ s4*rr2; // <= 87ffffe417fffff4
     const u64 x3 = s0*r3+ s1*r2 + s2*r1 + s3*r0 + s4*rr3; // <= 7fffffe61ffffff2
     const u32 x4 = s4 * (r0 & 3); // ...recover 2 bits    // <=                f
 
     // partial reduction modulo 2^130 - 5
     const u32 u5 = x4 + (x3 >> 32); // u5 <= 7ffffff5
     const u64 u0 = (u5 >>  2) * 5 + (x0 & 0xffffffff);
     const u64 u1 = (u0 >> 32)     + (x1 & 0xffffffff) + (x0 >> 32);
     const u64 u2 = (u1 >> 32)     + (x2 & 0xffffffff) + (x1 >> 32);
     const u64 u3 = (u2 >> 32)     + (x3 & 0xffffffff) + (x2 >> 32);
     const u64 u4 = (u3 >> 32)     + (u5 & 3);
 
     // Update the hash
     ctx->h[0] = (u32)u0; // u0 <= 1_9ffffff0
     ctx->h[1] = (u32)u1; // u1 <= 1_97ffffe0
     ctx->h[2] = (u32)u2; // u2 <= 1_8fffffe2
     ctx->h[3] = (u32)u3; // u3 <= 1_87ffffe4
     ctx->h[4] = (u32)u4; // u4 <=          4
 }
 
 // (re-)initialises the input counter and input buffer
 static void poly_clear_c(crypto_poly1305_ctx *ctx)
 {
     ZERO(ctx->c, 4);
     ctx->c_idx = 0;
 }
 
 static void poly_take_input(crypto_poly1305_ctx *ctx, u8 input)
 {
     size_t word = ctx->c_idx >> 2;
     size_t byte = ctx->c_idx & 3;
     ctx->c[word] |= (u32)input << (byte * 8);
     ctx->c_idx++;
 }
 
 static void poly_update(crypto_poly1305_ctx *ctx,
                         const u8 *message, size_t message_size)
 {
     FOR (i, 0, message_size) {
         poly_take_input(ctx, message[i]);
         if (ctx->c_idx == 16) {
             poly_block(ctx);
             poly_clear_c(ctx);
         }
     }
 }
 
 void crypto_poly1305_init(crypto_poly1305_ctx *ctx, const u8 key[32])
 {
     // Initial hash is zero
     ZERO(ctx->h, 5);
     // add 2^130 to every input block
     ctx->c[4] = 1;
     poly_clear_c(ctx);
     // load r and pad (r has some of its bits cleared)
     load32_le_buf(ctx->r  , key   , 4);
     load32_le_buf(ctx->pad, key+16, 4);
     FOR (i, 0, 1) { ctx->r[i] &= 0x0fffffff; }
     FOR (i, 1, 4) { ctx->r[i] &= 0x0ffffffc; }
 }
 
 void crypto_poly1305_update(crypto_poly1305_ctx *ctx,
                             const u8 *message, size_t message_size)
 {
     if (message_size == 0) {
         return;
     }
     // Align ourselves with block boundaries
     size_t aligned = MIN(align(ctx->c_idx, 16), message_size);
     poly_update(ctx, message, aligned);
     message      += aligned;
     message_size -= aligned;
 
     // Process the message block by block
     size_t nb_blocks = message_size >> 4;
     FOR (i, 0, nb_blocks) {
         load32_le_buf(ctx->c, message, 4);
         poly_block(ctx);
         message += 16;
     }
     if (nb_blocks > 0) {
         poly_clear_c(ctx);
     }
     message_size &= 15;
 
     // remaining bytes
     poly_update(ctx, message, message_size);
 }
 
 void crypto_poly1305_final(crypto_poly1305_ctx *ctx, u8 mac[16])
 {
     // Process the last block (if any)
     if (ctx->c_idx != 0) {
         // move the final 1 according to remaining input length
         // (We may add less than 2^130 to the last input block)
         ctx->c[4] = 0;
         poly_take_input(ctx, 1);
         // one last hash update
         poly_block(ctx);
     }
 
     // check if we should subtract 2^130-5 by performing the
     // corresponding carry propagation.
     u64 c = 5;
     FOR (i, 0, 4) {
         c  += ctx->h[i];
         c >>= 32;
     }
     c += ctx->h[4];
     c  = (c >> 2) * 5; // shift the carry back to the beginning
     // c now indicates how many times we should subtract 2^130-5 (0 or 1)
     FOR (i, 0, 4) {
         c += (u64)ctx->h[i] + ctx->pad[i];
         store32_le(mac + i*4, (u32)c);
         c = c >> 32;
     }
     WIPE_CTX(ctx);
 }
 
 void crypto_poly1305(u8     mac[16],  const u8 *message,
                      size_t message_size, const u8  key[32])
 {
     crypto_poly1305_ctx ctx;
     crypto_poly1305_init  (&ctx, key);
     crypto_poly1305_update(&ctx, message, message_size);
     crypto_poly1305_final (&ctx, mac);
 }
 
 ////////////////
 /// Blake2 b ///
 ////////////////
 static const u64 iv[8] = {
     0x6a09e667f3bcc908, 0xbb67ae8584caa73b,
     0x3c6ef372fe94f82b, 0xa54ff53a5f1d36f1,
     0x510e527fade682d1, 0x9b05688c2b3e6c1f,
     0x1f83d9abfb41bd6b, 0x5be0cd19137e2179,
 };
 
 // increment the input offset
 static void blake2b_incr(crypto_blake2b_ctx *ctx)
 {
     u64   *x = ctx->input_offset;
     size_t y = ctx->input_idx;
     x[0] += y;
     if (x[0] < y) {
         x[1]++;
     }
 }
 
 static void blake2b_compress(crypto_blake2b_ctx *ctx, int is_last_block)
 {
     static const u8 sigma[12][16] = {
         {  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15 },
         { 14, 10,  4,  8,  9, 15, 13,  6,  1, 12,  0,  2, 11,  7,  5,  3 },
         { 11,  8, 12,  0,  5,  2, 15, 13, 10, 14,  3,  6,  7,  1,  9,  4 },
         {  7,  9,  3,  1, 13, 12, 11, 14,  2,  6,  5, 10,  4,  0, 15,  8 },
         {  9,  0,  5,  7,  2,  4, 10, 15, 14,  1, 11, 12,  6,  8,  3, 13 },
         {  2, 12,  6, 10,  0, 11,  8,  3,  4, 13,  7,  5, 15, 14,  1,  9 },
         { 12,  5,  1, 15, 14, 13,  4, 10,  0,  7,  6,  3,  9,  2,  8, 11 },
         { 13, 11,  7, 14, 12,  1,  3,  9,  5,  0, 15,  4,  8,  6,  2, 10 },
         {  6, 15, 14,  9, 11,  3,  0,  8, 12,  2, 13,  7,  1,  4, 10,  5 },
         { 10,  2,  8,  4,  7,  6,  1,  5, 15, 11,  9, 14,  3, 12, 13,  0 },
         {  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15 },
         { 14, 10,  4,  8,  9, 15, 13,  6,  1, 12,  0,  2, 11,  7,  5,  3 },
     };
 
     // init work vector
     u64 v0 = ctx->hash[0];  u64 v8  = iv[0];
     u64 v1 = ctx->hash[1];  u64 v9  = iv[1];
     u64 v2 = ctx->hash[2];  u64 v10 = iv[2];
     u64 v3 = ctx->hash[3];  u64 v11 = iv[3];
     u64 v4 = ctx->hash[4];  u64 v12 = iv[4] ^ ctx->input_offset[0];
     u64 v5 = ctx->hash[5];  u64 v13 = iv[5] ^ ctx->input_offset[1];
     u64 v6 = ctx->hash[6];  u64 v14 = iv[6] ^ (u64)~(is_last_block - 1);
     u64 v7 = ctx->hash[7];  u64 v15 = iv[7];
 
     // mangle work vector
     u64 *input = ctx->input;
 #define BLAKE2_G(a, b, c, d, x, y)      \
     a += b + x;  d = rotr64(d ^ a, 32); \
     c += d;      b = rotr64(b ^ c, 24); \
     a += b + y;  d = rotr64(d ^ a, 16); \
     c += d;      b = rotr64(b ^ c, 63)
 #define BLAKE2_ROUND(i)                                                 \
     BLAKE2_G(v0, v4, v8 , v12, input[sigma[i][ 0]], input[sigma[i][ 1]]); \
     BLAKE2_G(v1, v5, v9 , v13, input[sigma[i][ 2]], input[sigma[i][ 3]]); \
     BLAKE2_G(v2, v6, v10, v14, input[sigma[i][ 4]], input[sigma[i][ 5]]); \
     BLAKE2_G(v3, v7, v11, v15, input[sigma[i][ 6]], input[sigma[i][ 7]]); \
     BLAKE2_G(v0, v5, v10, v15, input[sigma[i][ 8]], input[sigma[i][ 9]]); \
     BLAKE2_G(v1, v6, v11, v12, input[sigma[i][10]], input[sigma[i][11]]); \
     BLAKE2_G(v2, v7, v8 , v13, input[sigma[i][12]], input[sigma[i][13]]); \
     BLAKE2_G(v3, v4, v9 , v14, input[sigma[i][14]], input[sigma[i][15]])
 
 #ifdef BLAKE2_NO_UNROLLING
     FOR (i, 0, 12) {
         BLAKE2_ROUND(i);
     }
 #else
     BLAKE2_ROUND(0);  BLAKE2_ROUND(1);  BLAKE2_ROUND(2);  BLAKE2_ROUND(3);
     BLAKE2_ROUND(4);  BLAKE2_ROUND(5);  BLAKE2_ROUND(6);  BLAKE2_ROUND(7);
     BLAKE2_ROUND(8);  BLAKE2_ROUND(9);  BLAKE2_ROUND(10); BLAKE2_ROUND(11);
 #endif
 
     // update hash
     ctx->hash[0] ^= v0 ^ v8;   ctx->hash[1] ^= v1 ^ v9;
     ctx->hash[2] ^= v2 ^ v10;  ctx->hash[3] ^= v3 ^ v11;
     ctx->hash[4] ^= v4 ^ v12;  ctx->hash[5] ^= v5 ^ v13;
     ctx->hash[6] ^= v6 ^ v14;  ctx->hash[7] ^= v7 ^ v15;
 }
 
 static void blake2b_set_input(crypto_blake2b_ctx *ctx, u8 input, size_t index)
 {
     if (index == 0) {
         ZERO(ctx->input, 16);
     }
     size_t word = index >> 3;
     size_t byte = index & 7;
     ctx->input[word] |= (u64)input << (byte << 3);
 
 }
 
 static void blake2b_end_block(crypto_blake2b_ctx *ctx)
 {
     if (ctx->input_idx == 128) {  // If buffer is full,
         blake2b_incr(ctx);        // update the input offset
         blake2b_compress(ctx, 0); // and compress the (not last) block
         ctx->input_idx = 0;
     }
 }
 
 static void blake2b_update(crypto_blake2b_ctx *ctx,
                            const u8 *message, size_t message_size)
 {
     FOR (i, 0, message_size) {
         blake2b_end_block(ctx);
         blake2b_set_input(ctx, message[i], ctx->input_idx);
         ctx->input_idx++;
     }
 }
 
 void crypto_blake2b_general_init(crypto_blake2b_ctx *ctx, size_t hash_size,
                                  const u8           *key, size_t key_size)
 {
     // initial hash
     COPY(ctx->hash, iv, 8);
     ctx->hash[0] ^= 0x01010000 ^ (key_size << 8) ^ hash_size;
 
     ctx->input_offset[0] = 0;         // beginning of the input, no offset
     ctx->input_offset[1] = 0;         // beginning of the input, no offset
     ctx->hash_size       = hash_size; // remember the hash size we want
     ctx->input_idx       = 0;
 
     // if there is a key, the first block is that key (padded with zeroes)
     if (key_size > 0) {
         u8 key_block[128] = {0};
         COPY(key_block, key, key_size);
         // same as calling crypto_blake2b_update(ctx, key_block , 128)
         load64_le_buf(ctx->input, key_block, 16);
         ctx->input_idx = 128;
     }
 }
 
 void crypto_blake2b_init(crypto_blake2b_ctx *ctx)
 {
     crypto_blake2b_general_init(ctx, 64, 0, 0);
 }
 
 void crypto_blake2b_update(crypto_blake2b_ctx *ctx,
                            const u8 *message, size_t message_size)
 {
     if (message_size == 0) {
         return;
     }
     // Align ourselves with block boundaries
     size_t aligned = MIN(align(ctx->input_idx, 128), message_size);
     blake2b_update(ctx, message, aligned);
     message      += aligned;
     message_size -= aligned;
 
     // Process the message block by block
     FOR (i, 0, message_size >> 7) { // number of blocks
         blake2b_end_block(ctx);
         load64_le_buf(ctx->input, message, 16);
         message += 128;
         ctx->input_idx = 128;
     }
     message_size &= 127;
 
     // remaining bytes
     blake2b_update(ctx, message, message_size);
 }
 
 void crypto_blake2b_final(crypto_blake2b_ctx *ctx, u8 *hash)
 {
     // Pad the end of the block with zeroes
     FOR (i, ctx->input_idx, 128) {
         blake2b_set_input(ctx, 0, i);
     }
     blake2b_incr(ctx);        // update the input offset
     blake2b_compress(ctx, 1); // compress the last block
     size_t nb_words = ctx->hash_size >> 3;
     store64_le_buf(hash, ctx->hash, nb_words);
     FOR (i, nb_words << 3, ctx->hash_size) {
         hash[i] = (ctx->hash[i >> 3] >> (8 * (i & 7))) & 0xff;
     }
     WIPE_CTX(ctx);
 }
 
 void crypto_blake2b_general(u8       *hash   , size_t hash_size,
                             const u8 *key    , size_t key_size,
                             const u8 *message, size_t message_size)
 {
     crypto_blake2b_ctx ctx;
     crypto_blake2b_general_init(&ctx, hash_size, key, key_size);
     crypto_blake2b_update(&ctx, message, message_size);
     crypto_blake2b_final(&ctx, hash);
 }
 
 void crypto_blake2b(u8 hash[64], const u8 *message, size_t message_size)
 {
     crypto_blake2b_general(hash, 64, 0, 0, message, message_size);
 }
 
 static void blake2b_vtable_init(void *ctx) {
     crypto_blake2b_init(&((crypto_sign_ctx*)ctx)->hash);
 }
 static void blake2b_vtable_update(void *ctx, const u8 *m, size_t s) {
     crypto_blake2b_update(&((crypto_sign_ctx*)ctx)->hash, m, s);
 }
 static void blake2b_vtable_final(void *ctx, u8 *h) {
     crypto_blake2b_final(&((crypto_sign_ctx*)ctx)->hash, h);
 }
 const crypto_sign_vtable crypto_blake2b_vtable = {
     crypto_blake2b,
     blake2b_vtable_init,
     blake2b_vtable_update,
     blake2b_vtable_final,
     sizeof(crypto_sign_ctx),
 };
 
 ////////////////
 /// Argon2 i ///
 ////////////////
 // references to R, Z, Q etc. come from the spec
 
 // Argon2 operates on 1024 byte blocks.
 typedef struct { u64 a[128]; } block;
 
 static void wipe_block(block *b)
 {
     volatile u64* a = b->a;
     ZERO(a, 128);
 }
 
 // updates a Blake2 hash with a 32 bit word, little endian.
 static void blake_update_32(crypto_blake2b_ctx *ctx, u32 input)
 {
     u8 buf[4];
     store32_le(buf, input);
     crypto_blake2b_update(ctx, buf, 4);
     WIPE_BUFFER(buf);
 }
 
 static void load_block(block *b, const u8 bytes[1024])
 {
     load64_le_buf(b->a, bytes, 128);
 }
 
 static void store_block(u8 bytes[1024], const block *b)
 {
     store64_le_buf(bytes, b->a, 128);
 }
 
 static void copy_block(block *o,const block*in){FOR(i,0,128)o->a[i] = in->a[i];}
 static void  xor_block(block *o,const block*in){FOR(i,0,128)o->a[i]^= in->a[i];}
 
 // Hash with a virtually unlimited digest size.
 // Doesn't extract more entropy than the base hash function.
 // Mainly used for filling a whole kilobyte block with pseudo-random bytes.
 // (One could use a stream cipher with a seed hash as the key, but
 //  this would introduce another dependency —and point of failure.)
 static void extended_hash(u8       *digest, u32 digest_size,
                           const u8 *input , u32 input_size)
 {
     crypto_blake2b_ctx ctx;
     crypto_blake2b_general_init(&ctx, MIN(digest_size, 64), 0, 0);
     blake_update_32            (&ctx, digest_size);
     crypto_blake2b_update      (&ctx, input, input_size);
     crypto_blake2b_final       (&ctx, digest);
 
     if (digest_size > 64) {
         // the conversion to u64 avoids integer overflow on
         // ludicrously big hash sizes.
         u32 r   = (u32)(((u64)digest_size + 31) >> 5) - 2;
         u32 i   =  1;
         u32 in  =  0;
         u32 out = 32;
         while (i < r) {
             // Input and output overlap. This is intentional
             crypto_blake2b(digest + out, digest + in, 64);
             i   +=  1;
             in  += 32;
             out += 32;
         }
         crypto_blake2b_general(digest + out, digest_size - (32 * r),
                                0, 0, // no key
                                digest + in , 64);
     }
 }
 
 #define LSB(x) ((x) & 0xffffffff)
 #define G(a, b, c, d)                                            \
     a += b + 2 * LSB(a) * LSB(b);  d ^= a;  d = rotr64(d, 32);   \
     c += d + 2 * LSB(c) * LSB(d);  b ^= c;  b = rotr64(b, 24);   \
     a += b + 2 * LSB(a) * LSB(b);  d ^= a;  d = rotr64(d, 16);   \
     c += d + 2 * LSB(c) * LSB(d);  b ^= c;  b = rotr64(b, 63)
 #define ROUND(v0,  v1,  v2,  v3,  v4,  v5,  v6,  v7,    \
               v8,  v9, v10, v11, v12, v13, v14, v15)    \
     G(v0, v4,  v8, v12);  G(v1, v5,  v9, v13);          \
     G(v2, v6, v10, v14);  G(v3, v7, v11, v15);          \
     G(v0, v5, v10, v15);  G(v1, v6, v11, v12);          \
     G(v2, v7,  v8, v13);  G(v3, v4,  v9, v14)
 
 // Core of the compression function G.  Computes Z from R in place.
 static void g_rounds(block *work_block)
 {
     // column rounds (work_block = Q)
     for (int i = 0; i < 128; i += 16) {
         ROUND(work_block->a[i     ], work_block->a[i +  1],
               work_block->a[i +  2], work_block->a[i +  3],
               work_block->a[i +  4], work_block->a[i +  5],
               work_block->a[i +  6], work_block->a[i +  7],
               work_block->a[i +  8], work_block->a[i +  9],
               work_block->a[i + 10], work_block->a[i + 11],
               work_block->a[i + 12], work_block->a[i + 13],
               work_block->a[i + 14], work_block->a[i + 15]);
     }
     // row rounds (work_block = Z)
     for (int i = 0; i < 16; i += 2) {
         ROUND(work_block->a[i      ], work_block->a[i +   1],
               work_block->a[i +  16], work_block->a[i +  17],
               work_block->a[i +  32], work_block->a[i +  33],
               work_block->a[i +  48], work_block->a[i +  49],
               work_block->a[i +  64], work_block->a[i +  65],
               work_block->a[i +  80], work_block->a[i +  81],
               work_block->a[i +  96], work_block->a[i +  97],
               work_block->a[i + 112], work_block->a[i + 113]);
     }
 }
 
 // The compression function G (copy version for the first pass)
 static void g_copy(block *result, const block *x, const block *y, block* tmp)
 {
     copy_block(tmp   , x  ); // tmp    = X
     xor_block (tmp   , y  ); // tmp    = X ^ Y = R
     copy_block(result, tmp); // result = R         (only difference with g_xor)
     g_rounds  (tmp);         // tmp    = Z
     xor_block (result, tmp); // result = R ^ Z
 }
 
 // The compression function G (xor version for subsequent passes)
 static void g_xor(block *result, const block *x, const block *y, block *tmp)
 {
     copy_block(tmp   , x  ); // tmp    = X
     xor_block (tmp   , y  ); // tmp    = X ^ Y = R
     xor_block (result, tmp); // result = R ^ old   (only difference with g_copy)
     g_rounds  (tmp);         // tmp    = Z
     xor_block (result, tmp); // result = R ^ old ^ Z
 }
 
 // Unary version of the compression function.
 // The missing argument is implied zero.
 // Does the transformation in place.
 static void unary_g(block *work_block, block *tmp)
 {
     // work_block == R
     copy_block(tmp, work_block); // tmp        = R
     g_rounds  (work_block);      // work_block = Z
     xor_block (work_block, tmp); // work_block = Z ^ R
 }
 
 // Argon2i uses a kind of stream cipher to determine which reference
 // block it will take to synthesise the next block.  This context hold
 // that stream's state.  (It's very similar to Chacha20.  The block b
 // is analogous to Chacha's own pool)
 typedef struct {
     block b;
     u32 pass_number;
     u32 slice_number;
     u32 nb_blocks;
     u32 nb_iterations;
     u32 ctr;
     u32 offset;
 } gidx_ctx;
 
 // The block in the context will determine array indices. To avoid
 // timing attacks, it only depends on public information.  No looking
 // at a previous block to seed the next.  This makes offline attacks
 // easier, but timing attacks are the bigger threat in many settings.
 static void gidx_refresh(gidx_ctx *ctx)
 {
     // seed the beginning of the block...
     ctx->b.a[0] = ctx->pass_number;
     ctx->b.a[1] = 0;  // lane number (we have only one)
     ctx->b.a[2] = ctx->slice_number;
     ctx->b.a[3] = ctx->nb_blocks;
     ctx->b.a[4] = ctx->nb_iterations;
     ctx->b.a[5] = 1;  // type: Argon2i
     ctx->b.a[6] = ctx->ctr;
     ZERO(ctx->b.a + 7, 121); // ...then zero the rest out
 
     // Shuffle the block thus: ctx->b = G((G(ctx->b, zero)), zero)
     // (G "square" function), to get cheap pseudo-random numbers.
     block tmp;
     unary_g(&ctx->b, &tmp);
     unary_g(&ctx->b, &tmp);
     wipe_block(&tmp);
 }
 
 static void gidx_init(gidx_ctx *ctx,
                       u32 pass_number, u32 slice_number,
                       u32 nb_blocks,   u32 nb_iterations)
 {
     ctx->pass_number   = pass_number;
     ctx->slice_number  = slice_number;
     ctx->nb_blocks     = nb_blocks;
     ctx->nb_iterations = nb_iterations;
     ctx->ctr           = 0;
 
     // Offset from the beginning of the segment.  For the first slice
     // of the first pass, we start at the *third* block, so the offset
     // starts at 2, not 0.
     if (pass_number != 0 || slice_number != 0) {
         ctx->offset = 0;
     } else {
         ctx->offset = 2;
         ctx->ctr++;         // Compensates for missed lazy creation
         gidx_refresh(ctx);  // at the start of gidx_next()
     }
 }
 
 static u32 gidx_next(gidx_ctx *ctx)
 {
     // lazily creates the offset block we need
     if ((ctx->offset & 127) == 0) {
         ctx->ctr++;
         gidx_refresh(ctx);
     }
     u32 index  = ctx->offset & 127; // save index  for current call
     u32 offset = ctx->offset;       // save offset for current call
     ctx->offset++;                  // update offset for next call
 
     // Computes the area size.
     // Pass 0 : all already finished segments plus already constructed
     //          blocks in this segment
     // Pass 1+: 3 last segments plus already constructed
     //          blocks in this segment.  THE SPEC SUGGESTS OTHERWISE.
     //          I CONFORM TO THE REFERENCE IMPLEMENTATION.
     int first_pass  = ctx->pass_number == 0;
     u32 slice_size  = ctx->nb_blocks >> 2;
     u32 nb_segments = first_pass ? ctx->slice_number : 3;
     u32 area_size   = nb_segments * slice_size + offset - 1;
 
     // Computes the starting position of the reference area.
     // CONTRARY TO WHAT THE SPEC SUGGESTS, IT STARTS AT THE
     // NEXT SEGMENT, NOT THE NEXT BLOCK.
     u32 next_slice = ((ctx->slice_number + 1) & 3) * slice_size;
     u32 start_pos  = first_pass ? 0 : next_slice;
 
     // Generate offset from J1 (no need for J2, there's only one lane)
     u64 j1  = ctx->b.a[index] & 0xffffffff; // pseudo-random number
     u64 x   = (j1 * j1)       >> 32;
     u64 y   = (area_size * x) >> 32;
     u64 z   = (area_size - 1) - y;
     u64 ref = start_pos + z;                // ref < 2 * nb_blocks
     return (u32)(ref < ctx->nb_blocks ? ref : ref - ctx->nb_blocks);
 }
 
 // Main algorithm
 void crypto_argon2i_general(u8       *hash,      u32 hash_size,
                             void     *work_area, u32 nb_blocks,
                             u32 nb_iterations,
                             const u8 *password,  u32 password_size,
                             const u8 *salt,      u32 salt_size,
                             const u8 *key,       u32 key_size,
                             const u8 *ad,        u32 ad_size)
 {
     // work area seen as blocks (must be suitably aligned)
     block *blocks = (block*)work_area;
     {
         crypto_blake2b_ctx ctx;
         crypto_blake2b_init(&ctx);
 
         blake_update_32      (&ctx, 1            ); // p: number of threads
         blake_update_32      (&ctx, hash_size    );
         blake_update_32      (&ctx, nb_blocks    );
         blake_update_32      (&ctx, nb_iterations);
         blake_update_32      (&ctx, 0x13         ); // v: version number
         blake_update_32      (&ctx, 1            ); // y: Argon2i
         blake_update_32      (&ctx,           password_size);
         crypto_blake2b_update(&ctx, password, password_size);
         blake_update_32      (&ctx,           salt_size);
         crypto_blake2b_update(&ctx, salt,     salt_size);
         blake_update_32      (&ctx,           key_size);
         crypto_blake2b_update(&ctx, key,      key_size);
         blake_update_32      (&ctx,           ad_size);
         crypto_blake2b_update(&ctx, ad,       ad_size);
 
         u8 initial_hash[72]; // 64 bytes plus 2 words for future hashes
         crypto_blake2b_final(&ctx, initial_hash);
 
         // fill first 2 blocks
         block tmp_block;
         u8    hash_area[1024];
         store32_le(initial_hash + 64, 0); // first  additional word
         store32_le(initial_hash + 68, 0); // second additional word
         extended_hash(hash_area, 1024, initial_hash, 72);
         load_block(&tmp_block, hash_area);
         copy_block(blocks, &tmp_block);
 
         store32_le(initial_hash + 64, 1); // slight modification
         extended_hash(hash_area, 1024, initial_hash, 72);
         load_block(&tmp_block, hash_area);
         copy_block(blocks + 1, &tmp_block);
 
         WIPE_BUFFER(initial_hash);
         WIPE_BUFFER(hash_area);
         wipe_block(&tmp_block);
     }
 
     // Actual number of blocks
     nb_blocks -= nb_blocks & 3; // round down to 4 p (p == 1 thread)
     const u32 segment_size = nb_blocks >> 2;
 
     // fill (then re-fill) the rest of the blocks
     block tmp;
     gidx_ctx ctx; // public information, no need to wipe
     FOR_T (u32, pass_number, 0, nb_iterations) {
         int first_pass = pass_number == 0;
 
         FOR_T (u32, segment, 0, 4) {
             gidx_init(&ctx, pass_number, segment, nb_blocks, nb_iterations);
 
             // On the first segment of the first pass,
             // blocks 0 and 1 are already filled.
             // We use the offset to skip them.
             u32 start_offset  = first_pass && segment == 0 ? 2 : 0;
             u32 segment_start = segment * segment_size + start_offset;
             u32 segment_end   = (segment + 1) * segment_size;
             FOR_T (u32, current_block, segment_start, segment_end) {
                 u32 reference_block = gidx_next(&ctx);
                 u32 previous_block  = current_block == 0
                                     ? nb_blocks - 1
                                     : current_block - 1;
                 block *c = blocks + current_block;
                 block *p = blocks + previous_block;
                 block *r = blocks + reference_block;
                 if (first_pass) { g_copy(c, p, r, &tmp); }
                 else            { g_xor (c, p, r, &tmp); }
             }
         }
     }
     wipe_block(&tmp);
     u8 final_block[1024];
     store_block(final_block, blocks + (nb_blocks - 1));
 
     // wipe work area
     volatile u64 *p = (u64*)work_area;
     ZERO(p, 128 * nb_blocks);
 
     // hash the very last block with H' into the output hash
     extended_hash(hash, hash_size, final_block, 1024);
     WIPE_BUFFER(final_block);
 }
 
 void crypto_argon2i(u8   *hash,      u32 hash_size,
                     void *work_area, u32 nb_blocks, u32 nb_iterations,
                     const u8 *password,  u32 password_size,
                     const u8 *salt,      u32 salt_size)
 {
     crypto_argon2i_general(hash, hash_size, work_area, nb_blocks, nb_iterations,
                            password, password_size, salt , salt_size, 0,0,0,0);
 }
 
 ////////////////////////////////////
 /// Arithmetic modulo 2^255 - 19 ///
 ////////////////////////////////////
 //  Originally taken from SUPERCOP's ref10 implementation.
 //  A bit bigger than TweetNaCl, over 4 times faster.
 
 // field element
 typedef i32 fe[10];
 
 // field constants
 //
 // fe_one      : 1
 // sqrtm1      : sqrt(-1)
 // d           :     -121665 / 121666
 // D2          : 2 * -121665 / 121666
 // lop_x, lop_y: low order point in Edwards coordinates
 // ufactor     : -sqrt(-1) * 2
 // A2          : 486662^2  (A squared)
 static const fe fe_one  = {1};
 static const fe sqrtm1  = {-32595792, -7943725, 9377950, 3500415, 12389472,
                            -272473, -25146209, -2005654, 326686, 11406482,};
 static const fe d       = {-10913610, 13857413, -15372611, 6949391, 114729,
                            -8787816, -6275908, -3247719, -18696448, -12055116,};
 static const fe D2      = {-21827239, -5839606, -30745221, 13898782, 229458,
                            15978800, -12551817, -6495438, 29715968, 9444199,};
 static const fe lop_x   = {21352778, 5345713, 4660180, -8347857, 24143090,
                            14568123, 30185756, -12247770, -33528939, 8345319,};
 static const fe lop_y   = {-6952922, -1265500, 6862341, -7057498, -4037696,
                            -5447722, 31680899, -15325402, -19365852, 1569102,};
 static const fe ufactor = {-1917299, 15887451, -18755900, -7000830, -24778944,
                            544946, -16816446, 4011309, -653372, 10741468,};
 static const fe A2      = {12721188, 3529, 0, 0, 0, 0, 0, 0, 0, 0,};
 
 static void fe_0(fe h) {           ZERO(h  , 10); }
 static void fe_1(fe h) { h[0] = 1; ZERO(h+1,  9); }
 
 static void fe_copy(fe h,const fe f           ){FOR(i,0,10) h[i] =  f[i];      }
 static void fe_neg (fe h,const fe f           ){FOR(i,0,10) h[i] = -f[i];      }
 static void fe_add (fe h,const fe f,const fe g){FOR(i,0,10) h[i] = f[i] + g[i];}
 static void fe_sub (fe h,const fe f,const fe g){FOR(i,0,10) h[i] = f[i] - g[i];}
 
 static void fe_cswap(fe f, fe g, int b)
 {
     i32 mask = -b; // -1 = 0xffffffff
     FOR (i, 0, 10) {
         i32 x = (f[i] ^ g[i]) & mask;
         f[i] = f[i] ^ x;
         g[i] = g[i] ^ x;
     }
 }
 
 static void fe_ccopy(fe f, const fe g, int b)
 {
     i32 mask = -b; // -1 = 0xffffffff
     FOR (i, 0, 10) {
         i32 x = (f[i] ^ g[i]) & mask;
         f[i] = f[i] ^ x;
     }
 }
 
 
 // Signed carry propagation
 // ------------------------
 //
 // Let t be a number.  It can be uniquely decomposed thus:
 //
 //    t = h*2^26 + l
 //    such that -2^25 <= l < 2^25
 //
 // Let c = (t + 2^25) / 2^26            (rounded down)
 //     c = (h*2^26 + l + 2^25) / 2^26   (rounded down)
 //     c =  h   +   (l + 2^25) / 2^26   (rounded down)
 //     c =  h                           (exactly)
 // Because 0 <= l + 2^25 < 2^26
 //
 // Let u = t          - c*2^26
 //     u = h*2^26 + l - h*2^26
 //     u = l
 // Therefore, -2^25 <= u < 2^25
 //
 // Additionally, if |t| < x, then |h| < x/2^26 (rounded down)
 //
 // Notations:
 // - In C, 1<<25 means 2^25.
 // - In C, x>>25 means floor(x / (2^25)).
 // - All of the above applies with 25 & 24 as well as 26 & 25.
 //
 //
 // Note on negative right shifts
 // -----------------------------
 //
 // In C, x >> n, where x is a negative integer, is implementation
 // defined.  In practice, all platforms do arithmetic shift, which is
 // equivalent to division by 2^26, rounded down.  Some compilers, like
 // GCC, even guarantee it.
 //
 // If we ever stumble upon a platform that does not propagate the sign
 // bit (we won't), visible failures will show at the slightest test, and
 // the signed shifts can be replaced by the following:
 //
 //     typedef struct { i64 x:39; } s25;
 //     typedef struct { i64 x:38; } s26;
 //     i64 shift25(i64 x) { s25 s; s.x = ((u64)x)>>25; return s.x; }
 //     i64 shift26(i64 x) { s26 s; s.x = ((u64)x)>>26; return s.x; }
 //
 // Current compilers cannot optimise this, causing a 30% drop in
 // performance.  Fairly expensive for something that never happens.
 //
 //
 // Precondition
 // ------------
 //
 // |t0|       < 2^63
 // |t1|..|t9| < 2^62
 //
 // Algorithm
 // ---------
 // c   = t0 + 2^25 / 2^26   -- |c|  <= 2^36
 // t0 -= c * 2^26           -- |t0| <= 2^25
 // t1 += c                  -- |t1| <= 2^63
 //
 // c   = t4 + 2^25 / 2^26   -- |c|  <= 2^36
 // t4 -= c * 2^26           -- |t4| <= 2^25
 // t5 += c                  -- |t5| <= 2^63
 //
 // c   = t1 + 2^24 / 2^25   -- |c|  <= 2^38
 // t1 -= c * 2^25           -- |t1| <= 2^24
 // t2 += c                  -- |t2| <= 2^63
 //
 // c   = t5 + 2^24 / 2^25   -- |c|  <= 2^38
 // t5 -= c * 2^25           -- |t5| <= 2^24
 // t6 += c                  -- |t6| <= 2^63
 //
 // c   = t2 + 2^25 / 2^26   -- |c|  <= 2^37
 // t2 -= c * 2^26           -- |t2| <= 2^25        < 1.1 * 2^25  (final t2)
 // t3 += c                  -- |t3| <= 2^63
 //
 // c   = t6 + 2^25 / 2^26   -- |c|  <= 2^37
 // t6 -= c * 2^26           -- |t6| <= 2^25        < 1.1 * 2^25  (final t6)
 // t7 += c                  -- |t7| <= 2^63
 //
 // c   = t3 + 2^24 / 2^25   -- |c|  <= 2^38
 // t3 -= c * 2^25           -- |t3| <= 2^24        < 1.1 * 2^24  (final t3)
 // t4 += c                  -- |t4| <= 2^25 + 2^38 < 2^39
 //
 // c   = t7 + 2^24 / 2^25   -- |c|  <= 2^38
 // t7 -= c * 2^25           -- |t7| <= 2^24        < 1.1 * 2^24  (final t7)
 // t8 += c                  -- |t8| <= 2^63
 //
 // c   = t4 + 2^25 / 2^26   -- |c|  <= 2^13
 // t4 -= c * 2^26           -- |t4| <= 2^25        < 1.1 * 2^25  (final t4)
 // t5 += c                  -- |t5| <= 2^24 + 2^13 < 1.1 * 2^24  (final t5)
 //
 // c   = t8 + 2^25 / 2^26   -- |c|  <= 2^37
 // t8 -= c * 2^26           -- |t8| <= 2^25        < 1.1 * 2^25  (final t8)
 // t9 += c                  -- |t9| <= 2^63
 //
 // c   = t9 + 2^24 / 2^25   -- |c|  <= 2^38
 // t9 -= c * 2^25           -- |t9| <= 2^24        < 1.1 * 2^24  (final t9)
 // t0 += c * 19             -- |t0| <= 2^25 + 2^38*19 < 2^44
 //
 // c   = t0 + 2^25 / 2^26   -- |c|  <= 2^18
 // t0 -= c * 2^26           -- |t0| <= 2^25        < 1.1 * 2^25  (final t0)
 // t1 += c                  -- |t1| <= 2^24 + 2^18 < 1.1 * 2^24  (final t1)
 //
 // Postcondition
 // -------------
 //   |t0|, |t2|, |t4|, |t6|, |t8|  <  1.1 * 2^25
 //   |t1|, |t3|, |t5|, |t7|, |t9|  <  1.1 * 2^24
 #define FE_CARRY                                                        \
     i64 c;                                                              \
     c = (t0 + ((i64)1<<25)) >> 26;  t0 -= c * ((i64)1 << 26);  t1 += c; \
     c = (t4 + ((i64)1<<25)) >> 26;  t4 -= c * ((i64)1 << 26);  t5 += c; \
     c = (t1 + ((i64)1<<24)) >> 25;  t1 -= c * ((i64)1 << 25);  t2 += c; \
     c = (t5 + ((i64)1<<24)) >> 25;  t5 -= c * ((i64)1 << 25);  t6 += c; \
     c = (t2 + ((i64)1<<25)) >> 26;  t2 -= c * ((i64)1 << 26);  t3 += c; \
     c = (t6 + ((i64)1<<25)) >> 26;  t6 -= c * ((i64)1 << 26);  t7 += c; \
     c = (t3 + ((i64)1<<24)) >> 25;  t3 -= c * ((i64)1 << 25);  t4 += c; \
     c = (t7 + ((i64)1<<24)) >> 25;  t7 -= c * ((i64)1 << 25);  t8 += c; \
     c = (t4 + ((i64)1<<25)) >> 26;  t4 -= c * ((i64)1 << 26);  t5 += c; \
     c = (t8 + ((i64)1<<25)) >> 26;  t8 -= c * ((i64)1 << 26);  t9 += c; \
     c = (t9 + ((i64)1<<24)) >> 25;  t9 -= c * ((i64)1 << 25);  t0 += c * 19; \
     c = (t0 + ((i64)1<<25)) >> 26;  t0 -= c * ((i64)1 << 26);  t1 += c; \
     h[0]=(i32)t0;  h[1]=(i32)t1;  h[2]=(i32)t2;  h[3]=(i32)t3;  h[4]=(i32)t4; \
     h[5]=(i32)t5;  h[6]=(i32)t6;  h[7]=(i32)t7;  h[8]=(i32)t8;  h[9]=(i32)t9
 
 static void fe_frombytes(fe h, const u8 s[32])
 {
     i64 t0 =  load32_le(s);                        // t0 < 2^32
     i64 t1 =  load24_le(s +  4) << 6;              // t1 < 2^30
     i64 t2 =  load24_le(s +  7) << 5;              // t2 < 2^29
     i64 t3 =  load24_le(s + 10) << 3;              // t3 < 2^27
     i64 t4 =  load24_le(s + 13) << 2;              // t4 < 2^26
     i64 t5 =  load32_le(s + 16);                   // t5 < 2^32
     i64 t6 =  load24_le(s + 20) << 7;              // t6 < 2^31
     i64 t7 =  load24_le(s + 23) << 5;              // t7 < 2^29
     i64 t8 =  load24_le(s + 26) << 4;              // t8 < 2^28
     i64 t9 = (load24_le(s + 29) & 0x7fffff) << 2;  // t9 < 2^25
     FE_CARRY;                                      // Carry recondition OK
 }
 
 // Precondition
 //   |h[0]|, |h[2]|, |h[4]|, |h[6]|, |h[8]|  <  1.1 * 2^25
 //   |h[1]|, |h[3]|, |h[5]|, |h[7]|, |h[9]|  <  1.1 * 2^24
 //
 // Therefore, |h| < 2^255-19
 // There are two possibilities:
 //
 // - If h is positive, all we need to do is reduce its individual
 //   limbs down to their tight positive range.
 // - If h is negative, we also need to add 2^255-19 to it.
 //   Or just remove 19 and chop off any excess bit.
 static void fe_tobytes(u8 s[32], const fe h)
 {
     i32 t[10];
     COPY(t, h, 10);
     i32 q = (19 * t[9] + (((i32) 1) << 24)) >> 25;
     //                 |t9|                    < 1.1 * 2^24
     //  -1.1 * 2^24  <  t9                     < 1.1 * 2^24
     //  -21  * 2^24  <  19 * t9                < 21  * 2^24
     //  -2^29        <  19 * t9 + 2^24         < 2^29
     //  -2^29 / 2^25 < (19 * t9 + 2^24) / 2^25 < 2^29 / 2^25
     //  -16          < (19 * t9 + 2^24) / 2^25 < 16
     FOR (i, 0, 5) {
         q += t[2*i  ]; q >>= 26; // q = 0 or -1
         q += t[2*i+1]; q >>= 25; // q = 0 or -1
     }
     // q =  0 iff h >= 0
     // q = -1 iff h <  0
     // Adding q * 19 to h reduces h to its proper range.
     q *= 19;  // Shift carry back to the beginning
     FOR (i, 0, 5) {
         t[i*2  ] += q;  q = t[i*2  ] >> 26;  t[i*2  ] -= q * ((i32)1 << 26);
         t[i*2+1] += q;  q = t[i*2+1] >> 25;  t[i*2+1] -= q * ((i32)1 << 25);
     }
     // h is now fully reduced, and q represents the excess bit.
 
     store32_le(s +  0, ((u32)t[0] >>  0) | ((u32)t[1] << 26));
     store32_le(s +  4, ((u32)t[1] >>  6) | ((u32)t[2] << 19));
     store32_le(s +  8, ((u32)t[2] >> 13) | ((u32)t[3] << 13));
     store32_le(s + 12, ((u32)t[3] >> 19) | ((u32)t[4] <<  6));
     store32_le(s + 16, ((u32)t[5] >>  0) | ((u32)t[6] << 25));
     store32_le(s + 20, ((u32)t[6] >>  7) | ((u32)t[7] << 19));
     store32_le(s + 24, ((u32)t[7] >> 13) | ((u32)t[8] << 12));
     store32_le(s + 28, ((u32)t[8] >> 20) | ((u32)t[9] <<  6));
 
     WIPE_BUFFER(t);
 }
 
 // Precondition
 // -------------
 //   |f0|, |f2|, |f4|, |f6|, |f8|  <  1.65 * 2^26
 //   |f1|, |f3|, |f5|, |f7|, |f9|  <  1.65 * 2^25
 //
 //   |g0|, |g2|, |g4|, |g6|, |g8|  <  1.65 * 2^26
 //   |g1|, |g3|, |g5|, |g7|, |g9|  <  1.65 * 2^25
 static void fe_mul_small(fe h, const fe f, i32 g)
 {
     i64 t0 = f[0] * (i64) g;  i64 t1 = f[1] * (i64) g;
     i64 t2 = f[2] * (i64) g;  i64 t3 = f[3] * (i64) g;
     i64 t4 = f[4] * (i64) g;  i64 t5 = f[5] * (i64) g;
     i64 t6 = f[6] * (i64) g;  i64 t7 = f[7] * (i64) g;
     i64 t8 = f[8] * (i64) g;  i64 t9 = f[9] * (i64) g;
     // |t0|, |t2|, |t4|, |t6|, |t8|  <  1.65 * 2^26 * 2^31  < 2^58
     // |t1|, |t3|, |t5|, |t7|, |t9|  <  1.65 * 2^25 * 2^31  < 2^57
 
     FE_CARRY; // Carry precondition OK
 }
 
 // Precondition
 // -------------
 //   |f0|, |f2|, |f4|, |f6|, |f8|  <  1.65 * 2^26
 //   |f1|, |f3|, |f5|, |f7|, |f9|  <  1.65 * 2^25
 //
 //   |g0|, |g2|, |g4|, |g6|, |g8|  <  1.65 * 2^26
 //   |g1|, |g3|, |g5|, |g7|, |g9|  <  1.65 * 2^25
 static void fe_mul(fe h, const fe f, const fe g)
 {
     // Everything is unrolled and put in temporary variables.
     // We could roll the loop, but that would make curve25519 twice as slow.
     i32 f0 = f[0]; i32 f1 = f[1]; i32 f2 = f[2]; i32 f3 = f[3]; i32 f4 = f[4];
     i32 f5 = f[5]; i32 f6 = f[6]; i32 f7 = f[7]; i32 f8 = f[8]; i32 f9 = f[9];
     i32 g0 = g[0]; i32 g1 = g[1]; i32 g2 = g[2]; i32 g3 = g[3]; i32 g4 = g[4];
     i32 g5 = g[5]; i32 g6 = g[6]; i32 g7 = g[7]; i32 g8 = g[8]; i32 g9 = g[9];
     i32 F1 = f1*2; i32 F3 = f3*2; i32 F5 = f5*2; i32 F7 = f7*2; i32 F9 = f9*2;
     i32 G1 = g1*19;  i32 G2 = g2*19;  i32 G3 = g3*19;
     i32 G4 = g4*19;  i32 G5 = g5*19;  i32 G6 = g6*19;
     i32 G7 = g7*19;  i32 G8 = g8*19;  i32 G9 = g9*19;
     // |F1|, |F3|, |F5|, |F7|, |F9|  <  1.65 * 2^26
     // |G0|, |G2|, |G4|, |G6|, |G8|  <  2^31
     // |G1|, |G3|, |G5|, |G7|, |G9|  <  2^30
 
     i64 t0 = f0*(i64)g0 + F1*(i64)G9 + f2*(i64)G8 + F3*(i64)G7 + f4*(i64)G6
         +    F5*(i64)G5 + f6*(i64)G4 + F7*(i64)G3 + f8*(i64)G2 + F9*(i64)G1;
     i64 t1 = f0*(i64)g1 + f1*(i64)g0 + f2*(i64)G9 + f3*(i64)G8 + f4*(i64)G7
         +    f5*(i64)G6 + f6*(i64)G5 + f7*(i64)G4 + f8*(i64)G3 + f9*(i64)G2;
     i64 t2 = f0*(i64)g2 + F1*(i64)g1 + f2*(i64)g0 + F3*(i64)G9 + f4*(i64)G8
         +    F5*(i64)G7 + f6*(i64)G6 + F7*(i64)G5 + f8*(i64)G4 + F9*(i64)G3;
     i64 t3 = f0*(i64)g3 + f1*(i64)g2 + f2*(i64)g1 + f3*(i64)g0 + f4*(i64)G9
         +    f5*(i64)G8 + f6*(i64)G7 + f7*(i64)G6 + f8*(i64)G5 + f9*(i64)G4;
     i64 t4 = f0*(i64)g4 + F1*(i64)g3 + f2*(i64)g2 + F3*(i64)g1 + f4*(i64)g0
         +    F5*(i64)G9 + f6*(i64)G8 + F7*(i64)G7 + f8*(i64)G6 + F9*(i64)G5;
     i64 t5 = f0*(i64)g5 + f1*(i64)g4 + f2*(i64)g3 + f3*(i64)g2 + f4*(i64)g1
         +    f5*(i64)g0 + f6*(i64)G9 + f7*(i64)G8 + f8*(i64)G7 + f9*(i64)G6;
     i64 t6 = f0*(i64)g6 + F1*(i64)g5 + f2*(i64)g4 + F3*(i64)g3 + f4*(i64)g2
         +    F5*(i64)g1 + f6*(i64)g0 + F7*(i64)G9 + f8*(i64)G8 + F9*(i64)G7;
     i64 t7 = f0*(i64)g7 + f1*(i64)g6 + f2*(i64)g5 + f3*(i64)g4 + f4*(i64)g3
         +    f5*(i64)g2 + f6*(i64)g1 + f7*(i64)g0 + f8*(i64)G9 + f9*(i64)G8;
     i64 t8 = f0*(i64)g8 + F1*(i64)g7 + f2*(i64)g6 + F3*(i64)g5 + f4*(i64)g4
         +    F5*(i64)g3 + f6*(i64)g2 + F7*(i64)g1 + f8*(i64)g0 + F9*(i64)G9;
     i64 t9 = f0*(i64)g9 + f1*(i64)g8 + f2*(i64)g7 + f3*(i64)g6 + f4*(i64)g5
         +    f5*(i64)g4 + f6*(i64)g3 + f7*(i64)g2 + f8*(i64)g1 + f9*(i64)g0;
     // t0 < 0.67 * 2^61
     // t1 < 0.41 * 2^61
     // t2 < 0.52 * 2^61
     // t3 < 0.32 * 2^61
     // t4 < 0.38 * 2^61
     // t5 < 0.22 * 2^61
     // t6 < 0.23 * 2^61
     // t7 < 0.13 * 2^61
     // t8 < 0.09 * 2^61
     // t9 < 0.03 * 2^61
 
     FE_CARRY; // Everything below 2^62, Carry precondition OK
 }
 
 // Precondition
 // -------------
 //   |f0|, |f2|, |f4|, |f6|, |f8|  <  1.65 * 2^26
 //   |f1|, |f3|, |f5|, |f7|, |f9|  <  1.65 * 2^25
 //
 // Note: we could use fe_mul() for this, but this is significantly faster
 static void fe_sq(fe h, const fe f)
 {
     i32 f0 = f[0]; i32 f1 = f[1]; i32 f2 = f[2]; i32 f3 = f[3]; i32 f4 = f[4];
     i32 f5 = f[5]; i32 f6 = f[6]; i32 f7 = f[7]; i32 f8 = f[8]; i32 f9 = f[9];
     i32 f0_2  = f0*2;   i32 f1_2  = f1*2;   i32 f2_2  = f2*2;   i32 f3_2 = f3*2;
     i32 f4_2  = f4*2;   i32 f5_2  = f5*2;   i32 f6_2  = f6*2;   i32 f7_2 = f7*2;
     i32 f5_38 = f5*38;  i32 f6_19 = f6*19;  i32 f7_38 = f7*38;
     i32 f8_19 = f8*19;  i32 f9_38 = f9*38;
     // |f0_2| , |f2_2| , |f4_2| , |f6_2| , |f8_2|  <  1.65 * 2^27
     // |f1_2| , |f3_2| , |f5_2| , |f7_2| , |f9_2|  <  1.65 * 2^26
     // |f5_38|, |f6_19|, |f7_38|, |f8_19|, |f9_38| <  2^31
 
     i64 t0 = f0  *(i64)f0    + f1_2*(i64)f9_38 + f2_2*(i64)f8_19
         +    f3_2*(i64)f7_38 + f4_2*(i64)f6_19 + f5  *(i64)f5_38;
     i64 t1 = f0_2*(i64)f1    + f2  *(i64)f9_38 + f3_2*(i64)f8_19
         +    f4  *(i64)f7_38 + f5_2*(i64)f6_19;
     i64 t2 = f0_2*(i64)f2    + f1_2*(i64)f1    + f3_2*(i64)f9_38
         +    f4_2*(i64)f8_19 + f5_2*(i64)f7_38 + f6  *(i64)f6_19;
     i64 t3 = f0_2*(i64)f3    + f1_2*(i64)f2    + f4  *(i64)f9_38
         +    f5_2*(i64)f8_19 + f6  *(i64)f7_38;
     i64 t4 = f0_2*(i64)f4    + f1_2*(i64)f3_2  + f2  *(i64)f2
         +    f5_2*(i64)f9_38 + f6_2*(i64)f8_19 + f7  *(i64)f7_38;
     i64 t5 = f0_2*(i64)f5    + f1_2*(i64)f4    + f2_2*(i64)f3
         +    f6  *(i64)f9_38 + f7_2*(i64)f8_19;
     i64 t6 = f0_2*(i64)f6    + f1_2*(i64)f5_2  + f2_2*(i64)f4
         +    f3_2*(i64)f3    + f7_2*(i64)f9_38 + f8  *(i64)f8_19;
     i64 t7 = f0_2*(i64)f7    + f1_2*(i64)f6    + f2_2*(i64)f5
         +    f3_2*(i64)f4    + f8  *(i64)f9_38;
     i64 t8 = f0_2*(i64)f8    + f1_2*(i64)f7_2  + f2_2*(i64)f6
         +    f3_2*(i64)f5_2  + f4  *(i64)f4    + f9  *(i64)f9_38;
     i64 t9 = f0_2*(i64)f9    + f1_2*(i64)f8    + f2_2*(i64)f7
         +    f3_2*(i64)f6    + f4  *(i64)f5_2;
     // t0 < 0.67 * 2^61
     // t1 < 0.41 * 2^61
     // t2 < 0.52 * 2^61
     // t3 < 0.32 * 2^61
     // t4 < 0.38 * 2^61
     // t5 < 0.22 * 2^61
     // t6 < 0.23 * 2^61
     // t7 < 0.13 * 2^61
     // t8 < 0.09 * 2^61
     // t9 < 0.03 * 2^61
 
     FE_CARRY;
 }
 
 // h = 2 * (f^2)
 //
 // Precondition
 // -------------
 //   |f0|, |f2|, |f4|, |f6|, |f8|  <  1.65 * 2^26
 //   |f1|, |f3|, |f5|, |f7|, |f9|  <  1.65 * 2^25
 //
 // Note: we could implement fe_sq2() by copying fe_sq(), multiplying
 // each limb by 2, *then* perform the carry.  This saves one carry.
 // However, doing so with the stated preconditions does not work (t2
 // would overflow).  There are 3 ways to solve this:
 //
 // 1. Show that t2 actually never overflows (it really does not).
 // 2. Accept an additional carry, at a small lost of performance.
 // 3. Make sure the input of fe_sq2() is freshly carried.
 //
 // SUPERCOP ref10 relies on (1).
 // Monocypher chose (2) and (3), mostly to save code.
 static void fe_sq2(fe h, const fe f)
 {
     fe_sq(h, f);
     fe_mul_small(h, h, 2);
 }
 
 // This could be simplified, but it would be slower
 static void fe_pow22523(fe out, const fe z)
 {
     fe t0, t1, t2;
     fe_sq(t0, z);
     fe_sq(t1,t0);                   fe_sq(t1, t1);  fe_mul(t1, z, t1);
     fe_mul(t0, t0, t1);
     fe_sq(t0, t0);                                  fe_mul(t0, t1, t0);
     fe_sq(t1, t0);  FOR (i, 1,   5) fe_sq(t1, t1);  fe_mul(t0, t1, t0);
     fe_sq(t1, t0);  FOR (i, 1,  10) fe_sq(t1, t1);  fe_mul(t1, t1, t0);
     fe_sq(t2, t1);  FOR (i, 1,  20) fe_sq(t2, t2);  fe_mul(t1, t2, t1);
     fe_sq(t1, t1);  FOR (i, 1,  10) fe_sq(t1, t1);  fe_mul(t0, t1, t0);
     fe_sq(t1, t0);  FOR (i, 1,  50) fe_sq(t1, t1);  fe_mul(t1, t1, t0);
     fe_sq(t2, t1);  FOR (i, 1, 100) fe_sq(t2, t2);  fe_mul(t1, t2, t1);
     fe_sq(t1, t1);  FOR (i, 1,  50) fe_sq(t1, t1);  fe_mul(t0, t1, t0);
     fe_sq(t0, t0);  FOR (i, 1,   2) fe_sq(t0, t0);  fe_mul(out, t0, z);
     WIPE_BUFFER(t0);
     WIPE_BUFFER(t1);
     WIPE_BUFFER(t2);
 }
 
 // Inverting means multiplying by 2^255 - 21
 // 2^255 - 21 = (2^252 - 3) * 8 + 3
 // So we reuse the multiplication chain of fe_pow22523
 static void fe_invert(fe out, const fe z)
 {
     fe tmp;
     fe_pow22523(tmp, z);
     // tmp2^8 * z^3
     fe_sq(tmp, tmp);                        // 0
     fe_sq(tmp, tmp);  fe_mul(tmp, tmp, z);  // 1
     fe_sq(tmp, tmp);  fe_mul(out, tmp, z);  // 1
     WIPE_BUFFER(tmp);
 }
 
 //  Parity check.  Returns 0 if even, 1 if odd
 static int fe_isodd(const fe f)
 {
     u8 s[32];
     fe_tobytes(s, f);
     u8 isodd = s[0] & 1;
     WIPE_BUFFER(s);
     return isodd;
 }
 
 // Returns 1 if equal, 0 if not equal
 static int fe_isequal(const fe f, const fe g)
 {
     u8 fs[32];
     u8 gs[32];
     fe_tobytes(fs, f);
     fe_tobytes(gs, g);
     int isdifferent = crypto_verify32(fs, gs);
     WIPE_BUFFER(fs);
     WIPE_BUFFER(gs);
     return 1 + isdifferent;
 }
 
 // Inverse square root.
 // Returns true if x is a non zero square, false otherwise.
 // After the call:
 //   isr = sqrt(1/x)        if x is non-zero square.
 //   isr = sqrt(sqrt(-1)/x) if x is not a square.
 //   isr = 0                if x is zero.
 // We do not guarantee the sign of the square root.
 //
 // Notes:
 // Let quartic = x^((p-1)/4)
 //
 // x^((p-1)/2) = chi(x)
 // quartic^2   = chi(x)
 // quartic     = sqrt(chi(x))
 // quartic     = 1 or -1 or sqrt(-1) or -sqrt(-1)
 //
 // Note that x is a square if quartic is 1 or -1
 // There are 4 cases to consider:
 //
 // if   quartic         = 1  (x is a square)
 // then x^((p-1)/4)     = 1
 //      x^((p-5)/4) * x = 1
 //      x^((p-5)/4)     = 1/x
 //      x^((p-5)/8)     = sqrt(1/x) or -sqrt(1/x)
 //
 // if   quartic                = -1  (x is a square)
 // then x^((p-1)/4)            = -1
 //      x^((p-5)/4) * x        = -1
 //      x^((p-5)/4)            = -1/x
 //      x^((p-5)/8)            = sqrt(-1)   / sqrt(x)
 //      x^((p-5)/8) * sqrt(-1) = sqrt(-1)^2 / sqrt(x)
 //      x^((p-5)/8) * sqrt(-1) = -1/sqrt(x)
 //      x^((p-5)/8) * sqrt(-1) = -sqrt(1/x) or sqrt(1/x)
 //
 // if   quartic         = sqrt(-1)  (x is not a square)
 // then x^((p-1)/4)     = sqrt(-1)
 //      x^((p-5)/4) * x = sqrt(-1)
 //      x^((p-5)/4)     = sqrt(-1)/x
 //      x^((p-5)/8)     = sqrt(sqrt(-1)/x) or -sqrt(sqrt(-1)/x)
 //
 // Note that the product of two non-squares is always a square:
 //   For any non-squares a and b, chi(a) = -1 and chi(b) = -1.
 //   Since chi(x) = x^((p-1)/2), chi(a)*chi(b) = chi(a*b) = 1.
 //   Therefore a*b is a square.
 //
 //   Since sqrt(-1) and x are both non-squares, their product is a
 //   square, and we can compute their square root.
 //
 // if   quartic                = -sqrt(-1)  (x is not a square)
 // then x^((p-1)/4)            = -sqrt(-1)
 //      x^((p-5)/4) * x        = -sqrt(-1)
 //      x^((p-5)/4)            = -sqrt(-1)/x
 //      x^((p-5)/8)            = sqrt(-sqrt(-1)/x)
 //      x^((p-5)/8)            = sqrt( sqrt(-1)/x) * sqrt(-1)
 //      x^((p-5)/8) * sqrt(-1) = sqrt( sqrt(-1)/x) * sqrt(-1)^2
 //      x^((p-5)/8) * sqrt(-1) = sqrt( sqrt(-1)/x) * -1
 //      x^((p-5)/8) * sqrt(-1) = -sqrt(sqrt(-1)/x) or sqrt(sqrt(-1)/x)
 static int invsqrt(fe isr, const fe x)
 {
     fe check, quartic;
     fe_copy(check, x);
     fe_pow22523(isr, check);
     fe_sq (quartic, isr);
     fe_mul(quartic, quartic, check);
     fe_1  (check);          int p1 = fe_isequal(quartic, check);
     fe_neg(check, check );  int m1 = fe_isequal(quartic, check);
     fe_neg(check, sqrtm1);  int ms = fe_isequal(quartic, check);
     fe_mul(check, isr, sqrtm1);
     fe_ccopy(isr, check, m1 | ms);
     WIPE_BUFFER(quartic);
     WIPE_BUFFER(check);
     return p1 | m1;
 }
 
 // trim a scalar for scalar multiplication
 static void trim_scalar(u8 scalar[32])
 {
     scalar[ 0] &= 248;
     scalar[31] &= 127;
     scalar[31] |= 64;
 }
 
 // get bit from scalar at position i
 static int scalar_bit(const u8 s[32], int i)
 {
     if (i < 0) { return 0; } // handle -1 for sliding windows
     return (s[i>>3] >> (i&7)) & 1;
 }
 
 ///////////////
 /// X-25519 /// Taken from SUPERCOP's ref10 implementation.
 ///////////////
 static void scalarmult(u8 q[32], const u8 scalar[32], const u8 p[32],
                        int nb_bits)
 {
     // computes the scalar product
     fe x1;
     fe_frombytes(x1, p);
 
     // computes the actual scalar product (the result is in x2 and z2)
     fe x2, z2, x3, z3, t0, t1;
     // Montgomery ladder
     // In projective coordinates, to avoid divisions: x = X / Z
     // We don't care about the y coordinate, it's only 1 bit of information
     fe_1(x2);        fe_0(z2); // "zero" point
     fe_copy(x3, x1); fe_1(z3); // "one"  point
     int swap = 0;
     for (int pos = nb_bits-1; pos >= 0; --pos) {
         // constant time conditional swap before ladder step
         int b = scalar_bit(scalar, pos);
         swap ^= b; // xor trick avoids swapping at the end of the loop
         fe_cswap(x2, x3, swap);
         fe_cswap(z2, z3, swap);
         swap = b;  // anticipates one last swap after the loop
 
         // Montgomery ladder step: replaces (P2, P3) by (P2*2, P2+P3)
         // with differential addition
         fe_sub(t0, x3, z3);
         fe_sub(t1, x2, z2);
         fe_add(x2, x2, z2);
         fe_add(z2, x3, z3);
         fe_mul(z3, t0, x2);
         fe_mul(z2, z2, t1);
         fe_sq (t0, t1    );
         fe_sq (t1, x2    );
         fe_add(x3, z3, z2);
         fe_sub(z2, z3, z2);
         fe_mul(x2, t1, t0);
         fe_sub(t1, t1, t0);
         fe_sq (z2, z2    );
         fe_mul_small(z3, t1, 121666);
         fe_sq (x3, x3    );
         fe_add(t0, t0, z3);
         fe_mul(z3, x1, z2);
         fe_mul(z2, t1, t0);
     }
     // last swap is necessary to compensate for the xor trick
     // Note: after this swap, P3 == P2 + P1.
     fe_cswap(x2, x3, swap);
     fe_cswap(z2, z3, swap);
 
     // normalises the coordinates: x == X / Z
     fe_invert(z2, z2);
     fe_mul(x2, x2, z2);
     fe_tobytes(q, x2);
 
     WIPE_BUFFER(x1);
     WIPE_BUFFER(x2);  WIPE_BUFFER(z2);  WIPE_BUFFER(t0);
     WIPE_BUFFER(x3);  WIPE_BUFFER(z3);  WIPE_BUFFER(t1);
 }
 
 void crypto_x25519(u8       raw_shared_secret[32],
                    const u8 your_secret_key  [32],
                    const u8 their_public_key [32])
 {
     // restrict the possible scalar values
     u8 e[32];
     COPY(e, your_secret_key, 32);
     trim_scalar(e);
     scalarmult(raw_shared_secret, e, their_public_key, 255);
     WIPE_BUFFER(e);
 }
 
 void crypto_x25519_public_key(u8       public_key[32],
                               const u8 secret_key[32])
 {
     static const u8 base_point[32] = {9};
     crypto_x25519(public_key, secret_key, base_point);
 }
 
 ///////////////////////////
 /// Arithmetic modulo L ///
 ///////////////////////////
 static const u32 L[8] = {0x5cf5d3ed, 0x5812631a, 0xa2f79cd6, 0x14def9de,
                          0x00000000, 0x00000000, 0x00000000, 0x10000000,};
 
 //  p = a*b + p
 static void multiply(u32 p[16], const u32 a[8], const u32 b[8])
 {
     FOR (i, 0, 8) {
         u64 carry = 0;
         FOR (j, 0, 8) {
             carry  += p[i+j] + (u64)a[i] * b[j];
             p[i+j]  = (u32)carry;
             carry >>= 32;
         }
         p[i+8] = (u32)carry;
     }
 }
 
 static int is_above_l(const u32 x[8])
 {
     // We work with L directly, in a 2's complement encoding
     // (-L == ~L + 1)
     u64 carry = 1;
     FOR (i, 0, 8) {
         carry += (u64)x[i] + ~L[i];
         carry >>= 32;
     }
     return carry;
 }
 
 // Final reduction modulo L, by conditionally removing L.
 // if x < l     , then r = x
 // if l <= x 2*l, then r = x-l
 // otherwise the result will be wrong
 static void remove_l(u32 r[8], const u32 x[8])
 {
     u64 carry = is_above_l(x);
     u32 mask  = ~(u32)carry + 1; // carry == 0 or 1
     FOR (i, 0, 8) {
         carry += (u64)x[i] + (~L[i] & mask);
         r[i]   = (u32)carry;
         carry >>= 32;
     }
 }
 
 // Full reduction modulo L (Barrett reduction)
 static void mod_l(u8 reduced[32], const u32 x[16])
 {
     static const u32 r[9] = {0x0a2c131b,0xed9ce5a3,0x086329a7,0x2106215d,
                              0xffffffeb,0xffffffff,0xffffffff,0xffffffff,0xf,};
     // xr = x * r
     u32 xr[25] = {0};
     FOR (i, 0, 9) {
         u64 carry = 0;
         FOR (j, 0, 16) {
             carry  += xr[i+j] + (u64)r[i] * x[j];
             xr[i+j] = (u32)carry;
             carry >>= 32;
         }
         xr[i+16] = (u32)carry;
     }
     // xr = floor(xr / 2^512) * L
     // Since the result is guaranteed to be below 2*L,
     // it is enough to only compute the first 256 bits.
     // The division is performed by saying xr[i+16]. (16 * 32 = 512)
     ZERO(xr, 8);
     FOR (i, 0, 8) {
         u64 carry = 0;
         FOR (j, 0, 8-i) {
             carry   += xr[i+j] + (u64)xr[i+16] * L[j];
             xr[i+j] = (u32)carry;
             carry >>= 32;
         }
     }
     // xr = x - xr
     u64 carry = 1;
     FOR (i, 0, 8) {
         carry  += (u64)x[i] + ~xr[i];
         xr[i]   = (u32)carry;
         carry >>= 32;
     }
     // Final reduction modulo L (conditional subtraction)
     remove_l(xr, xr);
     store32_le_buf(reduced, xr, 8);
 
     WIPE_BUFFER(xr);
 }
 
 static void reduce(u8 r[64])
 {
     u32 x[16];
     load32_le_buf(x, r, 16);
     mod_l(r, x);
     WIPE_BUFFER(x);
 }
 
 // r = (a * b) + c
 static void mul_add(u8 r[32], const u8 a[32], const u8 b[32], const u8 c[32])
 {
     u32 A[8];  load32_le_buf(A, a, 8);
     u32 B[8];  load32_le_buf(B, b, 8);
     u32 p[16];
     load32_le_buf(p, c, 8);
     ZERO(p + 8, 8);
     multiply(p, A, B);
     mod_l(r, p);
     WIPE_BUFFER(p);
     WIPE_BUFFER(A);
     WIPE_BUFFER(B);
 }
 
 ///////////////
 /// Ed25519 ///
 ///////////////
 
 // Point (group element, ge) in a twisted Edwards curve,
 // in extended projective coordinates.
 // ge        : x  = X/Z, y  = Y/Z, T  = XY/Z
 // ge_cached : Yp = X+Y, Ym = X-Y, T2 = T*D2
 // ge_precomp: Z  = 1
 typedef struct { fe X;  fe Y;  fe Z; fe T;  } ge;
 typedef struct { fe Yp; fe Ym; fe Z; fe T2; } ge_cached;
 typedef struct { fe Yp; fe Ym;       fe T2; } ge_precomp;
 
 static void ge_zero(ge *p)
 {
     fe_0(p->X);
     fe_1(p->Y);
     fe_1(p->Z);
     fe_0(p->T);
 }
 
 static void ge_tobytes(u8 s[32], const ge *h)
 {
     fe recip, x, y;
     fe_invert(recip, h->Z);
     fe_mul(x, h->X, recip);
     fe_mul(y, h->Y, recip);
     fe_tobytes(s, y);
     s[31] ^= fe_isodd(x) << 7;
 
     WIPE_BUFFER(recip);
     WIPE_BUFFER(x);
     WIPE_BUFFER(y);
 }
 
 // h = s, where s is a point encoded in 32 bytes
 //
 // Variable time!  Inputs must not be secret!
 // => Use only to *check* signatures.
 //
 // From the specifications:
 //   The encoding of s contains y and the sign of x
 //   x = sqrt((y^2 - 1) / (d*y^2 + 1))
 // In extended coordinates:
 //   X = x, Y = y, Z = 1, T = x*y
 //
 //    Note that num * den is a square iff num / den is a square
 //    If num * den is not a square, the point was not on the curve.
 // From the above:
 //   Let num =   y^2 - 1
 //   Let den = d*y^2 + 1
 //   x = sqrt((y^2 - 1) / (d*y^2 + 1))
 //   x = sqrt(num / den)
 //   x = sqrt(num^2 / (num * den))
 //   x = num * sqrt(1 / (num * den))
 //
 // Therefore, we can just compute:
 //   num =   y^2 - 1
 //   den = d*y^2 + 1
 //   isr = invsqrt(num * den)  // abort if not square
 //   x   = num * isr
 // Finally, negate x if its sign is not as specified.
 static int ge_frombytes_vartime(ge *h, const u8 s[32])
 {
     fe_frombytes(h->Y, s);
     fe_1(h->Z);
     fe_sq (h->T, h->Y);        // t =   y^2
     fe_mul(h->X, h->T, d   );  // x = d*y^2
     fe_sub(h->T, h->T, h->Z);  // t =   y^2 - 1
     fe_add(h->X, h->X, h->Z);  // x = d*y^2 + 1
     fe_mul(h->X, h->T, h->X);  // x = (y^2 - 1) * (d*y^2 + 1)
     int is_square = invsqrt(h->X, h->X);
     if (!is_square) {
         return -1;             // Not on the curve, abort
     }
     fe_mul(h->X, h->T, h->X);  // x = sqrt((y^2 - 1) / (d*y^2 + 1))
     if (fe_isodd(h->X) != (s[31] >> 7)) {
         fe_neg(h->X, h->X);
     }
     fe_mul(h->T, h->X, h->Y);
     return 0;
 }
 
 static void ge_cache(ge_cached *c, const ge *p)
 {
     fe_add (c->Yp, p->Y, p->X);
     fe_sub (c->Ym, p->Y, p->X);
     fe_copy(c->Z , p->Z      );
     fe_mul (c->T2, p->T, D2  );
 }
 
 // Internal buffers are not wiped! Inputs must not be secret!
 // => Use only to *check* signatures.
 static void ge_add(ge *s, const ge *p, const ge_cached *q)
 {
     fe a, b;
     fe_add(a   , p->Y, p->X );
     fe_sub(b   , p->Y, p->X );
     fe_mul(a   , a   , q->Yp);
     fe_mul(b   , b   , q->Ym);
     fe_add(s->Y, a   , b    );
     fe_sub(s->X, a   , b    );
 
     fe_add(s->Z, p->Z, p->Z );
     fe_mul(s->Z, s->Z, q->Z );
     fe_mul(s->T, p->T, q->T2);
     fe_add(a   , s->Z, s->T );
     fe_sub(b   , s->Z, s->T );
 
     fe_mul(s->T, s->X, s->Y);
     fe_mul(s->X, s->X, b   );
     fe_mul(s->Y, s->Y, a   );
     fe_mul(s->Z, a   , b   );
 }
 
 // Internal buffers are not wiped! Inputs must not be secret!
 // => Use only to *check* signatures.
 static void ge_sub(ge *s, const ge *p, const ge_cached *q)
 {
     ge_cached neg;
     fe_copy(neg.Ym, q->Yp);
     fe_copy(neg.Yp, q->Ym);
     fe_copy(neg.Z , q->Z );
     fe_neg (neg.T2, q->T2);
     ge_add(s, p, &neg);
 }
 
 static void ge_madd(ge *s, const ge *p, const ge_precomp *q, fe a, fe b)
 {
     fe_add(a   , p->Y, p->X );
     fe_sub(b   , p->Y, p->X );
     fe_mul(a   , a   , q->Yp);
     fe_mul(b   , b   , q->Ym);
     fe_add(s->Y, a   , b    );
     fe_sub(s->X, a   , b    );
 
     fe_add(s->Z, p->Z, p->Z );
     fe_mul(s->T, p->T, q->T2);
     fe_add(a   , s->Z, s->T );
     fe_sub(b   , s->Z, s->T );
 
     fe_mul(s->T, s->X, s->Y);
     fe_mul(s->X, s->X, b   );
     fe_mul(s->Y, s->Y, a   );
     fe_mul(s->Z, a   , b   );
 }
 
 static void ge_msub(ge *s, const ge *p, const ge_precomp *q, fe a, fe b)
 {
     fe_add(a   , p->Y, p->X );
     fe_sub(b   , p->Y, p->X );
     fe_mul(a   , a   , q->Ym);
     fe_mul(b   , b   , q->Yp);
     fe_add(s->Y, a   , b    );
     fe_sub(s->X, a   , b    );
 
     fe_add(s->Z, p->Z, p->Z );
     fe_mul(s->T, p->T, q->T2);
     fe_sub(a   , s->Z, s->T );
     fe_add(b   , s->Z, s->T );
 
     fe_mul(s->T, s->X, s->Y);
     fe_mul(s->X, s->X, b   );
     fe_mul(s->Y, s->Y, a   );
     fe_mul(s->Z, a   , b   );
 }
 
 static void ge_double(ge *s, const ge *p, ge *q)
 {
     fe_sq (q->X, p->X);
     fe_sq (q->Y, p->Y);
     fe_sq2(q->Z, p->Z);
     fe_add(q->T, p->X, p->Y);
     fe_sq (s->T, q->T);
     fe_add(q->T, q->Y, q->X);
     fe_sub(q->Y, q->Y, q->X);
     fe_sub(q->X, s->T, q->T);
     fe_sub(q->Z, q->Z, q->Y);
 
     fe_mul(s->X, q->X , q->Z);
     fe_mul(s->Y, q->T , q->Y);
     fe_mul(s->Z, q->Y , q->Z);
     fe_mul(s->T, q->X , q->T);
 }
 
 // 5-bit signed window in cached format (Niels coordinates, Z=1)
 static const ge_precomp b_window[8] = {
     {{25967493,-14356035,29566456,3660896,-12694345,
       4014787,27544626,-11754271,-6079156,2047605,},
      {-12545711,934262,-2722910,3049990,-727428,
       9406986,12720692,5043384,19500929,-15469378,},
      {-8738181,4489570,9688441,-14785194,10184609,
       -12363380,29287919,11864899,-24514362,-4438546,},},
     {{15636291,-9688557,24204773,-7912398,616977,
       -16685262,27787600,-14772189,28944400,-1550024,},
      {16568933,4717097,-11556148,-1102322,15682896,
       -11807043,16354577,-11775962,7689662,11199574,},
      {30464156,-5976125,-11779434,-15670865,23220365,
       15915852,7512774,10017326,-17749093,-9920357,},},
     {{10861363,11473154,27284546,1981175,-30064349,
       12577861,32867885,14515107,-15438304,10819380,},
      {4708026,6336745,20377586,9066809,-11272109,
       6594696,-25653668,12483688,-12668491,5581306,},
      {19563160,16186464,-29386857,4097519,10237984,
       -4348115,28542350,13850243,-23678021,-15815942,},},
     {{5153746,9909285,1723747,-2777874,30523605,
       5516873,19480852,5230134,-23952439,-15175766,},
      {-30269007,-3463509,7665486,10083793,28475525,
       1649722,20654025,16520125,30598449,7715701,},
      {28881845,14381568,9657904,3680757,-20181635,
       7843316,-31400660,1370708,29794553,-1409300,},},
     {{-22518993,-6692182,14201702,-8745502,-23510406,
       8844726,18474211,-1361450,-13062696,13821877,},
      {-6455177,-7839871,3374702,-4740862,-27098617,
       -10571707,31655028,-7212327,18853322,-14220951,},
      {4566830,-12963868,-28974889,-12240689,-7602672,
       -2830569,-8514358,-10431137,2207753,-3209784,},},
     {{-25154831,-4185821,29681144,7868801,-6854661,
       -9423865,-12437364,-663000,-31111463,-16132436,},
      {25576264,-2703214,7349804,-11814844,16472782,
       9300885,3844789,15725684,171356,6466918,},
      {23103977,13316479,9739013,-16149481,817875,
       -15038942,8965339,-14088058,-30714912,16193877,},},
     {{-33521811,3180713,-2394130,14003687,-16903474,
       -16270840,17238398,4729455,-18074513,9256800,},
      {-25182317,-4174131,32336398,5036987,-21236817,
       11360617,22616405,9761698,-19827198,630305,},
      {-13720693,2639453,-24237460,-7406481,9494427,
       -5774029,-6554551,-15960994,-2449256,-14291300,},},
     {{-3151181,-5046075,9282714,6866145,-31907062,
       -863023,-18940575,15033784,25105118,-7894876,},
      {-24326370,15950226,-31801215,-14592823,-11662737,
       -5090925,1573892,-2625887,2198790,-15804619,},
      {-3099351,10324967,-2241613,7453183,-5446979,
       -2735503,-13812022,-16236442,-32461234,-12290683,},},
 };
 
 // Incremental sliding windows (left to right)
 // Based on Roberto Maria Avanzi[2005]
 typedef struct {
     i16 next_index; // position of the next signed digit
     i8  next_digit; // next signed digit (odd number below 2^window_width)
     u8  next_check; // point at which we must check for a new window
 } slide_ctx;
 
 static void slide_init(slide_ctx *ctx, const u8 scalar[32])
 {
     // scalar is guaranteed to be below L, either because we checked (s),
     // or because we reduced it modulo L (h_ram). L is under 2^253, so
     // so bits 253 to 255 are guaranteed to be zero. No need to test them.
     //
     // Note however that L is very close to 2^252, so bit 252 is almost
     // always zero.  If we were to start at bit 251, the tests wouldn't
     // catch the off-by-one error (constructing one that does would be
     // prohibitively expensive).
     //
     // We should still check bit 252, though.
     int i = 252;
     while (i > 0 && scalar_bit(scalar, i) == 0) {
         i--;
     }
     ctx->next_check = (u8)(i + 1);
     ctx->next_index = -1;
     ctx->next_digit = -1;
 }
 
 static int slide_step(slide_ctx *ctx, int width, int i, const u8 scalar[32])
 {
     if (i == ctx->next_check) {
         if (scalar_bit(scalar, i) == scalar_bit(scalar, i - 1)) {
             ctx->next_check--;
         } else {
             // compute digit of next window
             int w = MIN(width, i + 1);
             int v = -(scalar_bit(scalar, i) << (w-1));
             FOR_T (int, j, 0, w-1) {
                 v += scalar_bit(scalar, i-(w-1)+j) << j;
             }
             v += scalar_bit(scalar, i-w);
             int lsb = v & (~v + 1);            // smallest bit of v
             int s   = (   ((lsb & 0xAA) != 0)  // log2(lsb)
                        | (((lsb & 0xCC) != 0) << 1)
                        | (((lsb & 0xF0) != 0) << 2));
             ctx->next_index  = (i16)(i-(w-1)+s);
             ctx->next_digit  = (i8) (v >> s   );
             ctx->next_check -= (u8) w;
         }
     }
     return i == ctx->next_index ? ctx->next_digit: 0;
 }
 
 #define P_W_WIDTH 3 // Affects the size of the stack
 #define B_W_WIDTH 5 // Affects the size of the binary
 #define P_W_SIZE  (1<<(P_W_WIDTH-2))
 
 // P = [b]B + [p]P, where B is the base point
 //
 // Variable time! Internal buffers are not wiped! Inputs must not be secret!
 // => Use only to *check* signatures.
 static void ge_double_scalarmult_vartime(ge *P, const u8 p[32], const u8 b[32])
 {
     // cache P window for addition
     ge_cached cP[P_W_SIZE];
     {
         ge P2, tmp;
         ge_double(&P2, P, &tmp);
         ge_cache(&cP[0], P);
         FOR (i, 1, P_W_SIZE) {
             ge_add(&tmp, &P2, &cP[i-1]);
             ge_cache(&cP[i], &tmp);
         }
     }
 
     // Merged double and add ladder, fused with sliding
     slide_ctx p_slide;  slide_init(&p_slide, p);
     slide_ctx b_slide;  slide_init(&b_slide, b);
     int i = MAX(p_slide.next_check, b_slide.next_check);
     ge *sum = P;
     ge_zero(sum);
     while (i >= 0) {
         ge tmp;
         ge_double(sum, sum, &tmp);
         int p_digit = slide_step(&p_slide, P_W_WIDTH, i, p);
         int b_digit = slide_step(&b_slide, B_W_WIDTH, i, b);
         if (p_digit > 0) { ge_add(sum, sum, &cP[ p_digit / 2]); }
         if (p_digit < 0) { ge_sub(sum, sum, &cP[-p_digit / 2]); }
         fe t1, t2;
         if (b_digit > 0) { ge_madd(sum, sum, b_window +  b_digit/2, t1, t2); }
         if (b_digit < 0) { ge_msub(sum, sum, b_window + -b_digit/2, t1, t2); }
         i--;
     }
 }
 
 // R_check = s[B] - h_ram[pk], where B is the base point
 //
 // Variable time! Internal buffers are not wiped! Inputs must not be secret!
 // => Use only to *check* signatures.
 static int ge_r_check(u8 R_check[32], u8 s[32], u8 h_ram[32], u8 pk[32])
 {
     ge  A;      // not secret, not wiped
     u32 s32[8]; // not secret, not wiped
     load32_le_buf(s32, s, 8);
     if (ge_frombytes_vartime(&A, pk) ||         // A = pk
         is_above_l(s32)) {                      // prevent s malleability
         return -1;
     }
     fe_neg(A.X, A.X);
     fe_neg(A.T, A.T);                           // A = -pk
     ge_double_scalarmult_vartime(&A, h_ram, s); // A = [s]B - [h_ram]pk
     ge_tobytes(R_check, &A);                    // R_check = A
     return 0;
 }
 
 // 5-bit signed comb in cached format (Niels coordinates, Z=1)
 static const ge_precomp b_comb_low[8] = {
     {{-6816601,-2324159,-22559413,124364,18015490,
       8373481,19993724,1979872,-18549925,9085059,},
      {10306321,403248,14839893,9633706,8463310,
       -8354981,-14305673,14668847,26301366,2818560,},
      {-22701500,-3210264,-13831292,-2927732,-16326337,
       -14016360,12940910,177905,12165515,-2397893,},},
     {{-12282262,-7022066,9920413,-3064358,-32147467,
       2927790,22392436,-14852487,2719975,16402117,},
      {-7236961,-4729776,2685954,-6525055,-24242706,
       -15940211,-6238521,14082855,10047669,12228189,},
      {-30495588,-12893761,-11161261,3539405,-11502464,
       16491580,-27286798,-15030530,-7272871,-15934455,},},
     {{17650926,582297,-860412,-187745,-12072900,
       -10683391,-20352381,15557840,-31072141,-5019061,},
      {-6283632,-2259834,-4674247,-4598977,-4089240,
       12435688,-31278303,1060251,6256175,10480726,},
      {-13871026,2026300,-21928428,-2741605,-2406664,
       -8034988,7355518,15733500,-23379862,7489131,},},
     {{6883359,695140,23196907,9644202,-33430614,
       11354760,-20134606,6388313,-8263585,-8491918,},
      {-7716174,-13605463,-13646110,14757414,-19430591,
       -14967316,10359532,-11059670,-21935259,12082603,},
      {-11253345,-15943946,10046784,5414629,24840771,
       8086951,-6694742,9868723,15842692,-16224787,},},
     {{9639399,11810955,-24007778,-9320054,3912937,
       -9856959,996125,-8727907,-8919186,-14097242,},
      {7248867,14468564,25228636,-8795035,14346339,
       8224790,6388427,-7181107,6468218,-8720783,},
      {15513115,15439095,7342322,-10157390,18005294,
       -7265713,2186239,4884640,10826567,7135781,},},
     {{-14204238,5297536,-5862318,-6004934,28095835,
       4236101,-14203318,1958636,-16816875,3837147,},
      {-5511166,-13176782,-29588215,12339465,15325758,
       -15945770,-8813185,11075932,-19608050,-3776283,},
      {11728032,9603156,-4637821,-5304487,-7827751,
       2724948,31236191,-16760175,-7268616,14799772,},},
     {{-28842672,4840636,-12047946,-9101456,-1445464,
       381905,-30977094,-16523389,1290540,12798615,},
      {27246947,-10320914,14792098,-14518944,5302070,
       -8746152,-3403974,-4149637,-27061213,10749585,},
      {25572375,-6270368,-15353037,16037944,1146292,
       32198,23487090,9585613,24714571,-1418265,},},
     {{19844825,282124,-17583147,11004019,-32004269,
       -2716035,6105106,-1711007,-21010044,14338445,},
      {8027505,8191102,-18504907,-12335737,25173494,
       -5923905,15446145,7483684,-30440441,10009108,},
      {-14134701,-4174411,10246585,-14677495,33553567,
       -14012935,23366126,15080531,-7969992,7663473,},},
 };
 
 static const ge_precomp b_comb_high[8] = {
     {{33055887,-4431773,-521787,6654165,951411,
       -6266464,-5158124,6995613,-5397442,-6985227,},
      {4014062,6967095,-11977872,3960002,8001989,
       5130302,-2154812,-1899602,-31954493,-16173976,},
      {16271757,-9212948,23792794,731486,-25808309,
       -3546396,6964344,-4767590,10976593,10050757,},},
     {{2533007,-4288439,-24467768,-12387405,-13450051,
       14542280,12876301,13893535,15067764,8594792,},
      {20073501,-11623621,3165391,-13119866,13188608,
       -11540496,-10751437,-13482671,29588810,2197295,},
      {-1084082,11831693,6031797,14062724,14748428,
       -8159962,-20721760,11742548,31368706,13161200,},},
     {{2050412,-6457589,15321215,5273360,25484180,
       124590,-18187548,-7097255,-6691621,-14604792,},
      {9938196,2162889,-6158074,-1711248,4278932,
       -2598531,-22865792,-7168500,-24323168,11746309,},
      {-22691768,-14268164,5965485,9383325,20443693,
       5854192,28250679,-1381811,-10837134,13717818,},},
     {{-8495530,16382250,9548884,-4971523,-4491811,
       -3902147,6182256,-12832479,26628081,10395408,},
      {27329048,-15853735,7715764,8717446,-9215518,
       -14633480,28982250,-5668414,4227628,242148,},
      {-13279943,-7986904,-7100016,8764468,-27276630,
       3096719,29678419,-9141299,3906709,11265498,},},
     {{11918285,15686328,-17757323,-11217300,-27548967,
       4853165,-27168827,6807359,6871949,-1075745,},
      {-29002610,13984323,-27111812,-2713442,28107359,
       -13266203,6155126,15104658,3538727,-7513788,},
      {14103158,11233913,-33165269,9279850,31014152,
       4335090,-1827936,4590951,13960841,12787712,},},
     {{1469134,-16738009,33411928,13942824,8092558,
       -8778224,-11165065,1437842,22521552,-2792954,},
      {31352705,-4807352,-25327300,3962447,12541566,
       -9399651,-27425693,7964818,-23829869,5541287,},
      {-25732021,-6864887,23848984,3039395,-9147354,
       6022816,-27421653,10590137,25309915,-1584678,},},
     {{-22951376,5048948,31139401,-190316,-19542447,
       -626310,-17486305,-16511925,-18851313,-12985140,},
      {-9684890,14681754,30487568,7717771,-10829709,
       9630497,30290549,-10531496,-27798994,-13812825,},
      {5827835,16097107,-24501327,12094619,7413972,
       11447087,28057551,-1793987,-14056981,4359312,},},
     {{26323183,2342588,-21887793,-1623758,-6062284,
       2107090,-28724907,9036464,-19618351,-13055189,},
      {-29697200,14829398,-4596333,14220089,-30022969,
       2955645,12094100,-13693652,-5941445,7047569,},
      {-3201977,14413268,-12058324,-16417589,-9035655,
       -7224648,9258160,1399236,30397584,-5684634,},},
 };
 
 static void lookup_add(ge *p, ge_precomp *tmp_c, fe tmp_a, fe tmp_b,
                        const ge_precomp comb[8], const u8 scalar[32], int i)
 {
     u8 teeth = (u8)((scalar_bit(scalar, i)          ) +
                     (scalar_bit(scalar, i + 32) << 1) +
                     (scalar_bit(scalar, i + 64) << 2) +
                     (scalar_bit(scalar, i + 96) << 3));
     u8 high  = teeth >> 3;
     u8 index = (teeth ^ (high - 1)) & 7;
     FOR (j, 0, 8) {
         i32 select = 1 & (((j ^ index) - 1) >> 8);
         fe_ccopy(tmp_c->Yp, comb[j].Yp, select);
         fe_ccopy(tmp_c->Ym, comb[j].Ym, select);
         fe_ccopy(tmp_c->T2, comb[j].T2, select);
     }
     fe_neg(tmp_a, tmp_c->T2);
     fe_cswap(tmp_c->T2, tmp_a    , high ^ 1);
     fe_cswap(tmp_c->Yp, tmp_c->Ym, high ^ 1);
     ge_madd(p, p, tmp_c, tmp_a, tmp_b);
 }
 
 // p = [scalar]B, where B is the base point
 static void ge_scalarmult_base(ge *p, const u8 scalar[32])
 {
     // twin 4-bits signed combs, from Mike Hamburg's
     // Fast and compact elliptic-curve cryptography (2012)
     // 1 / 2 modulo L
     static const u8 half_mod_L[32] = {
         247,233,122,46,141,49,9,44,107,206,123,81,239,124,111,10,
         0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8, };
     // (2^256 - 1) / 2 modulo L
     static const u8 half_ones[32] = {
         142,74,204,70,186,24,118,107,184,231,190,57,250,173,119,99,
         255,255,255,255,255,255,255,255,255,255,255,255,255,255,255,7, };
 
     // All bits set form: 1 means 1, 0 means -1
     u8 s_scalar[32];
     mul_add(s_scalar, scalar, half_mod_L, half_ones);
 
     // Double and add ladder
     fe tmp_a, tmp_b;  // temporaries for addition
     ge_precomp tmp_c; // temporary for comb lookup
     ge tmp_d;         // temporary for doubling
     fe_1(tmp_c.Yp);
     fe_1(tmp_c.Ym);
     fe_0(tmp_c.T2);
 
     // Save a double on the first iteration
     ge_zero(p);
     lookup_add(p, &tmp_c, tmp_a, tmp_b, b_comb_low , s_scalar, 31);
     lookup_add(p, &tmp_c, tmp_a, tmp_b, b_comb_high, s_scalar, 31+128);
     // Regular double & add for the rest
     for (int i = 30; i >= 0; i--) {
         ge_double(p, p, &tmp_d);
         lookup_add(p, &tmp_c, tmp_a, tmp_b, b_comb_low , s_scalar, i);
         lookup_add(p, &tmp_c, tmp_a, tmp_b, b_comb_high, s_scalar, i+128);
     }
     // Note: we could save one addition at the end if we assumed the
     // scalar fit in 252 bit.  Which it does in practice if it is
     // selected at random.  However, non-random, non-hashed scalars
     // *can* overflow 252 bits in practice.  Better account for that
     // than leaving that kind of subtle corner case.
 
     WIPE_BUFFER(tmp_a);  WIPE_CTX(&tmp_d);
     WIPE_BUFFER(tmp_b);  WIPE_CTX(&tmp_c);
     WIPE_BUFFER(s_scalar);
 }
 
 void crypto_sign_public_key_custom_hash(u8       public_key[32],
                                         const u8 secret_key[32],
                                         const crypto_sign_vtable *hash)
 {
     u8 a[64];
     hash->hash(a, secret_key, 32);
     trim_scalar(a);
     ge A;
     ge_scalarmult_base(&A, a);
     ge_tobytes(public_key, &A);
     WIPE_BUFFER(a);
     WIPE_CTX(&A);
 }
 
 void crypto_sign_public_key(u8 public_key[32], const u8 secret_key[32])
 {
     crypto_sign_public_key_custom_hash(public_key, secret_key,
                                        &crypto_blake2b_vtable);
 }
 
 void crypto_sign_init_first_pass_custom_hash(crypto_sign_ctx_abstract *ctx,
                                              const u8 secret_key[32],
                                              const u8 public_key[32],
                                              const crypto_sign_vtable *hash)
 {
     ctx->hash  = hash; // set vtable
     u8 *a      = ctx->buf;
     u8 *prefix = ctx->buf + 32;
     ctx->hash->hash(a, secret_key, 32);
     trim_scalar(a);
 
     if (public_key == 0) {
         crypto_sign_public_key_custom_hash(ctx->pk, secret_key, ctx->hash);
     } else {
         COPY(ctx->pk, public_key, 32);
     }
 
     // Deterministic part of EdDSA: Construct a nonce by hashing the message
     // instead of generating a random number.
     // An actual random number would work just fine, and would save us
     // the trouble of hashing the message twice.  If we did that
     // however, the user could fuck it up and reuse the nonce.
     ctx->hash->init  (ctx);
     ctx->hash->update(ctx, prefix , 32);
 }
 
 void crypto_sign_init_first_pass(crypto_sign_ctx_abstract *ctx,
                                  const u8 secret_key[32],
                                  const u8 public_key[32])
 {
     crypto_sign_init_first_pass_custom_hash(ctx, secret_key, public_key,
                                             &crypto_blake2b_vtable);
 }
 
 void crypto_sign_update(crypto_sign_ctx_abstract *ctx,
                         const u8 *msg, size_t msg_size)
 {
     ctx->hash->update(ctx, msg, msg_size);
 }
 
 void crypto_sign_init_second_pass(crypto_sign_ctx_abstract *ctx)
 {
     u8 *r        = ctx->buf + 32;
     u8 *half_sig = ctx->buf + 64;
     ctx->hash->final(ctx, r);
     reduce(r);
 
     // first half of the signature = "random" nonce times the base point
     ge R;
     ge_scalarmult_base(&R, r);
     ge_tobytes(half_sig, &R);
     WIPE_CTX(&R);
 
     // Hash R, the public key, and the message together.
     // It cannot be done in parallel with the first hash.
     ctx->hash->init  (ctx);
     ctx->hash->update(ctx, half_sig, 32);
     ctx->hash->update(ctx, ctx->pk , 32);
 }
 
 void crypto_sign_final(crypto_sign_ctx_abstract *ctx, u8 signature[64])
 {
     u8 *a        = ctx->buf;
     u8 *r        = ctx->buf + 32;
     u8 *half_sig = ctx->buf + 64;
     u8  h_ram[64];
     ctx->hash->final(ctx, h_ram);
     reduce(h_ram);
     COPY(signature, half_sig, 32);
     mul_add(signature + 32, h_ram, a, r); // s = h_ram * a + r
     WIPE_BUFFER(h_ram);
     crypto_wipe(ctx, ctx->hash->ctx_size);
 }
 
 void crypto_sign(u8        signature[64],
                  const u8  secret_key[32],
                  const u8  public_key[32],
                  const u8 *message, size_t message_size)
 {
     crypto_sign_ctx ctx;
     crypto_sign_ctx_abstract *actx = (crypto_sign_ctx_abstract*)&ctx;
     crypto_sign_init_first_pass (actx, secret_key, public_key);
     crypto_sign_update          (actx, message, message_size);
     crypto_sign_init_second_pass(actx);
     crypto_sign_update          (actx, message, message_size);
     crypto_sign_final           (actx, signature);
 }
 
 void crypto_check_init_custom_hash(crypto_check_ctx_abstract *ctx,
                                    const u8 signature[64],
                                    const u8 public_key[32],
                                    const crypto_sign_vtable *hash)
 {
     ctx->hash = hash; // set vtable
     COPY(ctx->buf, signature , 64);
     COPY(ctx->pk , public_key, 32);
     ctx->hash->init  (ctx);
     ctx->hash->update(ctx, signature , 32);
     ctx->hash->update(ctx, public_key, 32);
 }
 
 void crypto_check_init(crypto_check_ctx_abstract *ctx, const u8 signature[64],
                        const u8 public_key[32])
 {
     crypto_check_init_custom_hash(ctx, signature, public_key,
                                   &crypto_blake2b_vtable);
 }
 
 void crypto_check_update(crypto_check_ctx_abstract *ctx,
                          const u8 *msg, size_t msg_size)
 {
     ctx->hash->update(ctx, msg, msg_size);
 }
 
 int crypto_check_final(crypto_check_ctx_abstract *ctx)
 {
     u8 h_ram[64];
     ctx->hash->final(ctx, h_ram);
     reduce(h_ram);
     u8 *R       = ctx->buf;      // R
     u8 *s       = ctx->buf + 32; // s
     u8 *R_check = ctx->pk;       // overwrite ctx->pk to save stack space
     if (ge_r_check(R_check, s, h_ram, ctx->pk)) {
         return -1;
     }
     return crypto_verify32(R, R_check); // R == R_check ? OK : fail
 }
 
 int crypto_check(const u8  signature[64], const u8 public_key[32],
                  const u8 *message, size_t message_size)
 {
     crypto_check_ctx ctx;
     crypto_check_ctx_abstract *actx = (crypto_check_ctx_abstract*)&ctx;
     crypto_check_init  (actx, signature, public_key);
     crypto_check_update(actx, message, message_size);
     return crypto_check_final(actx);
 }
 
 ///////////////////////
 /// EdDSA to X25519 ///
 ///////////////////////
 void crypto_from_eddsa_private(u8 x25519[32], const u8 eddsa[32])
 {
     u8 a[64];
     crypto_blake2b(a, eddsa, 32);
     COPY(x25519, a, 32);
     WIPE_BUFFER(a);
 }
 
 void crypto_from_eddsa_public(u8 x25519[32], const u8 eddsa[32])
 {
     fe t1, t2;
     fe_frombytes(t2, eddsa);
     fe_add(t1, fe_one, t2);
     fe_sub(t2, fe_one, t2);
     fe_invert(t2, t2);
     fe_mul(t1, t1, t2);
     fe_tobytes(x25519, t1);
     WIPE_BUFFER(t1);
     WIPE_BUFFER(t2);
 }
 
 /////////////////////////////////////////////
 /// Dirty ephemeral public key generation ///
 /////////////////////////////////////////////
 
 // Those functions generates a public key, *without* clearing the
 // cofactor.  Sending that key over the network leaks 3 bits of the
 // private key.  Use only to generate ephemeral keys that will be hidden
 // with crypto_curve_to_hidden().
 //
 // The public key is otherwise compatible with crypto_x25519() and
 // crypto_key_exchange() (those properly clear the cofactor).
 //
 // Note that the distribution of the resulting public keys is almost
 // uniform.  Flipping the sign of the v coordinate (not provided by this
 // function), covers the entire key space almost perfectly, where
 // "almost" means a 2^-128 bias (undetectable).  This uniformity is
 // needed to ensure the proper randomness of the resulting
 // representatives (once we apply crypto_curve_to_hidden()).
 //
 // Recall that Curve25519 has order C = 2^255 + e, with e < 2^128 (not
 // to be confused with the prime order of the main subgroup, L, which is
 // 8 times less than that).
 //
 // Generating all points would require us to multiply a point of order C
 // (the base point plus any point of order 8) by all scalars from 0 to
 // C-1.  Clamping limits us to scalars between 2^254 and 2^255 - 1. But
 // by negating the resulting point at random, we also cover scalars from
 // -2^255 + 1 to -2^254 (which modulo C is congruent to e+1 to 2^254 + e).
 //
 // In practice:
 // - Scalars from 0         to e + 1     are never generated
 // - Scalars from 2^255     to 2^255 + e are never generated
 // - Scalars from 2^254 + 1 to 2^254 + e are generated twice
 //
 // Since e < 2^128, detecting this bias requires observing over 2^100
 // representatives from a given source (this will never happen), *and*
 // recovering enough of the private key to determine that they do, or do
 // not, belong to the biased set (this practically requires solving
 // discrete logarithm, which is conjecturally intractable).
 //
 // In practice, this means the bias is impossible to detect.
 
 // s + (x*L) % 8*L
 // Guaranteed to fit in 256 bits iff s fits in 255 bits.
 //   L             < 2^253
 //   x%8           < 2^3
 //   L * (x%8)     < 2^255
 //   s             < 2^255
 //   s + L * (x%8) < 2^256
 static void add_xl(u8 s[32], u8 x)
 {
     u64 mod8  = x & 7;
     u64 carry = 0;
     FOR (i , 0, 8) {
         carry = carry + load32_le(s + 4*i) + L[i] * mod8;
         store32_le(s + 4*i, (u32)carry);
         carry >>= 32;
     }
 }
 
 // "Small" dirty ephemeral key.
 // Use if you need to shrink the size of the binary, and can afford to
 // slow down by a factor of two (compared to the fast version)
 //
 // This version works by decoupling the cofactor from the main factor.
 //
 // - The trimmed scalar determines the main factor
 // - The clamped bits of the scalar determine the cofactor.
 //
 // Cofactor and main factor are combined into a single scalar, which is
 // then multiplied by a point of order 8*L (unlike the base point, which
 // has prime order).  That "dirty" base point is the addition of the
 // regular base point (9), and a point of order 8.
 void crypto_x25519_dirty_small(u8 public_key[32], const u8 secret_key[32])
 {
     // Base point of order 8*L
     // Raw scalar multiplication with it does not clear the cofactor,
     // and the resulting public key will reveal 3 bits of the scalar.
     static const u8 dirty_base_point[32] = {
         0x34, 0xfc, 0x6c, 0xb7, 0xc8, 0xde, 0x58, 0x97, 0x77, 0x70, 0xd9, 0x52,
         0x16, 0xcc, 0xdc, 0x6c, 0x85, 0x90, 0xbe, 0xcd, 0x91, 0x9c, 0x07, 0x59,
         0x94, 0x14, 0x56, 0x3b, 0x4b, 0xa4, 0x47, 0x0f, };
     // separate the main factor & the cofactor of the scalar
     u8 scalar[32];
     COPY(scalar, secret_key, 32);
     trim_scalar(scalar);
 
     // Separate the main factor and the cofactor
     //
     // The scalar is trimmed, so its cofactor is cleared.  The three
     // least significant bits however still have a main factor.  We must
     // remove it for X25519 compatibility.
     //
     // We exploit the fact that 5*L = 1 (modulo 8)
     //   cofactor = lsb * 5 * L             (modulo 8*L)
     //   combined = scalar + cofactor       (modulo 8*L)
     //   combined = scalar + (lsb * 5 * L)  (modulo 8*L)
     add_xl(scalar, secret_key[0] * 5);
     scalarmult(public_key, scalar, dirty_base_point, 256);
     WIPE_BUFFER(scalar);
 }
 
 // "Fast" dirty ephemeral key
 // We use this one by default.
 //
 // This version works by performing a regular scalar multiplication,
 // then add a low order point.  The scalar multiplication is done in
 // Edwards space for more speed (*2 compared to the "small" version).
 // The cost is a bigger binary for programs that don't also sign messages.
 void crypto_x25519_dirty_fast(u8 public_key[32], const u8 secret_key[32])
 {
     u8 scalar[32];
     ge pk;
     COPY(scalar, secret_key, 32);
     trim_scalar(scalar);
     ge_scalarmult_base(&pk, scalar);
 
     // Select low order point
     // We're computing the [cofactor]lop scalar multiplication, where:
     //   cofactor = tweak & 7.
     //   lop      = (lop_x, lop_y)
     //   lop_x    = sqrt((sqrt(d + 1) + 1) / d)
     //   lop_y    = -lop_x * sqrtm1
     // Notes:
     // - A (single) Montgomery ladder would be twice as slow.
     // - An actual scalar multiplication would hurt performance.
     // - A full table lookup would take more code.
     u8 cofactor = secret_key[0] & 7;
     int a = (cofactor >> 2) & 1;
     int b = (cofactor >> 1) & 1;
     int c = (cofactor >> 0) & 1;
     fe t1, t2, t3;
     fe_0(t1);
     fe_ccopy(t1, sqrtm1, b);
     fe_ccopy(t1, lop_x , c);
     fe_neg  (t3, t1);
     fe_ccopy(t1, t3, a);
     fe_1(t2);
     fe_0(t3);
     fe_ccopy(t2, t3   , b);
     fe_ccopy(t2, lop_y, c);
     fe_neg  (t3, t2);
     fe_ccopy(t2, t3, a^b);
     ge_precomp low_order_point;
     fe_add(low_order_point.Yp, t2, t1);
     fe_sub(low_order_point.Ym, t2, t1);
     fe_mul(low_order_point.T2, t2, t1);
     fe_mul(low_order_point.T2, low_order_point.T2, D2);
 
     // Add low order point to the public key
     ge_madd(&pk, &pk, &low_order_point, t1, t2);
 
     // Convert to Montgomery u coordinate (we ignore the sign)
     fe_add(t1, pk.Z, pk.Y);
     fe_sub(t2, pk.Z, pk.Y);
     fe_invert(t2, t2);
     fe_mul(t1, t1, t2);
 
     fe_tobytes(public_key, t1);
 
     WIPE_BUFFER(t1);  WIPE_BUFFER(scalar);
     WIPE_BUFFER(t2);  WIPE_CTX(&pk);
     WIPE_BUFFER(t3);  WIPE_CTX(&low_order_point);
 }
 
 ///////////////////
 /// Elligator 2 ///
 ///////////////////
 static const fe A = {486662};
 
 // Elligator direct map
 //
 // Computes the point corresponding to a representative, encoded in 32
 // bytes (little Endian).  Since positive representatives fits in 254
 // bits, The two most significant bits are ignored.
 //
 // From the paper:
 // w = -A / (fe(1) + non_square * r^2)
 // e = chi(w^3 + A*w^2 + w)
 // u = e*w - (fe(1)-e)*(A//2)
 // v = -e * sqrt(u^3 + A*u^2 + u)
 //
 // We ignore v because we don't need it for X25519 (the Montgomery
 // ladder only uses u).
 //
 // Note that e is either 0, 1 or -1
 // if e = 0    u = 0  and v = 0
 // if e = 1    u = w
 // if e = -1   u = -w - A = w * non_square * r^2
 //
 // Let r1 = non_square * r^2
 // Let r2 = 1 + r1
 // Note that r2 cannot be zero, -1/non_square is not a square.
 // We can (tediously) verify that:
 //   w^3 + A*w^2 + w = (A^2*r1 - r2^2) * A / r2^3
 // Therefore:
 //   chi(w^3 + A*w^2 + w) = chi((A^2*r1 - r2^2) * (A / r2^3))
 //   chi(w^3 + A*w^2 + w) = chi((A^2*r1 - r2^2) * (A / r2^3)) * 1
 //   chi(w^3 + A*w^2 + w) = chi((A^2*r1 - r2^2) * (A / r2^3)) * chi(r2^6)
 //   chi(w^3 + A*w^2 + w) = chi((A^2*r1 - r2^2) * (A / r2^3)  *     r2^6)
 //   chi(w^3 + A*w^2 + w) = chi((A^2*r1 - r2^2) *  A * r2^3)
 // Corollary:
 //   e =  1 if (A^2*r1 - r2^2) *  A * r2^3) is a non-zero square
 //   e = -1 if (A^2*r1 - r2^2) *  A * r2^3) is not a square
 //   Note that w^3 + A*w^2 + w (and therefore e) can never be zero:
 //     w^3 + A*w^2 + w = w * (w^2 + A*w + 1)
 //     w^3 + A*w^2 + w = w * (w^2 + A*w + A^2/4 - A^2/4 + 1)
 //     w^3 + A*w^2 + w = w * (w + A/2)^2        - A^2/4 + 1)
 //     which is zero only if:
 //       w = 0                   (impossible)
 //       (w + A/2)^2 = A^2/4 - 1 (impossible, because A^2/4-1 is not a square)
 //
 // Let isr   = invsqrt((A^2*r1 - r2^2) *  A * r2^3)
 //     isr   = sqrt(1        / ((A^2*r1 - r2^2) *  A * r2^3)) if e =  1
 //     isr   = sqrt(sqrt(-1) / ((A^2*r1 - r2^2) *  A * r2^3)) if e = -1
 //
 // if e = 1
 //   let u1 = -A * (A^2*r1 - r2^2) * A * r2^2 * isr^2
 //       u1 = w
 //       u1 = u
 //
 // if e = -1
 //   let ufactor = -non_square * sqrt(-1) * r^2
 //   let vfactor = sqrt(ufactor)
 //   let u2 = -A * (A^2*r1 - r2^2) * A * r2^2 * isr^2 * ufactor
 //       u2 = w * -1 * -non_square * r^2
 //       u2 = w * non_square * r^2
 //       u2 = u
 void crypto_hidden_to_curve(uint8_t curve[32], const uint8_t hidden[32])
 {
     // Representatives are encoded in 254 bits.
     // The two most significant ones are random padding that must be ignored.
     u8 clamped[32];
     COPY(clamped, hidden, 32);
     clamped[31] &= 0x3f;
 
     fe r, u, t1, t2, t3;
     fe_frombytes(r, clamped);
     fe_sq2(t1, r);
     fe_add(u, t1, fe_one);
     fe_sq (t2, u);
     fe_mul(t3, A2, t1);
     fe_sub(t3, t3, t2);
     fe_mul(t3, t3, A);
     fe_mul(t1, t2, u);
     fe_mul(t1, t3, t1);
     int is_square = invsqrt(t1, t1);
     fe_sq(u, r);
     fe_mul(u, u, ufactor);
     fe_ccopy(u, fe_one, is_square);
     fe_sq (t1, t1);
     fe_mul(u, u, A);
     fe_mul(u, u, t3);
     fe_mul(u, u, t2);
     fe_mul(u, u, t1);
     fe_neg(u, u);
     fe_tobytes(curve, u);
 
     WIPE_BUFFER(t1);  WIPE_BUFFER(r);
     WIPE_BUFFER(t2);  WIPE_BUFFER(u);
     WIPE_BUFFER(t3);  WIPE_BUFFER(clamped);
 }
 
 // Elligator inverse map
 //
 // Computes the representative of a point, if possible.  If not, it does
 // nothing and returns -1.  Note that the success of the operation
 // depends only on the point (more precisely its u coordinate).  The
 // tweak parameter is used only upon success
 //
 // The tweak should be a random byte.  Beyond that, its contents are an
 // implementation detail. Currently, the tweak comprises:
 // - Bit  1  : sign of the v coordinate (0 if positive, 1 if negative)
 // - Bit  2-5: not used
 // - Bits 6-7: random padding
 //
 // From the paper:
 // Let sq = -non_square * u * (u+A)
 // if sq is not a square, or u = -A, there is no mapping
 // Assuming there is a mapping:
 //   if v is positive: r = sqrt(-(u+A) / u)
 //   if v is negative: r = sqrt(-u / (u+A))
 //
 // We compute isr = invsqrt(-non_square * u * (u+A))
 // if it wasn't a non-zero square, abort.
 // else, isr = sqrt(-1 / (non_square * u * (u+A))
 //
 // This causes us to abort if u is zero, even though we shouldn't. This
 // never happens in practice, because (i) a random point in the curve has
 // a negligible chance of being zero, and (ii) scalar multiplication with
 // a trimmed scalar *never* yields zero.
 //
 // Since:
 //   isr * (u+A) = sqrt(-1     / (non_square * u * (u+A)) * (u+A)
 //   isr * (u+A) = sqrt(-(u+A) / (non_square * u * (u+A))
 // and:
 //   isr = u = sqrt(-1 / (non_square * u * (u+A)) * u
 //   isr = u = sqrt(-u / (non_square * u * (u+A))
 // Therefore:
 //   if v is positive: r = isr * (u+A)
 //   if v is negative: r = isr * u
 int crypto_curve_to_hidden(u8 hidden[32], const u8 public_key[32], u8 tweak)
 {
     fe t1, t2, t3;
     fe_frombytes(t1, public_key);
 
     fe_add(t2, t1, A);
     fe_mul(t3, t1, t2);
     fe_mul_small(t3, t3, -2);
     int is_square = invsqrt(t3, t3);
     if (!is_square) {
         // The only variable time bit.  This ultimately reveals how many
         // tries it took us to find a representable key.
         // This does not affect security as long as we try keys at random.
         WIPE_BUFFER(t1);
         WIPE_BUFFER(t2);
         WIPE_BUFFER(t3);
         return -1;
     }
     fe_ccopy    (t1, t2, tweak & 1);
     fe_mul      (t3, t1, t3);
     fe_mul_small(t1, t3, 2);
     fe_neg      (t2, t3);
     fe_ccopy    (t3, t2, fe_isodd(t1));
     fe_tobytes(hidden, t3);
 
     // Pad with two random bits
     hidden[31] |= tweak & 0xc0;
 
     WIPE_BUFFER(t1);
     WIPE_BUFFER(t2);
     WIPE_BUFFER(t3);
     return 0;
 }
 
 void crypto_hidden_key_pair(u8 hidden[32], u8 secret_key[32], u8 seed[32])
 {
     u8 pk [32]; // public key
     u8 buf[64]; // seed + representative
     COPY(buf + 32, seed, 32);
     do {
         crypto_chacha20(buf, 0, 64, buf+32, zero);
         crypto_x25519_dirty_fast(pk, buf); // or the "small" version
     } while(crypto_curve_to_hidden(buf+32, pk, buf[32]));
     // Note that the return value of crypto_curve_to_hidden() is
     // independent from its tweak parameter.
     // Therefore, buf[32] is not actually reused.  Either we loop one
     // more time and buf[32] is used for the new seed, or we succeeded,
     // and buf[32] becomes the tweak parameter.
 
     crypto_wipe(seed, 32);
     COPY(hidden    , buf + 32, 32);
     COPY(secret_key, buf     , 32);
     WIPE_BUFFER(buf);
     WIPE_BUFFER(pk);
 }
 
 ////////////////////
 /// Key exchange ///
 ////////////////////
 void crypto_key_exchange(u8       shared_key[32],
                          const u8 your_secret_key [32],
                          const u8 their_public_key[32])
 {
     crypto_x25519(shared_key, your_secret_key, their_public_key);
     crypto_hchacha20(shared_key, shared_key, zero);
 }
 
 ///////////////////////
 /// Scalar division ///
 ///////////////////////
 
 // Montgomery reduction.
 // Divides x by (2^256), and reduces the result modulo L
 //
 // Precondition:
 //   x < L * 2^256
 // Constants:
 //   r = 2^256                 (makes division by r trivial)
 //   k = (r * (1/r) - 1) // L  (1/r is computed modulo L   )
 // Algorithm:
 //   s = (x * k) % r
 //   t = x + s*L      (t is always a multiple of r)
 //   u = (t/r) % L    (u is always below 2*L, conditional subtraction is enough)
 static void redc(u32 u[8], u32 x[16])
 {
     static const u32 k[8]  = { 0x12547e1b, 0xd2b51da3, 0xfdba84ff, 0xb1a206f2,
                                0xffa36bea, 0x14e75438, 0x6fe91836, 0x9db6c6f2,};
     static const u32 l[8]  = { 0x5cf5d3ed, 0x5812631a, 0xa2f79cd6, 0x14def9de,
                                0x00000000, 0x00000000, 0x00000000, 0x10000000,};
     // s = x * k (modulo 2^256)
     // This is cheaper than the full multiplication.
     u32 s[8] = {0};
     FOR (i, 0, 8) {
         u64 carry = 0;
         FOR (j, 0, 8-i) {
             carry  += s[i+j] + (u64)x[i] * k[j];
             s[i+j]  = (u32)carry;
             carry >>= 32;
         }
     }
     u32 t[16] = {0};
     multiply(t, s, l);
 
     // t = t + x
     u64 carry = 0;
     FOR (i, 0, 16) {
         carry  += (u64)t[i] + x[i];
         t[i]    = (u32)carry;
         carry >>= 32;
     }
 
     // u = (t / 2^256) % L
     // Note that t / 2^256 is always below 2*L,
     // So a constant time conditional subtraction is enough
     // We work with L directly, in a 2's complement encoding
     // (-L == ~L + 1)
     remove_l(u, t+8);
 
     WIPE_BUFFER(s);
     WIPE_BUFFER(t);
 }
 
 void crypto_x25519_inverse(u8 blind_salt [32], const u8 private_key[32],
                            const u8 curve_point[32])
 {
     static const  u8 Lm2[32] = { // L - 2
         0xeb, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2,
         0xde, 0xf9, 0xde, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
         0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10, };
     // 1 in Montgomery form
     u32 m_inv [8] = {0x8d98951d, 0xd6ec3174, 0x737dcf70, 0xc6ef5bf4,
                      0xfffffffe, 0xffffffff, 0xffffffff, 0x0fffffff,};
 
     u8 scalar[32];
     COPY(scalar, private_key, 32);
     trim_scalar(scalar);
 
     // Convert the scalar in Montgomery form
     // m_scl = scalar * 2^256 (modulo L)
     u32 m_scl[8];
     {
         u32 tmp[16];
         ZERO(tmp, 8);
         load32_le_buf(tmp+8, scalar, 8);
         mod_l(scalar, tmp);
         load32_le_buf(m_scl, scalar, 8);
         WIPE_BUFFER(tmp); // Wipe ASAP to save stack space
     }
 
     u32 product[16];
     for (int i = 252; i >= 0; i--) {
         ZERO(product, 16);
         multiply(product, m_inv, m_inv);
         redc(m_inv, product);
         if (scalar_bit(Lm2, i)) {
             ZERO(product, 16);
             multiply(product, m_inv, m_scl);
             redc(m_inv, product);
         }
     }
     // Convert the inverse *out* of Montgomery form
     // scalar = m_inv / 2^256 (modulo L)
     COPY(product, m_inv, 8);
     ZERO(product + 8, 8);
     redc(m_inv, product);
     store32_le_buf(scalar, m_inv, 8); // the *inverse* of the scalar
 
     // Clear the cofactor of scalar:
     //   cleared = scalar * (3*L + 1)      (modulo 8*L)
     //   cleared = scalar + scalar * 3 * L (modulo 8*L)
     // Note that (scalar * 3) is reduced modulo 8, so we only need the
     // first byte.
     add_xl(scalar, scalar[0] * 3);
 
     // Recall that 8*L < 2^256. However it is also very close to
     // 2^255. If we spanned the ladder over 255 bits, random tests
     // wouldn't catch the off-by-one error.
     scalarmult(blind_salt, scalar, curve_point, 256);
 
     WIPE_BUFFER(scalar);   WIPE_BUFFER(m_scl);
     WIPE_BUFFER(product);  WIPE_BUFFER(m_inv);
 }
 
 ////////////////////////////////
 /// Authenticated encryption ///
 ////////////////////////////////
 static void lock_auth(u8 mac[16], const u8  auth_key[32],
                       const u8 *ad         , size_t ad_size,
                       const u8 *cipher_text, size_t text_size)
 {
     u8 sizes[16]; // Not secret, not wiped
     store64_le(sizes + 0, ad_size);
     store64_le(sizes + 8, text_size);
     crypto_poly1305_ctx poly_ctx;           // auto wiped...
     crypto_poly1305_init  (&poly_ctx, auth_key);
     crypto_poly1305_update(&poly_ctx, ad         , ad_size);
     crypto_poly1305_update(&poly_ctx, zero       , align(ad_size, 16));
     crypto_poly1305_update(&poly_ctx, cipher_text, text_size);
     crypto_poly1305_update(&poly_ctx, zero       , align(text_size, 16));
     crypto_poly1305_update(&poly_ctx, sizes      , 16);
     crypto_poly1305_final (&poly_ctx, mac); // ...here
 }
 
 void crypto_lock_aead(u8 mac[16], u8 *cipher_text,
                       const u8  key[32], const u8  nonce[24],
                       const u8 *ad        , size_t ad_size,
                       const u8 *plain_text, size_t text_size)
 {
     u8 sub_key[32];
     u8 auth_key[64]; // "Wasting" the whole Chacha block is faster
     crypto_hchacha20(sub_key, key, nonce);
     crypto_chacha20(auth_key, 0, 64, sub_key, nonce + 16);
     crypto_chacha20_ctr(cipher_text, plain_text, text_size,
                         sub_key, nonce + 16, 1);
     lock_auth(mac, auth_key, ad, ad_size, cipher_text, text_size);
     WIPE_BUFFER(sub_key);
     WIPE_BUFFER(auth_key);
 }
 
 int crypto_unlock_aead(u8 *plain_text, const u8 key[32], const u8 nonce[24],
                        const u8  mac[16],
                        const u8 *ad         , size_t ad_size,
                        const u8 *cipher_text, size_t text_size)
 {
     u8 sub_key[32];
     u8 auth_key[64]; // "Wasting" the whole Chacha block is faster
     crypto_hchacha20(sub_key, key, nonce);
     crypto_chacha20(auth_key, 0, 64, sub_key, nonce + 16);
     u8 real_mac[16];
     lock_auth(real_mac, auth_key, ad, ad_size, cipher_text, text_size);
     WIPE_BUFFER(auth_key);
     if (crypto_verify16(mac, real_mac)) {
         WIPE_BUFFER(sub_key);
         WIPE_BUFFER(real_mac);
         return -1;
     }
     crypto_chacha20_ctr(plain_text, cipher_text, text_size,
                         sub_key, nonce + 16, 1);
     WIPE_BUFFER(sub_key);
     WIPE_BUFFER(real_mac);
     return 0;
 }
 
 void crypto_lock(u8 mac[16], u8 *cipher_text,
                  const u8 key[32], const u8 nonce[24],
                  const u8 *plain_text, size_t text_size)
 {
     crypto_lock_aead(mac, cipher_text, key, nonce, 0, 0, plain_text, text_size);
 }
 
 int crypto_unlock(u8 *plain_text,
                   const u8 key[32], const u8 nonce[24], const u8 mac[16],
                   const u8 *cipher_text, size_t text_size)
 {
     return crypto_unlock_aead(plain_text, key, nonce, mac, 0, 0,
                               cipher_text, text_size);
 }