Add Ed25519 algorithm.

Reinstate Ed25519 algorithm to curv25519.c this is largely just a copy of
the code from BoringSSL with some adjustments so it compiles under OpenSSL.

Reviewed-by: Rich Salz <rsalz@openssl.org>
(Merged from https://github.com/openssl/openssl/pull/3503)

2 files changed, 1335 insertions, 0 deletions

diff --git a/crypto/ec/curve25519.c b/crypto/ec/curve25519.c
index e535823c0f..a8b664ec26 100644
--- a/crypto/ec/curve25519.c
+++ b/crypto/ec/curve25519.c
@@ -14,6 +14,7 @@
 
 #include <string.h>
 #include "ec_lcl.h"
+#include <openssl/sha.h>
 
 
 /* fe means field element. Here the field is \Z/(2^255-19). An element t,
@@ -746,6 +747,30 @@ static void fe_cmov(fe f, const fe g, unsigned b) {
   }
 }
 
+/* return 0 if f == 0
+ * return 1 if f != 0
+ *
+ * Preconditions:
+ *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
+static int fe_isnonzero(const fe f) {
+  uint8_t s[32];
+  static const uint8_t zero[32] = {0};
+  fe_tobytes(s, f);
+
+  return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
+}
+
+/* return 1 if f is in {1,3,5,...,q-2}
+ * return 0 if f is in {0,2,4,...,q-1}
+ *
+ * Preconditions:
+ *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
+static int fe_isnegative(const fe f) {
+  uint8_t s[32];
+  fe_tobytes(s, f);
+  return s[0] & 1;
+}
+
 /* h = 2 * f * f
  * Can overlap h with f.
  *
@@ -898,6 +923,63 @@ static void fe_sq2(fe h, const fe f) {
   h[9] = h9;
 }
 
+static void fe_pow22523(fe out, const fe z) {
+  fe t0;
+  fe t1;
+  fe t2;
+  int i;
+
+  fe_sq(t0, z);
+  fe_sq(t1, t0);
+  for (i = 1; i < 2; ++i) {
+    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; i < 5; ++i) {
+    fe_sq(t1, t1);
+  }
+  fe_mul(t0, t1, t0);
+  fe_sq(t1, t0);
+  for (i = 1; i < 10; ++i) {
+    fe_sq(t1, t1);
+  }
+  fe_mul(t1, t1, t0);
+  fe_sq(t2, t1);
+  for (i = 1; i < 20; ++i) {
+    fe_sq(t2, t2);
+  }
+  fe_mul(t1, t2, t1);
+  fe_sq(t1, t1);
+  for (i = 1; i < 10; ++i) {
+    fe_sq(t1, t1);
+  }
+  fe_mul(t0, t1, t0);
+  fe_sq(t1, t0);
+  for (i = 1; i < 50; ++i) {
+    fe_sq(t1, t1);
+  }
+  fe_mul(t1, t1, t0);
+  fe_sq(t2, t1);
+  for (i = 1; i < 100; ++i) {
+    fe_sq(t2, t2);
+  }
+  fe_mul(t1, t2, t1);
+  fe_sq(t1, t1);
+  for (i = 1; i < 50; ++i) {
+    fe_sq(t1, t1);
+  }
+  fe_mul(t0, t1, t0);
+  fe_sq(t0, t0);
+  for (i = 1; i < 2; ++i) {
+    fe_sq(t0, t0);
+  }
+  fe_mul(out, t0, z);
+}
+
 /* ge means group element.
 
  * Here the group is the set of pairs (x,y) of field elements (see fe.h)
@@ -943,6 +1025,85 @@ typedef struct {
   fe T2d;
 } ge_cached;
 
+static void ge_tobytes(uint8_t *s, const ge_p2 *h) {
+  fe recip;
+  fe x;
+  fe 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_isnegative(x) << 7;
+}
+
+static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h) {
+  fe recip;
+  fe x;
+  fe 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_isnegative(x) << 7;
+}
+
+static const fe d = {-10913610, 13857413, -15372611, 6949391,   114729,
+                     -8787816,  -6275908, -3247719,  -18696448, -12055116};
+
+static const fe sqrtm1 = {-32595792, -7943725,  9377950,  3500415, 12389472,
+                          -272473,   -25146209, -2005654, 326686,  11406482};
+
+static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s) {
+  fe u;
+  fe v;
+  fe v3;
+  fe vxx;
+  fe check;
+
+  fe_frombytes(h->Y, s);
+  fe_1(h->Z);
+  fe_sq(u, h->Y);
+  fe_mul(v, u, d);
+  fe_sub(u, u, h->Z); /* u = y^2-1 */
+  fe_add(v, v, h->Z); /* v = dy^2+1 */
+
+  fe_sq(v3, v);
+  fe_mul(v3, v3, v); /* v3 = v^3 */
+  fe_sq(h->X, v3);
+  fe_mul(h->X, h->X, v);
+  fe_mul(h->X, h->X, u); /* x = uv^7 */
+
+  fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
+  fe_mul(h->X, h->X, v3);
+  fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */
+
+  fe_sq(vxx, h->X);
+  fe_mul(vxx, vxx, v);
+  fe_sub(check, vxx, u); /* vx^2-u */
+  if (fe_isnonzero(check)) {
+    fe_add(check, vxx, u); /* vx^2+u */
+    if (fe_isnonzero(check)) {
+      return -1;
+    }
+    fe_mul(h->X, h->X, sqrtm1);
+  }
+
+  if (fe_isnegative(h->X) != (s[31] >> 7)) {
+    fe_neg(h->X, h->X);
+  }
+
+  fe_mul(h->T, h->X, h->Y);
+  return 0;
+}
+
+static void ge_p2_0(ge_p2 *h) {
+  fe_0(h->X);
+  fe_1(h->Y);
+  fe_1(h->Z);
+}
+
 static void ge_p3_0(ge_p3 *h) {
   fe_0(h->X);
   fe_1(h->Y);
@@ -963,6 +1124,17 @@ static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) {
   fe_copy(r->Z, p->Z);
 }
 
+static const fe d2 = {-21827239, -5839606,  -30745221, 13898782, 229458,
+                      15978800,  -12551817, -6495438,  29715968, 9444199};
+
+/* r = p */
+static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p) {
+  fe_add(r->YplusX, p->Y, p->X);
+  fe_sub(r->YminusX, p->Y, p->X);
+  fe_copy(r->Z, p->Z);
+  fe_mul(r->T2d, p->T, d2);
+}
+
 /* r = p */
 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) {
   fe_mul(r->X, p->X, p->T);
@@ -1016,6 +1188,56 @@ static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
   fe_sub(r->T, t0, r->T);
 }
 
+/* r = p - q */
+static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
+  fe t0;
+
+  fe_add(r->X, p->Y, p->X);
+  fe_sub(r->Y, p->Y, p->X);
+  fe_mul(r->Z, r->X, q->yminusx);
+  fe_mul(r->Y, r->Y, q->yplusx);
+  fe_mul(r->T, q->xy2d, p->T);
+  fe_add(t0, p->Z, p->Z);
+  fe_sub(r->X, r->Z, r->Y);
+  fe_add(r->Y, r->Z, r->Y);
+  fe_sub(r->Z, t0, r->T);
+  fe_add(r->T, t0, r->T);
+}
+
+/* r = p + q */
+static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) {
+  fe t0;
+
+  fe_add(r->X, p->Y, p->X);
+  fe_sub(r->Y, p->Y, p->X);
+  fe_mul(r->Z, r->X, q->YplusX);
+  fe_mul(r->Y, r->Y, q->YminusX);
+  fe_mul(r->T, q->T2d, p->T);
+  fe_mul(r->X, p->Z, q->Z);
+  fe_add(t0, r->X, r->X);
+  fe_sub(r->X, r->Z, r->Y);
+  fe_add(r->Y, r->Z, r->Y);
+  fe_add(r->Z, t0, r->T);
+  fe_sub(r->T, t0, r->T);
+}
+
+/* r = p - q */
+static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) {
+  fe t0;
+
+  fe_add(r->X, p->Y, p->X);
+  fe_sub(r->Y, p->Y, p->X);
+  fe_mul(r->Z, r->X, q->YminusX);
+  fe_mul(r->Y, r->Y, q->YplusX);
+  fe_mul(r->T, q->T2d, p->T);
+  fe_mul(r->X, p->Z, q->Z);
+  fe_add(t0, r->X, r->X);
+  fe_sub(r->X, r->Z, r->Y);
+  fe_add(r->Y, r->Z, r->Y);
+  fe_sub(r->Z, t0, r->T);
+  fe_add(r->T, t0, r->T);
+}
+
 static uint8_t equal(signed char b, signed char c) {
   uint8_t ub = b;
   uint8_t uc = c;
@@ -3363,6 +3585,1110 @@ static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
   x25519_scalar_mult_generic(out, scalar, point);
 }
 
+static void slide(signed char *r, const uint8_t *a) {
+  int i;
+  int b;
+  int k;
+
+  for (i = 0; i < 256; ++i) {
+    r[i] = 1 & (a[i >> 3] >> (i & 7));
+  }
+
+  for (i = 0; i < 256; ++i) {
+    if (r[i]) {
+      for (b = 1; b <= 6 && i + b < 256; ++b) {
+        if (r[i + b]) {
+          if (r[i] + (r[i + b] << b) <= 15) {
+            r[i] += r[i + b] << b;
+            r[i + b] = 0;
+          } else if (r[i] - (r[i + b] << b) >= -15) {
+            r[i] -= r[i + b] << b;
+            for (k = i + b; k < 256; ++k) {
+              if (!r[k]) {
+                r[k] = 1;
+                break;
+              }
+              r[k] = 0;
+            }
+          } else {
+            break;
+          }
+        }
+      }
+    }
+  }
+}
+
+static const ge_precomp Bi[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},
+    },
+};
+
+/* r = a * A + b * B
+ * where a = a[0]+256*a[1]+...+256^31 a[31].
+ * and b = b[0]+256*b[1]+...+256^31 b[31].
+ * B is the Ed25519 base point (x,4/5) with x positive. */
+static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
+                                         const ge_p3 *A, const uint8_t *b) {
+  signed char aslide[256];
+  signed char bslide[256];
+  ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
+  ge_p1p1 t;
+  ge_p3 u;
+  ge_p3 A2;
+  int i;
+
+  slide(aslide, a);
+  slide(bslide, b);
+
+  ge_p3_to_cached(&Ai[0], A);
+  ge_p3_dbl(&t, A);
+  ge_p1p1_to_p3(&A2, &t);
+  ge_add(&t, &A2, &Ai[0]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[1], &u);
+  ge_add(&t, &A2, &Ai[1]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[2], &u);
+  ge_add(&t, &A2, &Ai[2]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[3], &u);
+  ge_add(&t, &A2, &Ai[3]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[4], &u);
+  ge_add(&t, &A2, &Ai[4]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[5], &u);
+  ge_add(&t, &A2, &Ai[5]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[6], &u);
+  ge_add(&t, &A2, &Ai[6]);
+  ge_p1p1_to_p3(&u, &t);
+  ge_p3_to_cached(&Ai[7], &u);
+
+  ge_p2_0(r);
+
+  for (i = 255; i >= 0; --i) {
+    if (aslide[i] || bslide[i]) {
+      break;
+    }
+  }
+
+  for (; i >= 0; --i) {
+    ge_p2_dbl(&t, r);
+
+    if (aslide[i] > 0) {
+      ge_p1p1_to_p3(&u, &t);
+      ge_add(&t, &u, &Ai[aslide[i] / 2]);
+    } else if (aslide[i] < 0) {
+      ge_p1p1_to_p3(&u, &t);
+      ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
+    }
+
+    if (bslide[i] > 0) {
+      ge_p1p1_to_p3(&u, &t);
+      ge_madd(&t, &u, &Bi[bslide[i] / 2]);
+    } else if (bslide[i] < 0) {
+      ge_p1p1_to_p3(&u, &t);
+      ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
+    }
+
+    ge_p1p1_to_p2(r, &t);
+  }
+}
+
+/* The set of scalars is \Z/l
+ * where l = 2^252 + 27742317777372353535851937790883648493. */
+
+/* Input:
+ *   s[0]+256*s[1]+...+256^63*s[63] = s
+ *
+ * Output:
+ *   s[0]+256*s[1]+...+256^31*s[31] = s mod l
+ *   where l = 2^252 + 27742317777372353535851937790883648493.
+ *   Overwrites s in place. */
+static void x25519_sc_reduce(uint8_t *s) {
+  int64_t s0 = 2097151 & load_3(s);
+  int64_t s1 = 2097151 & (load_4(s + 2) >> 5);
+  int64_t s2 = 2097151 & (load_3(s + 5) >> 2);
+  int64_t s3 = 2097151 & (load_4(s + 7) >> 7);
+  int64_t s4 = 2097151 & (load_4(s + 10) >> 4);
+  int64_t s5 = 2097151 & (load_3(s + 13) >> 1);
+  int64_t s6 = 2097151 & (load_4(s + 15) >> 6);
+  int64_t s7 = 2097151 & (load_3(s + 18) >> 3);
+  int64_t s8 = 2097151 & load_3(s + 21);
+  int64_t s9 = 2097151 & (load_4(s + 23) >> 5);
+  int64_t s10 = 2097151 & (load_3(s + 26) >> 2);
+  int64_t s11 = 2097151 & (load_4(s + 28) >> 7);
+  int64_t s12 = 2097151 & (load_4(s + 31) >> 4);
+  int64_t s13 = 2097151 & (load_3(s + 34) >> 1);
+  int64_t s14 = 2097151 & (load_4(s + 36) >> 6);
+  int64_t s15 = 2097151 & (load_3(s + 39) >> 3);
+  int64_t s16 = 2097151 & load_3(s + 42);
+  int64_t s17 = 2097151 & (load_4(s + 44) >> 5);
+  int64_t s18 = 2097151 & (load_3(s + 47) >> 2);
+  int64_t s19 = 2097151 & (load_4(s + 49) >> 7);
+  int64_t s20 = 2097151 & (load_4(s + 52) >> 4);
+  int64_t s21 = 2097151 & (load_3(s + 55) >> 1);
+  int64_t s22 = 2097151 & (load_4(s + 57) >> 6);
+  int64_t s23 = (load_4(s + 60) >> 3);
+  int64_t carry0;
+  int64_t carry1;
+  int64_t carry2;
+  int64_t carry3;
+  int64_t carry4;
+  int64_t carry5;
+  int64_t carry6;
+  int64_t carry7;
+  int64_t carry8;
+  int64_t carry9;
+  int64_t carry10;
+  int64_t carry11;
+  int64_t carry12;
+  int64_t carry13;
+  int64_t carry14;
+  int64_t carry15;
+  int64_t carry16;
+
+  s11 += s23 * 666643;
+  s12 += s23 * 470296;
+  s13 += s23 * 654183;
+  s14 -= s23 * 997805;
+  s15 += s23 * 136657;
+  s16 -= s23 * 683901;
+  s23 = 0;
+
+  s10 += s22 * 666643;
+  s11 += s22 * 470296;
+  s12 += s22 * 654183;
+  s13 -= s22 * 997805;
+  s14 += s22 * 136657;
+  s15 -= s22 * 683901;
+  s22 = 0;
+
+  s9 += s21 * 666643;
+  s10 += s21 * 470296;
+  s11 += s21 * 654183;
+  s12 -= s21 * 997805;
+  s13 += s21 * 136657;
+  s14 -= s21 * 683901;
+  s21 = 0;
+
+  s8 += s20 * 666643;
+  s9 += s20 * 470296;
+  s10 += s20 * 654183;
+  s11 -= s20 * 997805;
+  s12 += s20 * 136657;
+  s13 -= s20 * 683901;
+  s20 = 0;
+
+  s7 += s19 * 666643;
+  s8 += s19 * 470296;
+  s9 += s19 * 654183;
+  s10 -= s19 * 997805;
+  s11 += s19 * 136657;
+  s12 -= s19 * 683901;
+  s19 = 0;
+
+  s6 += s18 * 666643;
+  s7 += s18 * 470296;
+  s8 += s18 * 654183;
+  s9 -= s18 * 997805;
+  s10 += s18 * 136657;
+  s11 -= s18 * 683901;
+  s18 = 0;
+
+  carry6 = (s6 + (1 << 20)) >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry8 = (s8 + (1 << 20)) >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry10 = (s10 + (1 << 20)) >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+  carry12 = (s12 + (1 << 20)) >> 21;
+  s13 += carry12;
+  s12 -= carry12 << 21;
+  carry14 = (s14 + (1 << 20)) >> 21;
+  s15 += carry14;
+  s14 -= carry14 << 21;
+  carry16 = (s16 + (1 << 20)) >> 21;
+  s17 += carry16;
+  s16 -= carry16 << 21;
+
+  carry7 = (s7 + (1 << 20)) >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry9 = (s9 + (1 << 20)) >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry11 = (s11 + (1 << 20)) >> 21;
+  s12 += carry11;
+  s11 -= carry11 << 21;
+  carry13 = (s13 + (1 << 20)) >> 21;
+  s14 += carry13;
+  s13 -= carry13 << 21;
+  carry15 = (s15 + (1 << 20)) >> 21;
+  s16 += carry15;
+  s15 -= carry15 << 21;
+
+  s5 += s17 * 666643;
+  s6 += s17 * 470296;
+  s7 += s17 * 654183;
+  s8 -= s17 * 997805;
+  s9 += s17 * 136657;
+  s10 -= s17 * 683901;
+  s17 = 0;
+
+  s4 += s16 * 666643;
+  s5 += s16 * 470296;
+  s6 += s16 * 654183;
+  s7 -= s16 * 997805;
+  s8 += s16 * 136657;
+  s9 -= s16 * 683901;
+  s16 = 0;
+
+  s3 += s15 * 666643;
+  s4 += s15 * 470296;
+  s5 += s15 * 654183;
+  s6 -= s15 * 997805;
+  s7 += s15 * 136657;
+  s8 -= s15 * 683901;
+  s15 = 0;
+
+  s2 += s14 * 666643;
+  s3 += s14 * 470296;
+  s4 += s14 * 654183;
+  s5 -= s14 * 997805;
+  s6 += s14 * 136657;
+  s7 -= s14 * 683901;
+  s14 = 0;
+
+  s1 += s13 * 666643;
+  s2 += s13 * 470296;
+  s3 += s13 * 654183;
+  s4 -= s13 * 997805;
+  s5 += s13 * 136657;
+  s6 -= s13 * 683901;
+  s13 = 0;
+
+  s0 += s12 * 666643;
+  s1 += s12 * 470296;
+  s2 += s12 * 654183;
+  s3 -= s12 * 997805;
+  s4 += s12 * 136657;
+  s5 -= s12 * 683901;
+  s12 = 0;
+
+  carry0 = (s0 + (1 << 20)) >> 21;
+  s1 += carry0;
+  s0 -= carry0 << 21;
+  carry2 = (s2 + (1 << 20)) >> 21;
+  s3 += carry2;
+  s2 -= carry2 << 21;
+  carry4 = (s4 + (1 << 20)) >> 21;
+  s5 += carry4;
+  s4 -= carry4 << 21;
+  carry6 = (s6 + (1 << 20)) >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry8 = (s8 + (1 << 20)) >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry10 = (s10 + (1 << 20)) >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+
+  carry1 = (s1 + (1 << 20)) >> 21;
+  s2 += carry1;
+  s1 -= carry1 << 21;
+  carry3 = (s3 + (1 << 20)) >> 21;
+  s4 += carry3;
+  s3 -= carry3 << 21;
+  carry5 = (s5 + (1 << 20)) >> 21;
+  s6 += carry5;
+  s5 -= carry5 << 21;
+  carry7 = (s7 + (1 << 20)) >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry9 = (s9 + (1 << 20)) >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry11 = (s11 + (1 << 20)) >> 21;
+  s12 += carry11;
+  s11 -= carry11 << 21;
+
+  s0 += s12 * 666643;
+  s1 += s12 * 470296;
+  s2 += s12 * 654183;
+  s3 -= s12 * 997805;
+  s4 += s12 * 136657;
+  s5 -= s12 * 683901;
+  s12 = 0;
+
+  carry0 = s0 >> 21;
+  s1 += carry0;
+  s0 -= carry0 << 21;
+  carry1 = s1 >> 21;
+  s2 += carry1;
+  s1 -= carry1 << 21;
+  carry2 = s2 >> 21;
+  s3 += carry2;
+  s2 -= carry2 << 21;
+  carry3 = s3 >> 21;
+  s4 += carry3;
+  s3 -= carry3 << 21;
+  carry4 = s4 >> 21;
+  s5 += carry4;
+  s4 -= carry4 << 21;
+  carry5 = s5 >> 21;
+  s6 += carry5;
+  s5 -= carry5 << 21;
+  carry6 = s6 >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry7 = s7 >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry8 = s8 >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry9 = s9 >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry10 = s10 >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+  carry11 = s11 >> 21;
+  s12 += carry11;
+  s11 -= carry11 << 21;
+
+  s0 += s12 * 666643;
+  s1 += s12 * 470296;
+  s2 += s12 * 654183;
+  s3 -= s12 * 997805;
+  s4 += s12 * 136657;
+  s5 -= s12 * 683901;
+  s12 = 0;
+
+  carry0 = s0 >> 21;
+  s1 += carry0;
+  s0 -= carry0 << 21;
+  carry1 = s1 >> 21;
+  s2 += carry1;
+  s1 -= carry1 << 21;
+  carry2 = s2 >> 21;
+  s3 += carry2;
+  s2 -= carry2 << 21;
+  carry3 = s3 >> 21;
+  s4 += carry3;
+  s3 -= carry3 << 21;
+  carry4 = s4 >> 21;
+  s5 += carry4;
+  s4 -= carry4 << 21;
+  carry5 = s5 >> 21;
+  s6 += carry5;
+  s5 -= carry5 << 21;
+  carry6 = s6 >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry7 = s7 >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry8 = s8 >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry9 = s9 >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry10 = s10 >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+
+  s[0] = s0 >> 0;
+  s[1] = s0 >> 8;
+  s[2] = (s0 >> 16) | (s1 << 5);
+  s[3] = s1 >> 3;
+  s[4] = s1 >> 11;
+  s[5] = (s1 >> 19) | (s2 << 2);
+  s[6] = s2 >> 6;
+  s[7] = (s2 >> 14) | (s3 << 7);
+  s[8] = s3 >> 1;
+  s[9] = s3 >> 9;
+  s[10] = (s3 >> 17) | (s4 << 4);
+  s[11] = s4 >> 4;
+  s[12] = s4 >> 12;
+  s[13] = (s4 >> 20) | (s5 << 1);
+  s[14] = s5 >> 7;
+  s[15] = (s5 >> 15) | (s6 << 6);
+  s[16] = s6 >> 2;
+  s[17] = s6 >> 10;
+  s[18] = (s6 >> 18) | (s7 << 3);
+  s[19] = s7 >> 5;
+  s[20] = s7 >> 13;
+  s[21] = s8 >> 0;
+  s[22] = s8 >> 8;
+  s[23] = (s8 >> 16) | (s9 << 5);
+  s[24] = s9 >> 3;
+  s[25] = s9 >> 11;
+  s[26] = (s9 >> 19) | (s10 << 2);
+  s[27] = s10 >> 6;
+  s[28] = (s10 >> 14) | (s11 << 7);
+  s[29] = s11 >> 1;
+  s[30] = s11 >> 9;
+  s[31] = s11 >> 17;
+}
+
+/* Input:
+ *   a[0]+256*a[1]+...+256^31*a[31] = a
+ *   b[0]+256*b[1]+...+256^31*b[31] = b
+ *   c[0]+256*c[1]+...+256^31*c[31] = c
+ *
+ * Output:
+ *   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
+ *   where l = 2^252 + 27742317777372353535851937790883648493. */
+static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
+                      const uint8_t *c) {
+  int64_t a0 = 2097151 & load_3(a);
+  int64_t a1 = 2097151 & (load_4(a + 2) >> 5);
+  int64_t a2 = 2097151 & (load_3(a + 5) >> 2);
+  int64_t a3 = 2097151 & (load_4(a + 7) >> 7);
+  int64_t a4 = 2097151 & (load_4(a + 10) >> 4);
+  int64_t a5 = 2097151 & (load_3(a + 13) >> 1);
+  int64_t a6 = 2097151 & (load_4(a + 15) >> 6);
+  int64_t a7 = 2097151 & (load_3(a + 18) >> 3);
+  int64_t a8 = 2097151 & load_3(a + 21);
+  int64_t a9 = 2097151 & (load_4(a + 23) >> 5);
+  int64_t a10 = 2097151 & (load_3(a + 26) >> 2);
+  int64_t a11 = (load_4(a + 28) >> 7);
+  int64_t b0 = 2097151 & load_3(b);
+  int64_t b1 = 2097151 & (load_4(b + 2) >> 5);
+  int64_t b2 = 2097151 & (load_3(b + 5) >> 2);
+  int64_t b3 = 2097151 & (load_4(b + 7) >> 7);
+  int64_t b4 = 2097151 & (load_4(b + 10) >> 4);
+  int64_t b5 = 2097151 & (load_3(b + 13) >> 1);
+  int64_t b6 = 2097151 & (load_4(b + 15) >> 6);
+  int64_t b7 = 2097151 & (load_3(b + 18) >> 3);
+  int64_t b8 = 2097151 & load_3(b + 21);
+  int64_t b9 = 2097151 & (load_4(b + 23) >> 5);
+  int64_t b10 = 2097151 & (load_3(b + 26) >> 2);
+  int64_t b11 = (load_4(b + 28) >> 7);
+  int64_t c0 = 2097151 & load_3(c);
+  int64_t c1 = 2097151 & (load_4(c + 2) >> 5);
+  int64_t c2 = 2097151 & (load_3(c + 5) >> 2);
+  int64_t c3 = 2097151 & (load_4(c + 7) >> 7);
+  int64_t c4 = 2097151 & (load_4(c + 10) >> 4);
+  int64_t c5 = 2097151 & (load_3(c + 13) >> 1);
+  int64_t c6 = 2097151 & (load_4(c + 15) >> 6);
+  int64_t c7 = 2097151 & (load_3(c + 18) >> 3);
+  int64_t c8 = 2097151 & load_3(c + 21);
+  int64_t c9 = 2097151 & (load_4(c + 23) >> 5);
+  int64_t c10 = 2097151 & (load_3(c + 26) >> 2);
+  int64_t c11 = (load_4(c + 28) >> 7);
+  int64_t s0;
+  int64_t s1;
+  int64_t s2;
+  int64_t s3;
+  int64_t s4;
+  int64_t s5;
+  int64_t s6;
+  int64_t s7;
+  int64_t s8;
+  int64_t s9;
+  int64_t s10;
+  int64_t s11;
+  int64_t s12;
+  int64_t s13;
+  int64_t s14;
+  int64_t s15;
+  int64_t s16;
+  int64_t s17;
+  int64_t s18;
+  int64_t s19;
+  int64_t s20;
+  int64_t s21;
+  int64_t s22;
+  int64_t s23;
+  int64_t carry0;
+  int64_t carry1;
+  int64_t carry2;
+  int64_t carry3;
+  int64_t carry4;
+  int64_t carry5;
+  int64_t carry6;
+  int64_t carry7;
+  int64_t carry8;
+  int64_t carry9;
+  int64_t carry10;
+  int64_t carry11;
+  int64_t carry12;
+  int64_t carry13;
+  int64_t carry14;
+  int64_t carry15;
+  int64_t carry16;
+  int64_t carry17;
+  int64_t carry18;
+  int64_t carry19;
+  int64_t carry20;
+  int64_t carry21;
+  int64_t carry22;
+
+  s0 = c0 + a0 * b0;
+  s1 = c1 + a0 * b1 + a1 * b0;
+  s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
+  s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
+  s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
+  s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
+  s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
+  s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 +
+       a6 * b1 + a7 * b0;
+  s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 +
+       a6 * b2 + a7 * b1 + a8 * b0;
+  s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 +
+       a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
+  s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 +
+        a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
+  s11 = c11 + a0 * b11 + a1 * b10 + a2 * b9 + a3 * b8 + a4 * b7 + a5 * b6 +
+        a6 * b5 + a7 * b4 + a8 * b3 + a9 * b2 + a10 * b1 + a11 * b0;
+  s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 +
+        a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
+  s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 +
+        a9 * b4 + a10 * b3 + a11 * b2;
+  s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 +
+        a10 * b4 + a11 * b3;
+  s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 +
+        a11 * b4;
+  s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
+  s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
+  s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
+  s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
+  s20 = a9 * b11 + a10 * b10 + a11 * b9;
+  s21 = a10 * b11 + a11 * b10;
+  s22 = a11 * b11;
+  s23 = 0;
+
+  carry0 = (s0 + (1 << 20)) >> 21;
+  s1 += carry0;
+  s0 -= carry0 << 21;
+  carry2 = (s2 + (1 << 20)) >> 21;
+  s3 += carry2;
+  s2 -= carry2 << 21;
+  carry4 = (s4 + (1 << 20)) >> 21;
+  s5 += carry4;
+  s4 -= carry4 << 21;
+  carry6 = (s6 + (1 << 20)) >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry8 = (s8 + (1 << 20)) >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry10 = (s10 + (1 << 20)) >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+  carry12 = (s12 + (1 << 20)) >> 21;
+  s13 += carry12;
+  s12 -= carry12 << 21;
+  carry14 = (s14 + (1 << 20)) >> 21;
+  s15 += carry14;
+  s14 -= carry14 << 21;
+  carry16 = (s16 + (1 << 20)) >> 21;
+  s17 += carry16;
+  s16 -= carry16 << 21;
+  carry18 = (s18 + (1 << 20)) >> 21;
+  s19 += carry18;
+  s18 -= carry18 << 21;
+  carry20 = (s20 + (1 << 20)) >> 21;
+  s21 += carry20;
+  s20 -= carry20 << 21;
+  carry22 = (s22 + (1 << 20)) >> 21;
+  s23 += carry22;
+  s22 -= carry22 << 21;
+
+  carry1 = (s1 + (1 << 20)) >> 21;
+  s2 += carry1;
+  s1 -= carry1 << 21;
+  carry3 = (s3 + (1 << 20)) >> 21;
+  s4 += carry3;
+  s3 -= carry3 << 21;
+  carry5 = (s5 + (1 << 20)) >> 21;
+  s6 += carry5;
+  s5 -= carry5 << 21;
+  carry7 = (s7 + (1 << 20)) >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry9 = (s9 + (1 << 20)) >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry11 = (s11 + (1 << 20)) >> 21;
+  s12 += carry11;
+  s11 -= carry11 << 21;
+  carry13 = (s13 + (1 << 20)) >> 21;
+  s14 += carry13;
+  s13 -= carry13 << 21;
+  carry15 = (s15 + (1 << 20)) >> 21;
+  s16 += carry15;
+  s15 -= carry15 << 21;
+  carry17 = (s17 + (1 << 20)) >> 21;
+  s18 += carry17;
+  s17 -= carry17 << 21;
+  carry19 = (s19 + (1 << 20)) >> 21;
+  s20 += carry19;
+  s19 -= carry19 << 21;
+  carry21 = (s21 + (1 << 20)) >> 21;
+  s22 += carry21;
+  s21 -= carry21 << 21;
+
+  s11 += s23 * 666643;
+  s12 += s23 * 470296;
+  s13 += s23 * 654183;
+  s14 -= s23 * 997805;
+  s15 += s23 * 136657;
+  s16 -= s23 * 683901;
+  s23 = 0;
+
+  s10 += s22 * 666643;
+  s11 += s22 * 470296;
+  s12 += s22 * 654183;
+  s13 -= s22 * 997805;
+  s14 += s22 * 136657;
+  s15 -= s22 * 683901;
+  s22 = 0;
+
+  s9 += s21 * 666643;
+  s10 += s21 * 470296;
+  s11 += s21 * 654183;
+  s12 -= s21 * 997805;
+  s13 += s21 * 136657;
+  s14 -= s21 * 683901;
+  s21 = 0;
+
+  s8 += s20 * 666643;
+  s9 += s20 * 470296;
+  s10 += s20 * 654183;
+  s11 -= s20 * 997805;
+  s12 += s20 * 136657;
+  s13 -= s20 * 683901;
+  s20 = 0;
+
+  s7 += s19 * 666643;
+  s8 += s19 * 470296;
+  s9 += s19 * 654183;
+  s10 -= s19 * 997805;
+  s11 += s19 * 136657;
+  s12 -= s19 * 683901;
+  s19 = 0;
+
+  s6 += s18 * 666643;
+  s7 += s18 * 470296;
+  s8 += s18 * 654183;
+  s9 -= s18 * 997805;
+  s10 += s18 * 136657;
+  s11 -= s18 * 683901;
+  s18 = 0;
+
+  carry6 = (s6 + (1 << 20)) >> 21;
+  s7 += carry6;
+  s6 -= carry6 << 21;
+  carry8 = (s8 + (1 << 20)) >> 21;
+  s9 += carry8;
+  s8 -= carry8 << 21;
+  carry10 = (s10 + (1 << 20)) >> 21;
+  s11 += carry10;
+  s10 -= carry10 << 21;
+  carry12 = (s12 + (1 << 20)) >> 21;
+  s13 += carry12;
+  s12 -= carry12 << 21;
+  carry14 = (s14 + (1 << 20)) >> 21;
+  s15 += carry14;
+  s14 -= carry14 << 21;
+  carry16 = (s16 + (1 << 20)) >> 21;
+  s17 += carry16;
+  s16 -= carry16 << 21;
+
+  carry7 = (s7 + (1 << 20)) >> 21;
+  s8 += carry7;
+  s7 -= carry7 << 21;
+  carry9 = (s9 + (1 << 20)) >> 21;
+  s10 += carry9;
+  s9 -= carry9 << 21;
+  carry11 = (s11 + (1 << 20)) >> 21;
+  s12 += carry11;
+  s11 -= carry11 << 21;
+  carry13 = (s13 + (1 << 20)) >> 21;
+  s14 += carry13;
+  s13 -= carry13 << 21;
+  carry15 = (s15 + (1 << 20)) >> 21;
+  s16 += carry15;
+  s15 -= carry15 << 21;
+
+  s5 += s17 * 666643;
+  s6 += s17 * 470296;
+  s7 += s17 * 654183;
+  s8 -= s17 * 997805;
+  s9 += s17 * 136657;
+  s10 -= s17 * 683901;
+  s17 = 0;
+
+  s4 += s16 * 666643;
+  s5 += s16 * 470296;
+  s6 += s16 * 654183;
+  s7 -= s16 * 997805;
+  s8 += s16 * 136657;
+  s9 -= s16 * 683901;
+  s16 = 0;
+
+  s3 += s15 * 666643;
+  s4 += s15 * 470296;
+  s5 += s15 * 654183;
+  s6 -= s15 * 997805;
+  s7 += s15 * 136657;
+  s8 -= s15 * 683901;
+  s15 = 0;
+
+  s2 += s14 * 666643;
+  s3 += s14 * 470296;
+  s4 += s14 * 654183;
+  s5 -= s14 * 997805;
+  s6 += s14 * 136657;
+  s7 -= s14 * 683901;
+  s14 = 0;
+