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-rw-r--r--crypto/ec/curve448/arch_32/arch_intrinsics.h27
-rw-r--r--crypto/ec/curve448/arch_32/f_impl.c95
-rw-r--r--crypto/ec/curve448/arch_32/f_impl.h60
-rw-r--r--crypto/ec/curve448/curve448.c727
-rw-r--r--crypto/ec/curve448/curve448_lcl.h38
-rw-r--r--crypto/ec/curve448/curve448_tables.c475
-rw-r--r--crypto/ec/curve448/curve448utils.h78
-rw-r--r--crypto/ec/curve448/ed448.h195
-rw-r--r--crypto/ec/curve448/eddsa.c346
-rw-r--r--crypto/ec/curve448/f_generic.c204
-rw-r--r--crypto/ec/curve448/field.h168
-rw-r--r--crypto/ec/curve448/point_448.h301
-rw-r--r--crypto/ec/curve448/scalar.c235
-rw-r--r--crypto/ec/curve448/word.h81
14 files changed, 3030 insertions, 0 deletions
diff --git a/crypto/ec/curve448/arch_32/arch_intrinsics.h b/crypto/ec/curve448/arch_32/arch_intrinsics.h
new file mode 100644
index 000000000000..48081c77170b
--- /dev/null
+++ b/crypto/ec/curve448/arch_32/arch_intrinsics.h
@@ -0,0 +1,27 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_ARCH_32_ARCH_INTRINSICS_H
+# define HEADER_ARCH_32_ARCH_INTRINSICS_H
+
+#include "internal/constant_time_locl.h"
+
+# define ARCH_WORD_BITS 32
+
+#define word_is_zero(a) constant_time_is_zero_32(a)
+
+static ossl_inline uint64_t widemul(uint32_t a, uint32_t b)
+{
+ return ((uint64_t)a) * b;
+}
+
+#endif /* HEADER_ARCH_32_ARCH_INTRINSICS_H */
diff --git a/crypto/ec/curve448/arch_32/f_impl.c b/crypto/ec/curve448/arch_32/f_impl.c
new file mode 100644
index 000000000000..8a89d276edb6
--- /dev/null
+++ b/crypto/ec/curve448/arch_32/f_impl.c
@@ -0,0 +1,95 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2014 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#include "field.h"
+
+void gf_mul(gf_s * RESTRICT cs, const gf as, const gf bs)
+{
+ const uint32_t *a = as->limb, *b = bs->limb;
+ uint32_t *c = cs->limb;
+ uint64_t accum0 = 0, accum1 = 0, accum2 = 0;
+ uint32_t mask = (1 << 28) - 1;
+ uint32_t aa[8], bb[8];
+ int i, j;
+
+ for (i = 0; i < 8; i++) {
+ aa[i] = a[i] + a[i + 8];
+ bb[i] = b[i] + b[i + 8];
+ }
+
+ for (j = 0; j < 8; j++) {
+ accum2 = 0;
+ for (i = 0; i < j + 1; i++) {
+ accum2 += widemul(a[j - i], b[i]);
+ accum1 += widemul(aa[j - i], bb[i]);
+ accum0 += widemul(a[8 + j - i], b[8 + i]);
+ }
+ accum1 -= accum2;
+ accum0 += accum2;
+ accum2 = 0;
+ for (i = j + 1; i < 8; i++) {
+ accum0 -= widemul(a[8 + j - i], b[i]);
+ accum2 += widemul(aa[8 + j - i], bb[i]);
+ accum1 += widemul(a[16 + j - i], b[8 + i]);
+ }
+ accum1 += accum2;
+ accum0 += accum2;
+ c[j] = ((uint32_t)(accum0)) & mask;
+ c[j + 8] = ((uint32_t)(accum1)) & mask;
+ accum0 >>= 28;
+ accum1 >>= 28;
+ }
+
+ accum0 += accum1;
+ accum0 += c[8];
+ accum1 += c[0];
+ c[8] = ((uint32_t)(accum0)) & mask;
+ c[0] = ((uint32_t)(accum1)) & mask;
+
+ accum0 >>= 28;
+ accum1 >>= 28;
+ c[9] += ((uint32_t)(accum0));
+ c[1] += ((uint32_t)(accum1));
+}
+
+void gf_mulw_unsigned(gf_s * RESTRICT cs, const gf as, uint32_t b)
+{
+ const uint32_t *a = as->limb;
+ uint32_t *c = cs->limb;
+ uint64_t accum0 = 0, accum8 = 0;
+ uint32_t mask = (1 << 28) - 1;
+ int i;
+
+ assert(b <= mask);
+
+ for (i = 0; i < 8; i++) {
+ accum0 += widemul(b, a[i]);
+ accum8 += widemul(b, a[i + 8]);
+ c[i] = accum0 & mask;
+ accum0 >>= 28;
+ c[i + 8] = accum8 & mask;
+ accum8 >>= 28;
+ }
+
+ accum0 += accum8 + c[8];
+ c[8] = ((uint32_t)accum0) & mask;
+ c[9] += (uint32_t)(accum0 >> 28);
+
+ accum8 += c[0];
+ c[0] = ((uint32_t)accum8) & mask;
+ c[1] += (uint32_t)(accum8 >> 28);
+}
+
+void gf_sqr(gf_s * RESTRICT cs, const gf as)
+{
+ gf_mul(cs, as, as); /* Performs better with a dedicated square */
+}
diff --git a/crypto/ec/curve448/arch_32/f_impl.h b/crypto/ec/curve448/arch_32/f_impl.h
new file mode 100644
index 000000000000..bbde84a03897
--- /dev/null
+++ b/crypto/ec/curve448/arch_32/f_impl.h
@@ -0,0 +1,60 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2014-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_ARCH_32_F_IMPL_H
+# define HEADER_ARCH_32_F_IMPL_H
+
+# define GF_HEADROOM 2
+# define LIMB(x) ((x) & ((1 << 28) - 1)), ((x) >> 28)
+# define FIELD_LITERAL(a, b, c, d, e, f, g, h) \
+ {{LIMB(a), LIMB(b), LIMB(c), LIMB(d), LIMB(e), LIMB(f), LIMB(g), LIMB(h)}}
+
+# define LIMB_PLACE_VALUE(i) 28
+
+void gf_add_RAW(gf out, const gf a, const gf b)
+{
+ unsigned int i;
+
+ for (i = 0; i < NLIMBS; i++)
+ out->limb[i] = a->limb[i] + b->limb[i];
+}
+
+void gf_sub_RAW(gf out, const gf a, const gf b)
+{
+ unsigned int i;
+
+ for (i = 0; i < NLIMBS; i++)
+ out->limb[i] = a->limb[i] - b->limb[i];
+}
+
+void gf_bias(gf a, int amt)
+{
+ unsigned int i;
+ uint32_t co1 = ((1 << 28) - 1) * amt, co2 = co1 - amt;
+
+ for (i = 0; i < NLIMBS; i++)
+ a->limb[i] += (i == NLIMBS / 2) ? co2 : co1;
+}
+
+void gf_weak_reduce(gf a)
+{
+ uint32_t mask = (1 << 28) - 1;
+ uint32_t tmp = a->limb[NLIMBS - 1] >> 28;
+ unsigned int i;
+
+ a->limb[NLIMBS / 2] += tmp;
+ for (i = NLIMBS - 1; i > 0; i--)
+ a->limb[i] = (a->limb[i] & mask) + (a->limb[i - 1] >> 28);
+ a->limb[0] = (a->limb[0] & mask) + tmp;
+}
+
+#endif /* HEADER_ARCH_32_F_IMPL_H */
diff --git a/crypto/ec/curve448/curve448.c b/crypto/ec/curve448/curve448.c
new file mode 100644
index 000000000000..7dc68c8853e1
--- /dev/null
+++ b/crypto/ec/curve448/curve448.c
@@ -0,0 +1,727 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+#include <openssl/crypto.h>
+#include "word.h"
+#include "field.h"
+
+#include "point_448.h"
+#include "ed448.h"
+#include "curve448_lcl.h"
+
+#define COFACTOR 4
+
+#define C448_WNAF_FIXED_TABLE_BITS 5
+#define C448_WNAF_VAR_TABLE_BITS 3
+
+#define EDWARDS_D (-39081)
+
+static const curve448_scalar_t precomputed_scalarmul_adjustment = {
+ {
+ {
+ SC_LIMB(0xc873d6d54a7bb0cf), SC_LIMB(0xe933d8d723a70aad),
+ SC_LIMB(0xbb124b65129c96fd), SC_LIMB(0x00000008335dc163)
+ }
+ }
+};
+
+#define TWISTED_D (EDWARDS_D - 1)
+
+#define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
+
+/* Inverse. */
+static void gf_invert(gf y, const gf x, int assert_nonzero)
+{
+ mask_t ret;
+ gf t1, t2;
+
+ gf_sqr(t1, x); /* o^2 */
+ ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
+ (void)ret;
+ if (assert_nonzero)
+ assert(ret);
+ gf_sqr(t1, t2);
+ gf_mul(t2, t1, x); /* not direct to y in case of alias. */
+ gf_copy(y, t2);
+}
+
+/** identity = (0,1) */
+const curve448_point_t curve448_point_identity =
+ { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
+
+static void point_double_internal(curve448_point_t p, const curve448_point_t q,
+ int before_double)
+{
+ gf a, b, c, d;
+
+ gf_sqr(c, q->x);
+ gf_sqr(a, q->y);
+ gf_add_nr(d, c, a); /* 2+e */
+ gf_add_nr(p->t, q->y, q->x); /* 2+e */
+ gf_sqr(b, p->t);
+ gf_subx_nr(b, b, d, 3); /* 4+e */
+ gf_sub_nr(p->t, a, c); /* 3+e */
+ gf_sqr(p->x, q->z);
+ gf_add_nr(p->z, p->x, p->x); /* 2+e */
+ gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
+ if (GF_HEADROOM == 5)
+ gf_weak_reduce(a); /* or 1+e */
+ gf_mul(p->x, a, b);
+ gf_mul(p->z, p->t, a);
+ gf_mul(p->y, p->t, d);
+ if (!before_double)
+ gf_mul(p->t, b, d);
+}
+
+void curve448_point_double(curve448_point_t p, const curve448_point_t q)
+{
+ point_double_internal(p, q, 0);
+}
+
+/* Operations on [p]niels */
+static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
+{
+ gf_cond_swap(n->a, n->b, neg);
+ gf_cond_neg(n->c, neg);
+}
+
+static void pt_to_pniels(pniels_t b, const curve448_point_t a)
+{
+ gf_sub(b->n->a, a->y, a->x);
+ gf_add(b->n->b, a->x, a->y);
+ gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
+ gf_add(b->z, a->z, a->z);
+}
+
+static void pniels_to_pt(curve448_point_t e, const pniels_t d)
+{
+ gf eu;
+
+ gf_add(eu, d->n->b, d->n->a);
+ gf_sub(e->y, d->n->b, d->n->a);
+ gf_mul(e->t, e->y, eu);
+ gf_mul(e->x, d->z, e->y);
+ gf_mul(e->y, d->z, eu);
+ gf_sqr(e->z, d->z);
+}
+
+static void niels_to_pt(curve448_point_t e, const niels_t n)
+{
+ gf_add(e->y, n->b, n->a);
+ gf_sub(e->x, n->b, n->a);
+ gf_mul(e->t, e->y, e->x);
+ gf_copy(e->z, ONE);
+}
+
+static void add_niels_to_pt(curve448_point_t d, const niels_t e,
+ int before_double)
+{
+ gf a, b, c;
+
+ gf_sub_nr(b, d->y, d->x); /* 3+e */
+ gf_mul(a, e->a, b);
+ gf_add_nr(b, d->x, d->y); /* 2+e */
+ gf_mul(d->y, e->b, b);
+ gf_mul(d->x, e->c, d->t);
+ gf_add_nr(c, a, d->y); /* 2+e */
+ gf_sub_nr(b, d->y, a); /* 3+e */
+ gf_sub_nr(d->y, d->z, d->x); /* 3+e */
+ gf_add_nr(a, d->x, d->z); /* 2+e */
+ gf_mul(d->z, a, d->y);
+ gf_mul(d->x, d->y, b);
+ gf_mul(d->y, a, c);
+ if (!before_double)
+ gf_mul(d->t, b, c);
+}
+
+static void sub_niels_from_pt(curve448_point_t d, const niels_t e,
+ int before_double)
+{
+ gf a, b, c;
+
+ gf_sub_nr(b, d->y, d->x); /* 3+e */
+ gf_mul(a, e->b, b);
+ gf_add_nr(b, d->x, d->y); /* 2+e */
+ gf_mul(d->y, e->a, b);
+ gf_mul(d->x, e->c, d->t);
+ gf_add_nr(c, a, d->y); /* 2+e */
+ gf_sub_nr(b, d->y, a); /* 3+e */
+ gf_add_nr(d->y, d->z, d->x); /* 2+e */
+ gf_sub_nr(a, d->z, d->x); /* 3+e */
+ gf_mul(d->z, a, d->y);
+ gf_mul(d->x, d->y, b);
+ gf_mul(d->y, a, c);
+ if (!before_double)
+ gf_mul(d->t, b, c);
+}
+
+static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
+ int before_double)
+{
+ gf L0;
+
+ gf_mul(L0, p->z, pn->z);
+ gf_copy(p->z, L0);
+ add_niels_to_pt(p, pn->n, before_double);
+}
+
+static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
+ int before_double)
+{
+ gf L0;
+
+ gf_mul(L0, p->z, pn->z);
+ gf_copy(p->z, L0);
+ sub_niels_from_pt(p, pn->n, before_double);
+}
+
+c448_bool_t curve448_point_eq(const curve448_point_t p,
+ const curve448_point_t q)
+{
+ mask_t succ;
+ gf a, b;
+
+ /* equality mod 2-torsion compares x/y */
+ gf_mul(a, p->y, q->x);
+ gf_mul(b, q->y, p->x);
+ succ = gf_eq(a, b);
+
+ return mask_to_bool(succ);
+}
+
+c448_bool_t curve448_point_valid(const curve448_point_t p)
+{
+ mask_t out;
+ gf a, b, c;
+
+ gf_mul(a, p->x, p->y);
+ gf_mul(b, p->z, p->t);
+ out = gf_eq(a, b);
+ gf_sqr(a, p->x);
+ gf_sqr(b, p->y);
+ gf_sub(a, b, a);
+ gf_sqr(b, p->t);
+ gf_mulw(c, b, TWISTED_D);
+ gf_sqr(b, p->z);
+ gf_add(b, b, c);
+ out &= gf_eq(a, b);
+ out &= ~gf_eq(p->z, ZERO);
+ return mask_to_bool(out);
+}
+
+static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
+ const niels_t * table,
+ int nelts, int idx)
+{
+ constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
+}
+
+void curve448_precomputed_scalarmul(curve448_point_t out,
+ const curve448_precomputed_s * table,
+ const curve448_scalar_t scalar)
+{
+ unsigned int i, j, k;
+ const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
+ niels_t ni;
+ curve448_scalar_t scalar1x;
+
+ curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
+ curve448_scalar_halve(scalar1x, scalar1x);
+
+ for (i = s; i > 0; i--) {
+ if (i != s)
+ point_double_internal(out, out, 0);
+
+ for (j = 0; j < n; j++) {
+ int tab = 0;
+ mask_t invert;
+
+ for (k = 0; k < t; k++) {
+ unsigned int bit = (i - 1) + s * (k + j * t);
+
+ if (bit < C448_SCALAR_BITS)
+ tab |=
+ (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
+ }
+
+ invert = (tab >> (t - 1)) - 1;
+ tab ^= invert;
+ tab &= (1 << (t - 1)) - 1;
+
+ constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
+ 1 << (t - 1), tab);
+
+ cond_neg_niels(ni, invert);
+ if ((i != s) || j != 0)
+ add_niels_to_pt(out, ni, j == n - 1 && i != 1);
+ else
+ niels_to_pt(out, ni);
+ }
+ }
+
+ OPENSSL_cleanse(ni, sizeof(ni));
+ OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
+}
+
+void curve448_point_mul_by_ratio_and_encode_like_eddsa(
+ uint8_t enc[EDDSA_448_PUBLIC_BYTES],
+ const curve448_point_t p)
+{
+ gf x, y, z, t;
+ curve448_point_t q;
+
+ /* The point is now on the twisted curve. Move it to untwisted. */
+ curve448_point_copy(q, p);
+
+ {
+ /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
+ gf u;
+
+ gf_sqr(x, q->x);
+ gf_sqr(t, q->y);
+ gf_add(u, x, t);
+ gf_add(z, q->y, q->x);
+ gf_sqr(y, z);
+ gf_sub(y, y, u);
+ gf_sub(z, t, x);
+ gf_sqr(x, q->z);
+ gf_add(t, x, x);
+ gf_sub(t, t, z);
+ gf_mul(x, t, y);
+ gf_mul(y, z, u);
+ gf_mul(z, u, t);
+ OPENSSL_cleanse(u, sizeof(u));
+ }
+
+ /* Affinize */
+ gf_invert(z, z, 1);
+ gf_mul(t, x, z);
+ gf_mul(x, y, z);
+
+ /* Encode */
+ enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
+ gf_serialize(enc, x, 1);
+ enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
+
+ OPENSSL_cleanse(x, sizeof(x));
+ OPENSSL_cleanse(y, sizeof(y));
+ OPENSSL_cleanse(z, sizeof(z));
+ OPENSSL_cleanse(t, sizeof(t));
+ curve448_point_destroy(q);
+}
+
+c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
+ curve448_point_t p,
+ const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
+{
+ uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
+ mask_t low;
+ mask_t succ;
+
+ memcpy(enc2, enc, sizeof(enc2));
+
+ low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
+ enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
+
+ succ = gf_deserialize(p->y, enc2, 1, 0);
+ succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
+
+ gf_sqr(p->x, p->y);
+ gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
+ gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
+ gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
+
+ gf_mul(p->x, p->z, p->t);
+ succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
+
+ gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
+ gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
+ gf_copy(p->z, ONE);
+
+ {
+ gf a, b, c, d;
+
+ /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
+ gf_sqr(c, p->x);
+ gf_sqr(a, p->y);
+ gf_add(d, c, a);
+ gf_add(p->t, p->y, p->x);
+ gf_sqr(b, p->t);
+ gf_sub(b, b, d);
+ gf_sub(p->t, a, c);
+ gf_sqr(p->x, p->z);
+ gf_add(p->z, p->x, p->x);
+ gf_sub(a, p->z, d);
+ gf_mul(p->x, a, b);
+ gf_mul(p->z, p->t, a);
+ gf_mul(p->y, p->t, d);
+ gf_mul(p->t, b, d);
+ OPENSSL_cleanse(a, sizeof(a));
+ OPENSSL_cleanse(b, sizeof(b));
+ OPENSSL_cleanse(c, sizeof(c));
+ OPENSSL_cleanse(d, sizeof(d));
+ }
+
+ OPENSSL_cleanse(enc2, sizeof(enc2));
+ assert(curve448_point_valid(p) || ~succ);
+
+ return c448_succeed_if(mask_to_bool(succ));
+}
+
+c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
+ const uint8_t base[X_PUBLIC_BYTES],
+ const uint8_t scalar[X_PRIVATE_BYTES])
+{
+ gf x1, x2, z2, x3, z3, t1, t2;
+ int t;
+ mask_t swap = 0;
+ mask_t nz;
+
+ (void)gf_deserialize(x1, base, 1, 0);
+ gf_copy(x2, ONE);
+ gf_copy(z2, ZERO);
+ gf_copy(x3, x1);
+ gf_copy(z3, ONE);
+
+ for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
+ uint8_t sb = scalar[t / 8];
+ mask_t k_t;
+
+ /* Scalar conditioning */
+ if (t / 8 == 0)
+ sb &= -(uint8_t)COFACTOR;
+ else if (t == X_PRIVATE_BITS - 1)
+ sb = -1;
+
+ k_t = (sb >> (t % 8)) & 1;
+ k_t = 0 - k_t; /* set to all 0s or all 1s */
+
+ swap ^= k_t;
+ gf_cond_swap(x2, x3, swap);
+ gf_cond_swap(z2, z3, swap);
+ swap = k_t;
+
+ /*
+ * The "_nr" below skips coefficient reduction. In the following
+ * comments, "2+e" is saying that the coefficients are at most 2+epsilon
+ * times the reduction limit.
+ */
+ gf_add_nr(t1, x2, z2); /* A = x2 + z2 */ /* 2+e */
+ gf_sub_nr(t2, x2, z2); /* B = x2 - z2 */ /* 3+e */
+ gf_sub_nr(z2, x3, z3); /* D = x3 - z3 */ /* 3+e */
+ gf_mul(x2, t1, z2); /* DA */
+ gf_add_nr(z2, z3, x3); /* C = x3 + z3 */ /* 2+e */
+ gf_mul(x3, t2, z2); /* CB */
+ gf_sub_nr(z3, x2, x3); /* DA-CB */ /* 3+e */
+ gf_sqr(z2, z3); /* (DA-CB)^2 */
+ gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
+ gf_add_nr(z2, x2, x3); /* (DA+CB) */ /* 2+e */
+ gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
+
+ gf_sqr(z2, t1); /* AA = A^2 */
+ gf_sqr(t1, t2); /* BB = B^2 */
+ gf_mul(x2, z2, t1); /* x2 = AA*BB */
+ gf_sub_nr(t2, z2, t1); /* E = AA-BB */ /* 3+e */
+
+ gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
+ gf_add_nr(t1, t1, z2); /* AA + a24*E */ /* 2+e */
+ gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
+ }
+
+ /* Finish */
+ gf_cond_swap(x2, x3, swap);
+ gf_cond_swap(z2, z3, swap);
+ gf_invert(z2, z2, 0);
+ gf_mul(x1, x2, z2);
+ gf_serialize(out, x1, 1);
+ nz = ~gf_eq(x1, ZERO);
+
+ OPENSSL_cleanse(x1, sizeof(x1));
+ OPENSSL_cleanse(x2, sizeof(x2));
+ OPENSSL_cleanse(z2, sizeof(z2));
+ OPENSSL_cleanse(x3, sizeof(x3));
+ OPENSSL_cleanse(z3, sizeof(z3));
+ OPENSSL_cleanse(t1, sizeof(t1));
+ OPENSSL_cleanse(t2, sizeof(t2));
+
+ return c448_succeed_if(mask_to_bool(nz));
+}
+
+void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
+ out[X_PUBLIC_BYTES],
+ const curve448_point_t p)
+{
+ curve448_point_t q;
+
+ curve448_point_copy(q, p);
+ gf_invert(q->t, q->x, 0); /* 1/x */
+ gf_mul(q->z, q->t, q->y); /* y/x */
+ gf_sqr(q->y, q->z); /* (y/x)^2 */
+ gf_serialize(out, q->y, 1);
+ curve448_point_destroy(q);
+}
+
+void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
+ const uint8_t scalar[X_PRIVATE_BYTES])
+{
+ /* Scalar conditioning */
+ uint8_t scalar2[X_PRIVATE_BYTES];
+ curve448_scalar_t the_scalar;
+ curve448_point_t p;
+ unsigned int i;
+
+ memcpy(scalar2, scalar, sizeof(scalar2));
+ scalar2[0] &= -(uint8_t)COFACTOR;
+
+ scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
+ scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
+
+ curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
+
+ /* Compensate for the encoding ratio */
+ for (i = 1; i < X448_ENCODE_RATIO; i <<= 1)
+ curve448_scalar_halve(the_scalar, the_scalar);
+
+ curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
+ curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
+ curve448_point_destroy(p);
+}
+
+/* Control for variable-time scalar multiply algorithms. */
+struct smvt_control {
+ int power, addend;
+};
+
+#if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ > 3))
+# define NUMTRAILINGZEROS __builtin_ctz
+#else
+# define NUMTRAILINGZEROS numtrailingzeros
+static uint32_t numtrailingzeros(uint32_t i)
+{
+ uint32_t tmp;
+ uint32_t num = 31;
+
+ if (i == 0)
+ return 32;
+
+ tmp = i << 16;
+ if (tmp != 0) {
+ i = tmp;
+ num -= 16;
+ }
+ tmp = i << 8;
+ if (tmp != 0) {
+ i = tmp;
+ num -= 8;
+ }
+ tmp = i << 4;
+ if (tmp != 0) {
+ i = tmp;
+ num -= 4;
+ }
+ tmp = i << 2;
+ if (tmp != 0) {
+ i = tmp;
+ num -= 2;
+ }
+ tmp = i << 1;
+ if (tmp != 0)
+ num--;
+
+ return num;
+}
+#endif
+
+static int recode_wnaf(struct smvt_control *control,
+ /* [nbits/(table_bits + 1) + 3] */
+ const curve448_scalar_t scalar,
+ unsigned int table_bits)
+{
+ unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
+ int position = table_size - 1; /* at the end */
+ uint64_t current = scalar->limb[0] & 0xFFFF;
+ uint32_t mask = (1 << (table_bits + 1)) - 1;
+ unsigned int w;
+ const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
+ unsigned int n, i;
+
+ /* place the end marker */
+ control[position].power = -1;
+ control[position].addend = 0;
+ position--;
+
+ /*
+ * PERF: Could negate scalar if it's large. But then would need more cases
+ * in the actual code that uses it, all for an expected reduction of like
+ * 1/5 op. Probably not worth it.
+ */
+
+ for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
+ if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
+ /* Refill the 16 high bits of current */
+ current += (uint32_t)((scalar->limb[w / B_OVER_16]
+ >> (16 * (w % B_OVER_16))) << 16);
+ }
+
+ while (current & 0xFFFF) {
+ uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
+ uint32_t odd = (uint32_t)current >> pos;
+ int32_t delta = odd & mask;
+
+ assert(position >= 0);
+ if (odd & (1 << (table_bits + 1)))
+ delta -= (1 << (table_bits + 1));
+ current -= delta * (1 << pos);
+ control[position].power = pos + 16 * (w - 1);
+ control[position].addend = delta;
+ position--;
+ }
+ current >>= 16;
+ }
+ assert(current == 0);
+
+ position++;
+ n = table_size - position;
+ for (i = 0; i < n; i++)
+ control[i] = control[i + position];
+
+ return n - 1;
+}
+
+static void prepare_wnaf_table(pniels_t * output,
+ const curve448_point_t working,
+ unsigned int tbits)
+{
+ curve448_point_t tmp;
+ int i;
+ pniels_t twop;
+
+ pt_to_pniels(output[0], working);
+
+ if (tbits == 0)
+ return;
+
+ curve448_point_double(tmp, working);
+ pt_to_pniels(twop, tmp);
+
+ add_pniels_to_pt(tmp, output[0], 0);
+ pt_to_pniels(output[1], tmp);
+
+ for (i = 2; i < 1 << tbits; i++) {
+ add_pniels_to_pt(tmp, twop, 0);
+ pt_to_pniels(output[i], tmp);
+ }
+
+ curve448_point_destroy(tmp);
+ OPENSSL_cleanse(twop, sizeof(twop));
+}
+
+void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
+ const curve448_scalar_t scalar1,
+ const curve448_point_t base2,
+ const curve448_scalar_t scalar2)
+{
+ const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
+ const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
+ struct smvt_control control_var[C448_SCALAR_BITS /
+ (C448_WNAF_VAR_TABLE_BITS + 1) + 3];
+ struct smvt_control control_pre[C448_SCALAR_BITS /
+ (C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
+ int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
+ int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
+ pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
+ int contp = 0, contv = 0, i;
+
+ prepare_wnaf_table(precmp_var, base2, table_bits_var);
+ i = control_var[0].power;
+
+ if (i < 0) {
+ curve448_point_copy(combo, curve448_point_identity);
+ return;
+ }
+ if (i > control_pre[0].power) {
+ pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
+ contv++;
+ } else if (i == control_pre[0].power && i >= 0) {
+ pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
+ add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
+ i);
+ contv++;
+ contp++;
+ } else {
+ i = control_pre[0].power;
+ niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
+ contp++;
+ }
+
+ for (i--; i >= 0; i--) {
+ int cv = (i == control_var[contv].power);
+ int cp = (i == control_pre[contp].power);
+
+ point_double_internal(combo, combo, i && !(cv || cp));
+
+ if (cv) {
+ assert(control_var[contv].addend);
+
+ if (control_var[contv].addend > 0)
+ add_pniels_to_pt(combo,
+ precmp_var[control_var[contv].addend >> 1],
+ i && !cp);
+ else
+ sub_pniels_from_pt(combo,
+ precmp_var[(-control_var[contv].addend)
+ >> 1], i && !cp);
+ contv++;
+ }
+
+ if (cp) {
+ assert(control_pre[contp].addend);
+
+ if (control_pre[contp].addend > 0)
+ add_niels_to_pt(combo,
+ curve448_wnaf_base[control_pre[contp].addend
+ >> 1], i);
+ else
+ sub_niels_from_pt(combo,
+ curve448_wnaf_base[(-control_pre
+ [contp].addend) >> 1], i);
+ contp++;
+ }
+ }
+
+ /* This function is non-secret, but whatever this is cheap. */
+ OPENSSL_cleanse(control_var, sizeof(control_var));
+ OPENSSL_cleanse(control_pre, sizeof(control_pre));
+ OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
+
+ assert(contv == ncb_var);
+ (void)ncb_var;
+ assert(contp == ncb_pre);
+ (void)ncb_pre;
+}
+
+void curve448_point_destroy(curve448_point_t point)
+{
+ OPENSSL_cleanse(point, sizeof(curve448_point_t));
+}
+
+int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
+ const uint8_t peer_public_value[56])
+{
+ return x448_int(out_shared_key, peer_public_value, private_key)
+ == C448_SUCCESS;
+}
+
+void X448_public_from_private(uint8_t out_public_value[56],
+ const uint8_t private_key[56])
+{
+ x448_derive_public_key(out_public_value, private_key);
+}
diff --git a/crypto/ec/curve448/curve448_lcl.h b/crypto/ec/curve448/curve448_lcl.h
new file mode 100644
index 000000000000..2bc3bd84c86d
--- /dev/null
+++ b/crypto/ec/curve448/curve448_lcl.h
@@ -0,0 +1,38 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ */
+#ifndef HEADER_CURVE448_LCL_H
+# define HEADER_CURVE448_LCL_H
+# include "curve448utils.h"
+
+int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
+ const uint8_t peer_public_value[56]);
+
+void X448_public_from_private(uint8_t out_public_value[56],
+ const uint8_t private_key[56]);
+
+int ED448_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
+ const uint8_t public_key[57], const uint8_t private_key[57],
+ const uint8_t *context, size_t context_len);
+
+int ED448_verify(const uint8_t *message, size_t message_len,
+ const uint8_t signature[114], const uint8_t public_key[57],
+ const uint8_t *context, size_t context_len);
+
+int ED448ph_sign(uint8_t *out_sig, const uint8_t hash[64],
+ const uint8_t public_key[57], const uint8_t private_key[57],
+ const uint8_t *context, size_t context_len);
+
+int ED448ph_verify(const uint8_t hash[64], const uint8_t signature[114],
+ const uint8_t public_key[57], const uint8_t *context,
+ size_t context_len);
+
+int ED448_public_from_private(uint8_t out_public_key[57],
+ const uint8_t private_key[57]);
+
+#endif /* HEADER_CURVE448_LCL_H */
diff --git a/crypto/ec/curve448/curve448_tables.c b/crypto/ec/curve448/curve448_tables.c
new file mode 100644
index 000000000000..a1185b1eee6a
--- /dev/null
+++ b/crypto/ec/curve448/curve448_tables.c
@@ -0,0 +1,475 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+#include "field.h"
+
+#include "point_448.h"
+
+static const curve448_precomputed_s curve448_precomputed_base_table = {
+ {
+ {{
+ {FIELD_LITERAL(0x00cc3b062366f4cc,0x003d6e34e314aa3c,0x00d51c0a7521774d,0x0094e060eec6ab8b,0x00d21291b4d80082,0x00befed12b55ef1e,0x00c3dd2df5c94518,0x00e0a7b112b8d4e6)},
+ {FIELD_LITERAL(0x0019eb5608d8723a,0x00d1bab52fb3aedb,0x00270a7311ebc90c,0x0037c12b91be7f13,0x005be16cd8b5c704,0x003e181acda888e1,0x00bc1f00fc3fc6d0,0x00d3839bfa319e20)},
+ {FIELD_LITERAL(0x003caeb88611909f,0x00ea8b378c4df3d4,0x00b3295b95a5a19a,0x00a65f97514bdfb5,0x00b39efba743cab1,0x0016ba98b862fd2d,0x0001508812ee71d7,0x000a75740eea114a)},
+ }}, {{
+ {FIELD_LITERAL(0x00ebcf0eb649f823,0x00166d332e98ea03,0x0059ddf64f5cd5f6,0x0047763123d9471b,0x00a64065c53ef62f,0x00978e44c480153d,0x000b5b2a0265f194,0x0046a24b9f32965a)},
+ {FIELD_LITERAL(0x00b9eef787034df0,0x0020bc24de3390cd,0x000022160bae99bb,0x00ae66e886e97946,0x0048d4bbe02cbb8b,0x0072ba97b34e38d4,0x00eae7ec8f03e85a,0x005ba92ecf808b2c)},
+ {FIELD_LITERAL(0x00c9cfbbe74258fd,0x00843a979ea9eaa7,0x000cbb4371cfbe90,0x0059bac8f7f0a628,0x004b3dff882ff530,0x0011869df4d90733,0x00595aa71f4abfc2,0x0070e2d38990c2e6)},
+ }}, {{
+ {FIELD_LITERAL(0x00de2010c0a01733,0x00c739a612e24297,0x00a7212643141d7c,0x00f88444f6b67c11,0x00484b7b16ec28f2,0x009c1b8856af9c68,0x00ff4669591fe9d6,0x0054974be08a32c8)},
+ {FIELD_LITERAL(0x0010de3fd682ceed,0x008c07642d83ca4e,0x0013bb064e00a1cc,0x009411ae27870e11,0x00ea8e5b4d531223,0x0032fe7d2aaece2e,0x00d989e243e7bb41,0x000fe79a508e9b8b)},
+ {FIELD_LITERAL(0x005e0426b9bfc5b1,0x0041a5b1d29ee4fa,0x0015b0def7774391,0x00bc164f1f51af01,0x00d543b0942797b9,0x003c129b6398099c,0x002b114c6e5adf18,0x00b4e630e4018a7b)},
+ }}, {{
+ {FIELD_LITERAL(0x00d490afc95f8420,0x00b096bf50c1d9b9,0x00799fd707679866,0x007c74d9334afbea,0x00efaa8be80ff4ed,0x0075c4943bb81694,0x00c21c2fca161f36,0x00e77035d492bfee)},
+ {FIELD_LITERAL(0x006658a190dd6661,0x00e0e9bab38609a6,0x0028895c802237ed,0x006a0229c494f587,0x002dcde96c9916b7,0x00d158822de16218,0x00173b917a06856f,0x00ca78a79ae07326)},
+ {FIELD_LITERAL(0x00e35bfc79caced4,0x0087238a3e1fe3bb,0x00bcbf0ff4ceff5b,0x00a19c1c94099b91,0x0071e102b49db976,0x0059e3d004eada1e,0x008da78afa58a47e,0x00579c8ebf269187)},
+ }}, {{
+ {FIELD_LITERAL(0x00a16c2905eee75f,0x009d4bcaea2c7e1d,0x00d3bd79bfad19df,0x0050da745193342c,0x006abdb8f6b29ab1,0x00a24fe0a4fef7ef,0x0063730da1057dfb,0x00a08c312c8eb108)},
+ {FIELD_LITERAL(0x00b583be005375be,0x00a40c8f8a4e3df4,0x003fac4a8f5bdbf7,0x00d4481d872cd718,0x004dc8749cdbaefe,0x00cce740d5e5c975,0x000b1c1f4241fd21,0x00a76de1b4e1cd07)},
+ {FIELD_LITERAL(0x007a076500d30b62,0x000a6e117b7f090f,0x00c8712ae7eebd9a,0x000fbd6c1d5f6ff7,0x003a7977246ebf11,0x00166ed969c6600e,0x00aa42e469c98bec,0x00dc58f307cf0666)},
+ }}, {{
+ {FIELD_LITERAL(0x004b491f65a9a28b,0x006a10309e8a55b7,0x00b67210185187ef,0x00cf6497b12d9b8f,0x0085778c56e2b1ba,0x0015b4c07a814d85,0x00686479e62da561,0x008de5d88f114916)},
+ {FIELD_LITERAL(0x00e37c88d6bba7b1,0x003e4577e1b8d433,0x0050d8ea5f510ec0,0x0042fc9f2da9ef59,0x003bd074c1141420,0x00561b8b7b68774e,0x00232e5e5d1013a3,0x006b7f2cb3d7e73f)},
+ {FIELD_LITERAL(0x004bdd0f0b41e6a0,0x001773057c405d24,0x006029f99915bd97,0x006a5ba70a17fe2f,0x0046111977df7e08,0x004d8124c89fb6b7,0x00580983b2bb2724,0x00207bf330d6f3fe)},
+ }}, {{
+ {FIELD_LITERAL(0x007efdc93972a48b,0x002f5e50e78d5fee,0x0080dc11d61c7fe5,0x0065aa598707245b,0x009abba2300641be,0x000c68787656543a,0x00ffe0fef2dc0a17,0x00007ffbd6cb4f3a)},
+ {FIELD_LITERAL(0x0036012f2b836efc,0x00458c126d6b5fbc,0x00a34436d719ad1e,0x0097be6167117dea,0x0009c219c879cff3,0x0065564493e60755,0x00993ac94a8cdec0,0x002d4885a4d0dbaf)},
+ {FIELD_LITERAL(0x00598b60b4c068ba,0x00c547a0be7f1afd,0x009582164acf12af,0x00af4acac4fbbe40,0x005f6ca7c539121a,0x003b6e752ebf9d66,0x00f08a30d5cac5d4,0x00e399bb5f97c5a9)},
+ }}, {{
+ {FIELD_LITERAL(0x007445a0409c0a66,0x00a65c369f3829c0,0x0031d248a4f74826,0x006817f34defbe8e,0x00649741d95ebf2e,0x00d46466ab16b397,0x00fdc35703bee414,0x00343b43334525f8)},
+ {FIELD_LITERAL(0x001796bea93f6401,0x00090c5a42e85269,0x00672412ba1252ed,0x001201d47b6de7de,0x006877bccfe66497,0x00b554fd97a4c161,0x009753f42dbac3cf,0x00e983e3e378270a)},
+ {FIELD_LITERAL(0x00ac3eff18849872,0x00f0eea3bff05690,0x00a6d72c21dd505d,0x001b832642424169,0x00a6813017b540e5,0x00a744bd71b385cd,0x0022a7d089130a7b,0x004edeec9a133486)},
+ }}, {{
+ {FIELD_LITERAL(0x00b2d6729196e8a9,0x0088a9bb2031cef4,0x00579e7787dc1567,0x0030f49feb059190,0x00a0b1d69c7f7d8f,0x0040bdcc6d9d806f,0x00d76c4037edd095,0x00bbf24376415dd7)},
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+ }}, {{
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+ }}, {{
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+ }}, {{
+ {FIELD_LITERAL(0x006c59c9ae744185,0x009fc32f1b4282cd,0x004d6348ca59b1ac,0x00105376881be067,0x00af4096013147dc,0x004abfb5a5cb3124,0x000d2a7f8626c354,0x009c6ed568e07431)},
+ {FIELD_LITERAL(0x00e828333c297f8b,0x009ef3cf8c3f7e1f,0x00ab45f8fff31cb9,0x00c8b4178cb0b013,0x00d0c50dd3260a3f,0x0097126ac257f5bc,0x0042376cc90c705a,0x001d96fdb4a1071e)},
+ {FIELD_LITERAL(0x00542d44d89ee1a8,0x00306642e0442d98,0x0090853872b87338,0x002362cbf22dc044,0x002c222adff663b8,0x0067c924495fcb79,0x000e621d983c977c,0x00df77a9eccb66fb)},
+ }}, {{
+ {FIELD_LITERAL(0x002809e4bbf1814a,0x00b9e854f9fafb32,0x00d35e67c10f7a67,0x008f1bcb76e748cf,0x004224d9515687d2,0x005ba0b774e620c4,0x00b5e57db5d54119,0x00e15babe5683282)},
+ {FIELD_LITERAL(0x00832d02369b482c,0x00cba52ff0d93450,0x003fa9c908d554db,0x008d1e357b54122f,0x00abd91c2dc950c6,0x007eff1df4c0ec69,0x003f6aeb13fb2d31,0x00002d6179fc5b2c)},
+ {FIELD_LITERAL(0x0046c9eda81c9c89,0x00b60cb71c8f62fc,0x0022f5a683baa558,0x00f87319fccdf997,0x009ca09b51ce6a22,0x005b12baf4af7d77,0x008a46524a1e33e2,0x00035a77e988be0d)},
+ }}, {{
+ {FIELD_LITERAL(0x00a7efe46a7dbe2f,0x002f66fd55014fe7,0x006a428afa1ff026,0x0056caaa9604ab72,0x0033f3bcd7fac8ae,0x00ccb1aa01c86764,0x00158d1edf13bf40,0x009848ee76fcf3b4)},
+ {FIELD_LITERAL(0x00a9e7730a819691,0x00d9cc73c4992b70,0x00e299bde067de5a,0x008c314eb705192a,0x00e7226f17e8a3cc,0x0029dfd956e65a47,0x0053a8e839073b12,0x006f942b2ab1597e)},
+ {FIELD_LITERAL(0x001c3d780ecd5e39,0x0094f247fbdcc5fe,0x00d5c786fd527764,0x00b6f4da74f0db2a,0x0080f1f8badcd5fc,0x00f36a373ad2e23b,0x00f804f9f4343bf2,0x00d1af40ec623982)},
+ }}, {{
+ {FIELD_LITERAL(0x0082aeace5f1b144,0x00f68b3108cf4dd3,0x00634af01dde3020,0x000beab5df5c2355,0x00e8b790d1b49b0b,0x00e48d15854e36f4,0x0040ab2d95f3db9f,0x002711c4ed9e899a)},
+ {FIELD_LITERAL(0x0039343746531ebe,0x00c8509d835d429d,0x00e79eceff6b0018,0x004abfd31e8efce5,0x007bbfaaa1e20210,0x00e3be89c193e179,0x001c420f4c31d585,0x00f414a315bef5ae)},
+ {FIELD_LITERAL(0x007c296a24990df8,0x00d5d07525a75588,0x00dd8e113e94b7e7,0x007bbc58febe0cc8,0x0029f51af9bfcad3,0x007e9311ec7ab6f3,0x009a884de1676343,0x0050d5f2dce84be9)},
+ }}, {{
+ {FIELD_LITERAL(0x005fa020cca2450a,0x00491c29db6416d8,0x0037cefe3f9f9a85,0x003d405230647066,0x0049e835f0fdbe89,0x00feb78ac1a0815c,0x00828e4b32dc9724,0x00db84f2dc8d6fd4)},
+ {FIELD_LITERAL(0x0098cddc8b39549a,0x006da37e3b05d22c,0x00ce633cfd4eb3cb,0x00fda288ef526acd,0x0025338878c5d30a,0x00f34438c4e5a1b4,0x00584efea7c310f1,0x0041a551f1b660ad)},
+ {FIELD_LITERAL(0x00d7f7a8fbd6437a,0x0062872413bf3753,0x00ad4bbcb43c584b,0x007fe49be601d7e3,0x0077c659789babf4,0x00eb45fcb06a741b,0x005ce244913f9708,0x0088426401736326)},
+ }}, {{
+ {FIELD_LITERAL(0x007bf562ca768d7c,0x006c1f3a174e387c,0x00f024b447fee939,0x007e7af75f01143f,0x003adb70b4eed89d,0x00e43544021ad79a,0x0091f7f7042011f6,0x0093c1a1ee3a0ddc)},
+ {FIELD_LITERAL(0x00a0b68ec1eb72d2,0x002c03235c0d45a0,0x00553627323fe8c5,0x006186e94b17af94,0x00a9906196e29f14,0x0025b3aee6567733,0x007e0dd840080517,0x0018eb5801a4ba93)},
+ {FIELD_LITERAL(0x00d7fe7017bf6a40,0x006e3f0624be0c42,0x00ffbba205358245,0x00f9fc2cf8194239,0x008d93b37bf15b4e,0x006ddf2e38be8e95,0x002b6e79bf5fcff9,0x00ab355da425e2de)},
+ }}, {{
+ {FIELD_LITERAL(0x00938f97e20be973,0x0099141a36aaf306,0x0057b0ca29e545a1,0x0085db571f9fbc13,0x008b333c554b4693,0x0043ab6ef3e241cb,0x0054fb20aa1e5c70,0x00be0ff852760adf)},
+ {FIELD_LITERAL(0x003973d8938971d6,0x002aca26fa80c1f5,0x00108af1faa6b513,0x00daae275d7924e6,0x0053634ced721308,0x00d2355fe0bbd443,0x00357612b2d22095,0x00f9bb9dd4136cf3)},
+ {FIELD_LITERAL(0x002bff12cf5e03a5,0x001bdb1fa8a19cf8,0x00c91c6793f84d39,0x00f869f1b2eba9af,0x0059bc547dc3236b,0x00d91611d6d38689,0x00e062daaa2c0214,0x00ed3c047cc2bc82)},
+ }}, {{
+ {FIELD_LITERAL(0x000050d70c32b31a,0x001939d576d437b3,0x00d709e598bf9fe6,0x00a885b34bd2ee9e,0x00dd4b5c08ab1a50,0x0091bebd50b55639,0x00cf79ff64acdbc6,0x006067a39d826336)},
+ {FIELD_LITERAL(0x0062dd0fb31be374,0x00fcc96b84c8e727,0x003f64f1375e6ae3,0x0057d9b6dd1af004,0x00d6a167b1103c7b,0x00dd28f3180fb537,0x004ff27ad7167128,0x008934c33461f2ac)},
+ {FIELD_LITERAL(0x0065b472b7900043,0x00ba7efd2ff1064b,0x000b67d6c4c3020f,0x0012d28469f4e46d,0x0031c32939703ec7,0x00b49f0bce133066,0x00f7e10416181d47,0x005c90f51867eecc)},
+ }}, {{
+ {FIELD_LITERAL(0x0051207abd179101,0x00fc2a5c20d9c5da,0x00fb9d5f2701b6df,0x002dd040fdea82b8,0x00f163b0738442ff,0x00d9736bd68855b8,0x00e0d8e93005e61c,0x00df5a40b3988570)},
+ {FIELD_LITERAL(0x0006918f5dfce6dc,0x00d4bf1c793c57fb,0x0069a3f649435364,0x00e89a50e5b0cd6e,0x00b9f6a237e973af,0x006d4ed8b104e41d,0x00498946a3924cd2,0x00c136ec5ac9d4f7)},
+ {FIELD_LITERAL(0x0011a9c290ac5336,0x002b9a2d4a6a6533,0x009a8a68c445d937,0x00361b27b07e5e5c,0x003c043b1755b974,0x00b7eb66cf1155ee,0x0077af5909eefff2,0x0098f609877cc806)},
+ }}, {{
+ {FIELD_LITERAL(0x00ab13af436bf8f4,0x000bcf0a0dac8574,0x00d50c864f705045,0x00c40e611debc842,0x0085010489bd5caa,0x007c5050acec026f,0x00f67d943c8da6d1,0x00de1da0278074c6)},
+ {FIELD_LITERAL(0x00b373076597455f,0x00e83f1af53ac0f5,0x0041f63c01dc6840,0x0097dea19b0c6f4b,0x007f9d63b4c1572c,0x00e692d492d0f5f0,0x00cbcb392e83b4ad,0x0069c0f39ed9b1a8)},
+ {FIELD_LITERAL(0x00861030012707c9,0x009fbbdc7fd4aafb,0x008f591d6b554822,0x00df08a41ea18ade,0x009d7d83e642abea,0x0098c71bda3b78ff,0x0022c89e7021f005,0x0044d29a3fe1e3c4)},
+ }}, {{
+ {FIELD_LITERAL(0x00e748cd7b5c52f2,0x00ea9df883f89cc3,0x0018970df156b6c7,0x00c5a46c2a33a847,0x00cbde395e32aa09,0x0072474ebb423140,0x00fb00053086a23d,0x001dafcfe22d4e1f)},
+ {FIELD_LITERAL(0x00c903ee6d825540,0x00add6c4cf98473e,0x007636efed4227f1,0x00905124ae55e772,0x00e6b38fab12ed53,0x0045e132b863fe55,0x003974662edb366a,0x00b1787052be8208)},
+ {FIELD_LITERAL(0x00a614b00d775c7c,0x00d7c78941cc7754,0x00422dd68b5dabc4,0x00a6110f0167d28b,0x00685a309c252886,0x00b439ffd5143660,0x003656e29ee7396f,0x00c7c9b9ed5ad854)},
+ }}, {{
+ {FIELD_LITERAL(0x0040f7e7c5b37bf2,0x0064e4dc81181bba,0x00a8767ae2a366b6,0x001496b4f90546f2,0x002a28493f860441,0x0021f59513049a3a,0x00852d369a8b7ee3,0x00dd2e7d8b7d30a9)},
+ {FIELD_LITERAL(0x00006e34a35d9fbc,0x00eee4e48b2f019a,0x006b344743003a5f,0x00541d514f04a7e3,0x00e81f9ee7647455,0x005e2b916c438f81,0x00116f8137b7eff0,0x009bd3decc7039d1)},
+ {FIELD_LITERAL(0x0005d226f434110d,0x00af8288b8ef21d5,0x004a7a52ef181c8c,0x00be0b781b4b06de,0x00e6e3627ded07e1,0x00e43aa342272b8b,0x00e86ab424577d84,0x00fb292c566e35bb)},
+ }}, {{
+ {FIELD_LITERAL(0x00334f5303ea1222,0x00dfb3dbeb0a5d3e,0x002940d9592335c1,0x00706a7a63e8938a,0x005a533558bc4caf,0x00558e33192022a9,0x00970d9faf74c133,0x002979fcb63493ca)},
+ {FIELD_LITERAL(0x00e38abece3c82ab,0x005a51f18a2c7a86,0x009dafa2e86d592e,0x00495a62eb688678,0x00b79df74c0eb212,0x0023e8cc78b75982,0x005998cb91075e13,0x00735aa9ba61bc76)},
+ {FIELD_LITERAL(0x00d9f7a82ddbe628,0x00a1fc782889ae0f,0x0071ffda12d14b66,0x0037cf4eca7fb3d5,0x00c80bc242c58808,0x0075bf8c2d08c863,0x008d41f31afc52a7,0x00197962ecf38741)},
+ }}, {{
+ {FIELD_LITERAL(0x006e9f475cccf2ee,0x00454b9cd506430c,0x00224a4fb79ee479,0x0062e3347ef0b5e2,0x0034fd2a3512232a,0x00b8b3cb0f457046,0x00eb20165daa38ec,0x00128eebc2d9c0f7)},
+ {FIELD_LITERAL(0x00bfc5fa1e4ea21f,0x00c21d7b6bb892e6,0x00cf043f3acf0291,0x00c13f2f849b3c90,0x00d1a97ebef10891,0x0061e130a445e7fe,0x0019513fdedbf22b,0x001d60c813bff841)},
+ {FIELD_LITERAL(0x0019561c7fcf0213,0x00e3dca6843ebd77,0x0068ea95b9ca920e,0x009bdfb70f253595,0x00c68f59186aa02a,0x005aee1cca1c3039,0x00ab79a8a937a1ce,0x00b9a0e549959e6f)},
+ }}, {{
+ {FIELD_LITERAL(0x00c79e0b6d97dfbd,0x00917c71fd2bc6e8,0x00db7529ccfb63d8,0x00be5be957f17866,0x00a9e11fdc2cdac1,0x007b91a8e1f44443,0x00a3065e4057d80f,0x004825f5b8d5f6d4)},
+ {FIELD_LITERAL(0x003e4964fa8a8fc8,0x00f6a1cdbcf41689,0x00943cb18fe7fda7,0x00606dafbf34440a,0x005d37a86399c789,0x00e79a2a69417403,0x00fe34f7e68b8866,0x0011f448ed2df10e)},
+ {FIELD_LITERAL(0x00f1f57efcc1fcc4,0x00513679117de154,0x002e5b5b7c86d8c3,0x009f6486561f9cfb,0x00169e74b0170cf7,0x00900205af4af696,0x006acfddb77853f3,0x00df184c90f31068)},
+ }}, {{
+ {FIELD_LITERAL(0x00b37396c3320791,0x00fc7b67175c5783,0x00c36d2cd73ecc38,0x0080ebcc0b328fc5,0x0043a5b22b35d35d,0x00466c9f1713c9da,0x0026ad346dcaa8da,0x007c684e701183a6)},
+ {FIELD_LITERAL(0x00fd579ffb691713,0x00b76af4f81c412d,0x00f239de96110f82,0x00e965fb437f0306,0x00ca7e9436900921,0x00e487f1325fa24a,0x00633907de476380,0x00721c62ac5b8ea0)},
+ {FIELD_LITERAL(0x00c0d54e542eb4f9,0x004ed657171c8dcf,0x00b743a4f7c2a39b,0x00fd9f93ed6cc567,0x00307fae3113e58b,0x0058aa577c93c319,0x00d254556f35b346,0x00491aada2203f0d)},
+ }}, {{
+ {FIELD_LITERAL(0x00dff3103786ff34,0x000144553b1f20c3,0x0095613baeb930e4,0x00098058275ea5d4,0x007cd1402b046756,0x0074d74e4d58aee3,0x005f93fc343ff69b,0x00873df17296b3b0)},
+ {FIELD_LITERAL(0x00c4a1fb48635413,0x00b5dd54423ad59f,0x009ff5d53fd24a88,0x003c98d267fc06a7,0x002db7cb20013641,0x00bd1d6716e191f2,0x006dbc8b29094241,0x0044bbf233dafa2c)},
+ {FIELD_LITERAL(0x0055838d41f531e6,0x00bf6a2dd03c81b2,0x005827a061c4839e,0x0000de2cbb36aac3,0x002efa29d9717478,0x00f9e928cc8a77ba,0x00c134b458def9ef,0x00958a182223fc48)},
+ }}, {{
+ {FIELD_LITERAL(0x000a9ee23c06881f,0x002c727d3d871945,0x00f47d971512d24a,0x00671e816f9ef31a,0x00883af2cfaad673,0x00601f98583d6c9a,0x00b435f5adc79655,0x00ad87b71c04bff2)},
+ {FIELD_LITERAL(0x007860d99db787cf,0x00fda8983018f4a8,0x008c8866bac4743c,0x00ef471f84c82a3f,0x00abea5976d3b8e7,0x00714882896cd015,0x00b49fae584ddac5,0x008e33a1a0b69c81)},
+ {FIELD_LITERAL(0x007b6ee2c9e8a9ec,0x002455dbbd89d622,0x006490cf4eaab038,0x00d925f6c3081561,0x00153b3047de7382,0x003b421f8bdceb6f,0x00761a4a5049da78,0x00980348c5202433)},
+ }}, {{
+ {FIELD_LITERAL(0x007f8a43da97dd5c,0x00058539c800fc7b,0x0040f3cf5a28414a,0x00d68dd0d95283d6,0x004adce9da90146e,0x00befa41c7d4f908,0x007603bc2e3c3060,0x00bdf360ab3545db)},
+ {FIELD_LITERAL(0x00eebfd4e2312cc3,0x00474b2564e4fc8c,0x003303ef14b1da9b,0x003c93e0e66beb1d,0x0013619b0566925a,0x008817c24d901bf3,0x00b62bd8898d218b,0x0075a7716f1e88a2)},
+ {FIELD_LITERAL(0x0009218da1e6890f,0x0026907f5fd02575,0x004dabed5f19d605,0x003abf181870249d,0x00b52fd048cc92c4,0x00b6dd51e415a5c5,0x00d9eb82bd2b4014,0x002c865a43b46b43)},
+ }}, {{
+ {FIELD_LITERAL(0x0070047189452f4c,0x00f7ad12e1ce78d5,0x00af1ba51ec44a8b,0x005f39f63e667cd6,0x00058eac4648425e,0x00d7fdab42bea03b,0x0028576a5688de15,0x00af973209e77c10)},
+ {FIELD_LITERAL(0x00c338b915d8fef0,0x00a893292045c39a,0x0028ab4f2eba6887,0x0060743cb519fd61,0x0006213964093ac0,0x007c0b7a43f6266d,0x008e3557c4fa5bda,0x002da976de7b8d9d)},
+ {FIELD_LITERAL(0x0048729f8a8b6dcd,0x00fe23b85cc4d323,0x00e7384d16e4db0e,0x004a423970678942,0x00ec0b763345d4ba,0x00c477b9f99ed721,0x00c29dad3777b230,0x001c517b466f7df6)},
+ }}, {{
+ {FIELD_LITERAL(0x006366c380f7b574,0x001c7d1f09ff0438,0x003e20a7301f5b22,0x00d3efb1916d28f6,0x0049f4f81060ce83,0x00c69d91ea43ced1,0x002b6f3e5cd269ed,0x005b0fb22ce9ec65)},
+ {FIELD_LITERAL(0x00aa2261022d883f,0x00ebcca4548010ac,0x002528512e28a437,0x0070ca7676b66082,0x0084bda170f7c6d3,0x00581b4747c9b8bb,0x005c96a01061c7e2,0x00fb7c4a362b5273)},
+ {FIELD_LITERAL(0x00c30020eb512d02,0x0060f288283a4d26,0x00b7ed13becde260,0x0075ebb74220f6e9,0x00701079fcfe8a1f,0x001c28fcdff58938,0x002e4544b8f4df6b,0x0060c5bc4f1a7d73)},
+ }}, {{
+ {FIELD_LITERAL(0x00ae307cf069f701,0x005859f222dd618b,0x00212d6c46ec0b0d,0x00a0fe4642afb62d,0x00420d8e4a0a8903,0x00a80ff639bdf7b0,0x0019bee1490b5d8e,0x007439e4b9c27a86)},
+ {FIELD_LITERAL(0x00a94700032a093f,0x0076e96c225216e7,0x00a63a4316e45f91,0x007d8bbb4645d3b2,0x00340a6ff22793eb,0x006f935d4572aeb7,0x00b1fb69f00afa28,0x009e8f3423161ed3)},
+ {FIELD_LITERAL(0x009ef49c6b5ced17,0x00a555e6269e9f0a,0x007e6f1d79ec73b5,0x009ac78695a32ac4,0x0001d77fbbcd5682,0x008cea1fee0aaeed,0x00f42bea82a53462,0x002e46ab96cafcc9)},
+ }}, {{
+ {FIELD_LITERAL(0x0051cfcc5885377a,0x00dce566cb1803ca,0x00430c7643f2c7d4,0x00dce1a1337bdcc0,0x0010d5bd7283c128,0x003b1b547f9b46fe,0x000f245e37e770ab,0x007b72511f022b37)},
+ {FIELD_LITERAL(0x0060db815bc4786c,0x006fab25beedc434,0x00c610d06084797c,0x000c48f08537bec0,0x0031aba51c5b93da,0x007968fa6e01f347,0x0030070da52840c6,0x00c043c225a4837f)},
+ {FIELD_LITERAL(0x001bcfd00649ee93,0x006dceb47e2a0fd5,0x00f2cebda0cf8fd0,0x00b6b9d9d1fbdec3,0x00815262e6490611,0x00ef7f5ce3176760,0x00e49cd0c998d58b,0x005fc6cc269ba57c)},
+ }}, {{
+ {FIELD_LITERAL(0x008940211aa0d633,0x00addae28136571d,0x00d68fdbba20d673,0x003bc6129bc9e21a,0x000346cf184ebe9a,0x0068774d741ebc7f,0x0019d5e9e6966557,0x0003cbd7f981b651)},
+ {FIELD_LITERAL(0x004a2902926f8d3f,0x00ad79b42637ab75,0x0088f60b90f2d4e8,0x0030f54ef0e398c4,0x00021dc9bf99681e,0x007ebf66fde74ee3,0x004ade654386e9a4,0x00e7485066be4c27)},
+ {FIELD_LITERAL(0x00445f1263983be0,0x004cf371dda45e6a,0x00744a89d5a310e7,0x001f20ce4f904833,0x00e746edebe66e29,0x000912ab1f6c153d,0x00f61d77d9b2444c,0x0001499cd6647610)},
+ }}
+ }
+};
+const struct curve448_precomputed_s *curve448_precomputed_base
+ = &curve448_precomputed_base_table;
+
+static const niels_t curve448_wnaf_base_table[32] = {
+ {{
+ {FIELD_LITERAL(0x00303cda6feea532,0x00860f1d5a3850e4,0x00226b9fa4728ccd,0x00e822938a0a0c0c,0x00263a61c9ea9216,0x001204029321b828,0x006a468360983c65,0x0002846f0a782143)},
+ {FIELD_LITERAL(0x00303cda6feea532,0x00860f1d5a3850e4,0x00226b9fa4728ccd,0x006822938a0a0c0c,0x00263a61c9ea9215,0x001204029321b828,0x006a468360983c65,0x0082846f0a782143)},
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+ }}, {{
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+ }}, {{
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+ {FIELD_LITERAL(0x0044588d1a81d680,0x00b0a9097208e4f8,0x00212605350dc57e,0x0028717cd2871123,0x00fb083c100fd979,0x0045a056ce063fdf,0x00a5d604b4dd6a41,0x001dabc08ba4e236)},
+ }}, {{
+ {FIELD_LITERAL(0x00c4887198d7a7fa,0x00244f98fb45784a,0x0045911e15a15d01,0x001d323d374c0966,0x00967c3915196562,0x0039373abd2f3c67,0x000d2c5614312423,0x0041cf2215442ce3)},
+ {FIELD_LITERAL(0x008ede889ada7f06,0x001611e91de2e135,0x00fdb9a458a471b9,0x00563484e03710d1,0x0031cc81925e3070,0x0062c97b3af80005,0x00fa733eea28edeb,0x00e82457e1ebbc88)},
+ {FIELD_LITERAL(0x006a0df5fe9b6f59,0x00a0d4ff46040d92,0x004a7cedb6f93250,0x00d1df8855b8c357,0x00e73a46086fd058,0x0048fb0add6dfe59,0x001e03a28f1b4e3d,0x00a871c993308d76)},
+ }}, {{
+ {FIELD_LITERAL(0x0030dbb2d1766ec8,0x00586c0ad138555e,0x00d1a34f9e91c77c,0x0063408ad0e89014,0x00d61231b05f6f5b,0x0009abf569f5fd8a,0x00aec67a110f1c43,0x0031d1a790938dd7)},
+ {FIELD_LITERAL(0x006cded841e2a862,0x00198d60af0ab6fb,0x0018f09db809e750,0x004e6ac676016263,0x00eafcd1620969cb,0x002c9784ca34917d,0x0054f00079796de7,0x00d9fab5c5972204)},
+ {FIELD_LITERAL(0x004bd0fee2438a83,0x00b571e62b0f83bd,0x0059287d7ce74800,0x00fb3631b645c3f0,0x00a018e977f78494,0x0091e27065c27b12,0x007696c1817165e0,0x008c40be7c45ba3a)},
+ }}, {{
+ {FIELD_LITERAL(0x00a0f326327cb684,0x001c7d0f672680ff,0x008c1c81ffb112d1,0x00f8f801674eddc8,0x00e926d5d48c2a9d,0x005bd6d954c6fe9a,0x004c6b24b4e33703,0x00d05eb5c09105cc)},
+ {FIELD_LITERAL(0x00d61731caacf2cf,0x002df0c7609e01c5,0x00306172208b1e2b,0x00b413fe4fb2b686,0x00826d360902a221,0x003f8d056e67e7f7,0x0065025b0175e989,0x00369add117865eb)},
+ {FIELD_LITERAL(0x00aaf895aec2fa11,0x000f892bc313eb52,0x005b1c794dad050b,0x003f8ec4864cec14,0x00af81058d0b90e5,0x00ebe43e183997bb,0x00a9d610f9f3e615,0x007acd8eec2e88d3)},
+ }}, {{
+ {FIELD_LITERAL(0x0049b2fab13812a3,0x00846db32cd60431,0x000177fa578c8d6c,0x00047d0e2ad4bc51,0x00b158ba38d1e588,0x006a45daad79e3f3,0x000997b93cab887b,0x00c47ea42fa23dc3)},
+ {FIELD_LITERAL(0x0012b6fef7aeb1ca,0x009412768194b6a7,0x00ff0d351f23ab93,0x007e8a14c1aff71b,0x006c1c0170c512bc,0x0016243ea02ab2e5,0x007bb6865b303f3e,0x0015ce6b29b159f4)},
+ {FIELD_LITERAL(0x009961cd02e68108,0x00e2035d3a1d0836,0x005d51f69b5e1a1d,0x004bccb4ea36edcd,0x0069be6a7aeef268,0x0063f4dd9de8d5a7,0x006283783092ca35,0x0075a31af2c35409)},
+ }}, {{
+ {FIELD_LITERAL(0x00c412365162e8cf,0x00012283fb34388a,0x003e6543babf39e2,0x00eead6b3a804978,0x0099c0314e8b326f,0x00e98e0a8d477a4f,0x00d2eb96b127a687,0x00ed8d7df87571bb)},
+ {FIELD_LITERAL(0x00777463e308cacf,0x00c8acb93950132d,0x00ebddbf4ca48b2c,0x0026ad7ca0795a0a,0x00f99a3d9a715064,0x000d60bcf9d4dfcc,0x005e65a73a437a06,0x0019d536a8db56c8)},
+ {FIELD_LITERAL(0x00192d7dd558d135,0x0027cd6a8323ffa7,0x00239f1a412dc1e7,0x0046b4b3be74fc5c,0x0020c47a2bef5bce,0x00aa17e48f43862b,0x00f7e26c96342e5f,0x0008011c530f39a9)},
+ }}, {{
+ {FIELD_LITERAL(0x00aad4ac569bf0f1,0x00a67adc90b27740,0x0048551369a5751a,0x0031252584a3306a,0x0084e15df770e6fc,0x00d7bba1c74b5805,0x00a80ef223af1012,0x0089c85ceb843a34)},
+ {FIELD_LITERAL(0x00c4545be4a54004,0x0099e11f60357e6c,0x001f3936d19515a6,0x007793df84341a6e,0x0051061886717ffa,0x00e9b0a660b28f85,0x0044ea685892de0d,0x000257d2a1fda9d9)},
+ {FIELD_LITERAL(0x007e8b01b24ac8a8,0x006cf3b0b5ca1337,0x00f1607d3e36a570,0x0039b7fab82991a1,0x00231777065840c5,0x00998e5afdd346f9,0x00b7dc3e64acc85f,0x00baacc748013ad6)},
+ }}, {{
+ {FIELD_LITERAL(0x008ea6a4177580bf,0x005fa1953e3f0378,0x005fe409ac74d614,0x00452327f477e047,0x00a4018507fb6073,0x007b6e71951caac8,0x0012b42ab8a6ce91,0x0080eca677294ab7)},
+ {FIELD_LITERAL(0x00a53edc023ba69b,0x00c6afa83ddde2e8,0x00c3f638b307b14e,0x004a357a64414062,0x00e4d94d8b582dc9,0x001739caf71695b7,0x0012431b2ae28de1,0x003b6bc98682907c)},
+ {FIELD_LITERAL(0x008a9a93be1f99d6,0x0079fa627cc699c8,0x00b0cfb134ba84c8,0x001c4b778249419a,0x00df4ab3d9c44f40,0x009f596e6c1a9e3c,0x001979c0df237316,0x00501e953a919b87)},
+ }}
+};
+const niels_t *curve448_wnaf_base = curve448_wnaf_base_table;
diff --git a/crypto/ec/curve448/curve448utils.h b/crypto/ec/curve448/curve448utils.h
new file mode 100644
index 000000000000..9bf837993c94
--- /dev/null
+++ b/crypto/ec/curve448/curve448utils.h
@@ -0,0 +1,78 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_CURVE448UTILS_H
+# define HEADER_CURVE448UTILS_H
+
+# include <openssl/e_os2.h>
+
+/*
+ * Internal word types. Somewhat tricky. This could be decided separately per
+ * platform. However, the structs do need to be all the same size and
+ * alignment on a given platform to support dynamic linking, since even if you
+ * header was built with eg arch_neon, you might end up linking a library built
+ * with arch_arm32.
+ */
+# ifndef C448_WORD_BITS
+# if (defined(__SIZEOF_INT128__) && (__SIZEOF_INT128__ == 16)) \
+ && !defined(__sparc__)
+# define C448_WORD_BITS 64 /* The number of bits in a word */
+# else
+# define C448_WORD_BITS 32 /* The number of bits in a word */
+# endif
+# endif
+
+# if C448_WORD_BITS == 64
+/* Word size for internal computations */
+typedef uint64_t c448_word_t;
+/* Signed word size for internal computations */
+typedef int64_t c448_sword_t;
+/* "Boolean" type, will be set to all-zero or all-one (i.e. -1u) */
+typedef uint64_t c448_bool_t;
+/* Double-word size for internal computations */
+typedef __uint128_t c448_dword_t;
+/* Signed double-word size for internal computations */
+typedef __int128_t c448_dsword_t;
+# elif C448_WORD_BITS == 32
+/* Word size for internal computations */
+typedef uint32_t c448_word_t;
+/* Signed word size for internal computations */
+typedef int32_t c448_sword_t;
+/* "Boolean" type, will be set to all-zero or all-one (i.e. -1u) */
+typedef uint32_t c448_bool_t;
+/* Double-word size for internal computations */
+typedef uint64_t c448_dword_t;
+/* Signed double-word size for internal computations */
+typedef int64_t c448_dsword_t;
+# else
+# error "Only supporting C448_WORD_BITS = 32 or 64 for now"
+# endif
+
+/* C448_TRUE = -1 so that C448_TRUE & x = x */
+# define C448_TRUE (0 - (c448_bool_t)1)
+
+/* C448_FALSE = 0 so that C448_FALSE & x = 0 */
+# define C448_FALSE 0
+
+/* Another boolean type used to indicate success or failure. */
+typedef enum {
+ C448_SUCCESS = -1, /**< The operation succeeded. */
+ C448_FAILURE = 0 /**< The operation failed. */
+} c448_error_t;
+
+/* Return success if x is true */
+static ossl_inline c448_error_t c448_succeed_if(c448_bool_t x)
+{
+ return (c448_error_t) x;
+}
+
+#endif /* __C448_COMMON_H__ */
diff --git a/crypto/ec/curve448/ed448.h b/crypto/ec/curve448/ed448.h
new file mode 100644
index 000000000000..5fe939e8e19d
--- /dev/null
+++ b/crypto/ec/curve448/ed448.h
@@ -0,0 +1,195 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_ED448_H
+# define HEADER_ED448_H
+
+# include "point_448.h"
+
+/* Number of bytes in an EdDSA public key. */
+# define EDDSA_448_PUBLIC_BYTES 57
+
+/* Number of bytes in an EdDSA private key. */
+# define EDDSA_448_PRIVATE_BYTES EDDSA_448_PUBLIC_BYTES
+
+/* Number of bytes in an EdDSA private key. */
+# define EDDSA_448_SIGNATURE_BYTES (EDDSA_448_PUBLIC_BYTES + \
+ EDDSA_448_PRIVATE_BYTES)
+
+/* EdDSA encoding ratio. */
+# define C448_EDDSA_ENCODE_RATIO 4
+
+/* EdDSA decoding ratio. */
+# define C448_EDDSA_DECODE_RATIO (4 / 4)
+
+/*
+ * EdDSA key generation. This function uses a different (non-Decaf) encoding.
+ *
+ * pubkey (out): The public key.
+ * privkey (in): The private key.
+ */
+c448_error_t c448_ed448_derive_public_key(
+ uint8_t pubkey [EDDSA_448_PUBLIC_BYTES],
+ const uint8_t privkey [EDDSA_448_PRIVATE_BYTES]);
+
+/*
+ * EdDSA signing.
+ *
+ * signature (out): The signature.
+ * privkey (in): The private key.
+ * pubkey (in): The public key.
+ * message (in): The message to sign.
+ * message_len (in): The length of the message.
+ * prehashed (in): Nonzero if the message is actually the hash of something
+ * you want to sign.
+ * context (in): A "context" for this signature of up to 255 bytes.
+ * context_len (in): Length of the context.
+ *
+ * For Ed25519, it is unsafe to use the same key for both prehashed and
+ * non-prehashed messages, at least without some very careful protocol-level
+ * disambiguation. For Ed448 it is safe.
+ */
+c448_error_t c448_ed448_sign(
+ uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t privkey[EDDSA_448_PRIVATE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t *message, size_t message_len,
+ uint8_t prehashed, const uint8_t *context,
+ size_t context_len);
+
+/*
+ * EdDSA signing with prehash.
+ *
+ * signature (out): The signature.
+ * privkey (in): The private key.
+ * pubkey (in): The public key.
+ * hash (in): The hash of the message. This object will not be modified by the
+ * call.
+ * context (in): A "context" for this signature of up to 255 bytes. Must be the
+ * same as what was used for the prehash.
+ * context_len (in): Length of the context.
+ *
+ * For Ed25519, it is unsafe to use the same key for both prehashed and
+ * non-prehashed messages, at least without some very careful protocol-level
+ * disambiguation. For Ed448 it is safe.
+ */
+c448_error_t c448_ed448_sign_prehash(
+ uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t privkey[EDDSA_448_PRIVATE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t hash[64],
+ const uint8_t *context,
+ size_t context_len);
+
+/*
+ * EdDSA signature verification.
+ *
+ * Uses the standard (i.e. less-strict) verification formula.
+ *
+ * signature (in): The signature.
+ * pubkey (in): The public key.
+ * message (in): The message to verify.
+ * message_len (in): The length of the message.
+ * prehashed (in): Nonzero if the message is actually the hash of something you
+ * want to verify.
+ * context (in): A "context" for this signature of up to 255 bytes.
+ * context_len (in): Length of the context.
+ *
+ * For Ed25519, it is unsafe to use the same key for both prehashed and
+ * non-prehashed messages, at least without some very careful protocol-level
+ * disambiguation. For Ed448 it is safe.
+ */
+c448_error_t c448_ed448_verify(const uint8_t
+ signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t
+ pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t *message, size_t message_len,
+ uint8_t prehashed, const uint8_t *context,
+ uint8_t context_len);
+
+/*
+ * EdDSA signature verification.
+ *
+ * Uses the standard (i.e. less-strict) verification formula.
+ *
+ * signature (in): The signature.
+ * pubkey (in): The public key.
+ * hash (in): The hash of the message. This object will not be modified by the
+ * call.
+ * context (in): A "context" for this signature of up to 255 bytes. Must be the
+ * same as what was used for the prehash.
+ * context_len (in): Length of the context.
+ *
+ * For Ed25519, it is unsafe to use the same key for both prehashed and
+ * non-prehashed messages, at least without some very careful protocol-level
+ * disambiguation. For Ed448 it is safe.
+ */
+c448_error_t c448_ed448_verify_prehash(
+ const uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t hash[64],
+ const uint8_t *context,
+ uint8_t context_len);
+
+/*
+ * EdDSA point encoding. Used internally, exposed externally.
+ * Multiplies by C448_EDDSA_ENCODE_RATIO first.
+ *
+ * The multiplication is required because the EdDSA encoding represents
+ * the cofactor information, but the Decaf encoding ignores it (which
+ * is the whole point). So if you decode from EdDSA and re-encode to
+ * EdDSA, the cofactor info must get cleared, because the intermediate
+ * representation doesn't track it.
+ *
+ * The way we handle this is to multiply by C448_EDDSA_DECODE_RATIO when
+ * decoding, and by C448_EDDSA_ENCODE_RATIO when encoding. The product of
+ * these ratios is always exactly the cofactor 4, so the cofactor ends up
+ * cleared one way or another. But exactly how that shakes out depends on the
+ * base points specified in RFC 8032.
+ *
+ * The upshot is that if you pass the Decaf/Ristretto base point to
+ * this function, you will get C448_EDDSA_ENCODE_RATIO times the
+ * EdDSA base point.
+ *
+ * enc (out): The encoded point.
+ * p (in): The point.
+ */
+void curve448_point_mul_by_ratio_and_encode_like_eddsa(
+ uint8_t enc [EDDSA_448_PUBLIC_BYTES],
+ const curve448_point_t p);
+
+/*
+ * EdDSA point decoding. Multiplies by C448_EDDSA_DECODE_RATIO, and
+ * ignores cofactor information.
+ *
+ * See notes on curve448_point_mul_by_ratio_and_encode_like_eddsa
+ *
+ * enc (out): The encoded point.
+ * p (in): The point.
+ */
+c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
+ curve448_point_t p,
+ const uint8_t enc[EDDSA_448_PUBLIC_BYTES]);
+
+/*
+ * EdDSA to ECDH private key conversion
+ * Using the appropriate hash function, hash the EdDSA private key
+ * and keep only the lower bytes to get the ECDH private key
+ *
+ * x (out): The ECDH private key as in RFC7748
+ * ed (in): The EdDSA private key
+ */
+c448_error_t c448_ed448_convert_private_key_to_x448(
+ uint8_t x[X448_PRIVATE_BYTES],
+ const uint8_t ed[EDDSA_448_PRIVATE_BYTES]);
+
+#endif /* HEADER_ED448_H */
diff --git a/crypto/ec/curve448/eddsa.c b/crypto/ec/curve448/eddsa.c
new file mode 100644
index 000000000000..909413a535a8
--- /dev/null
+++ b/crypto/ec/curve448/eddsa.c
@@ -0,0 +1,346 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+#include <string.h>
+#include <openssl/crypto.h>
+#include <openssl/evp.h>
+#include "curve448_lcl.h"
+#include "word.h"
+#include "ed448.h"
+#include "internal/numbers.h"
+
+#define COFACTOR 4
+
+static c448_error_t oneshot_hash(uint8_t *out, size_t outlen,
+ const uint8_t *in, size_t inlen)
+{
+ EVP_MD_CTX *hashctx = EVP_MD_CTX_new();
+
+ if (hashctx == NULL)
+ return C448_FAILURE;
+
+ if (!EVP_DigestInit_ex(hashctx, EVP_shake256(), NULL)
+ || !EVP_DigestUpdate(hashctx, in, inlen)
+ || !EVP_DigestFinalXOF(hashctx, out, outlen)) {
+ EVP_MD_CTX_free(hashctx);
+ return C448_FAILURE;
+ }
+
+ EVP_MD_CTX_free(hashctx);
+ return C448_SUCCESS;
+}
+
+static void clamp(uint8_t secret_scalar_ser[EDDSA_448_PRIVATE_BYTES])
+{
+ secret_scalar_ser[0] &= -COFACTOR;
+ secret_scalar_ser[EDDSA_448_PRIVATE_BYTES - 1] = 0;
+ secret_scalar_ser[EDDSA_448_PRIVATE_BYTES - 2] |= 0x80;
+}
+
+static c448_error_t hash_init_with_dom(EVP_MD_CTX *hashctx, uint8_t prehashed,
+ uint8_t for_prehash,
+ const uint8_t *context,
+ size_t context_len)
+{
+ const char *dom_s = "SigEd448";
+ uint8_t dom[2];
+
+ if (context_len > UINT8_MAX)
+ return C448_FAILURE;
+
+ dom[0] = (uint8_t)(2 - (prehashed == 0 ? 1 : 0)
+ - (for_prehash == 0 ? 1 : 0));
+ dom[1] = (uint8_t)context_len;
+
+ if (!EVP_DigestInit_ex(hashctx, EVP_shake256(), NULL)
+ || !EVP_DigestUpdate(hashctx, dom_s, strlen(dom_s))
+ || !EVP_DigestUpdate(hashctx, dom, sizeof(dom))
+ || !EVP_DigestUpdate(hashctx, context, context_len))
+ return C448_FAILURE;
+
+ return C448_SUCCESS;
+}
+
+/* In this file because it uses the hash */
+c448_error_t c448_ed448_convert_private_key_to_x448(
+ uint8_t x[X448_PRIVATE_BYTES],
+ const uint8_t ed [EDDSA_448_PRIVATE_BYTES])
+{
+ /* pass the private key through oneshot_hash function */
+ /* and keep the first X448_PRIVATE_BYTES bytes */
+ return oneshot_hash(x, X448_PRIVATE_BYTES, ed,
+ EDDSA_448_PRIVATE_BYTES);
+}
+
+c448_error_t c448_ed448_derive_public_key(
+ uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t privkey[EDDSA_448_PRIVATE_BYTES])
+{
+ /* only this much used for keygen */
+ uint8_t secret_scalar_ser[EDDSA_448_PRIVATE_BYTES];
+ curve448_scalar_t secret_scalar;
+ unsigned int c;
+ curve448_point_t p;
+
+ if (!oneshot_hash(secret_scalar_ser, sizeof(secret_scalar_ser), privkey,
+ EDDSA_448_PRIVATE_BYTES))
+ return C448_FAILURE;
+
+ clamp(secret_scalar_ser);
+
+ curve448_scalar_decode_long(secret_scalar, secret_scalar_ser,
+ sizeof(secret_scalar_ser));
+
+ /*
+ * Since we are going to mul_by_cofactor during encoding, divide by it
+ * here. However, the EdDSA base point is not the same as the decaf base
+ * point if the sigma isogeny is in use: the EdDSA base point is on
+ * Etwist_d/(1-d) and the decaf base point is on Etwist_d, and when
+ * converted it effectively picks up a factor of 2 from the isogenies. So
+ * we might start at 2 instead of 1.
+ */
+ for (c = 1; c < C448_EDDSA_ENCODE_RATIO; c <<= 1)
+ curve448_scalar_halve(secret_scalar, secret_scalar);
+
+ curve448_precomputed_scalarmul(p, curve448_precomputed_base, secret_scalar);
+
+ curve448_point_mul_by_ratio_and_encode_like_eddsa(pubkey, p);
+
+ /* Cleanup */
+ curve448_scalar_destroy(secret_scalar);
+ curve448_point_destroy(p);
+ OPENSSL_cleanse(secret_scalar_ser, sizeof(secret_scalar_ser));
+
+ return C448_SUCCESS;
+}
+
+c448_error_t c448_ed448_sign(
+ uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t privkey[EDDSA_448_PRIVATE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t *message, size_t message_len,
+ uint8_t prehashed, const uint8_t *context,
+ size_t context_len)
+{
+ curve448_scalar_t secret_scalar;
+ EVP_MD_CTX *hashctx = EVP_MD_CTX_new();
+ c448_error_t ret = C448_FAILURE;
+ curve448_scalar_t nonce_scalar;
+ uint8_t nonce_point[EDDSA_448_PUBLIC_BYTES] = { 0 };
+ unsigned int c;
+ curve448_scalar_t challenge_scalar;
+
+ if (hashctx == NULL)
+ return C448_FAILURE;
+
+ {
+ /*
+ * Schedule the secret key, First EDDSA_448_PRIVATE_BYTES is serialised
+ * secret scalar,next EDDSA_448_PRIVATE_BYTES bytes is the seed.
+ */
+ uint8_t expanded[EDDSA_448_PRIVATE_BYTES * 2];
+
+ if (!oneshot_hash(expanded, sizeof(expanded), privkey,
+ EDDSA_448_PRIVATE_BYTES))
+ goto err;
+ clamp(expanded);
+ curve448_scalar_decode_long(secret_scalar, expanded,
+ EDDSA_448_PRIVATE_BYTES);
+
+ /* Hash to create the nonce */
+ if (!hash_init_with_dom(hashctx, prehashed, 0, context, context_len)
+ || !EVP_DigestUpdate(hashctx,
+ expanded + EDDSA_448_PRIVATE_BYTES,
+ EDDSA_448_PRIVATE_BYTES)
+ || !EVP_DigestUpdate(hashctx, message, message_len)) {
+ OPENSSL_cleanse(expanded, sizeof(expanded));
+ goto err;
+ }
+ OPENSSL_cleanse(expanded, sizeof(expanded));
+ }
+
+ /* Decode the nonce */
+ {
+ uint8_t nonce[2 * EDDSA_448_PRIVATE_BYTES];
+
+ if (!EVP_DigestFinalXOF(hashctx, nonce, sizeof(nonce)))
+ goto err;
+ curve448_scalar_decode_long(nonce_scalar, nonce, sizeof(nonce));
+ OPENSSL_cleanse(nonce, sizeof(nonce));
+ }
+
+ {
+ /* Scalarmul to create the nonce-point */
+ curve448_scalar_t nonce_scalar_2;
+ curve448_point_t p;
+
+ curve448_scalar_halve(nonce_scalar_2, nonce_scalar);
+ for (c = 2; c < C448_EDDSA_ENCODE_RATIO; c <<= 1)
+ curve448_scalar_halve(nonce_scalar_2, nonce_scalar_2);
+
+ curve448_precomputed_scalarmul(p, curve448_precomputed_base,
+ nonce_scalar_2);
+ curve448_point_mul_by_ratio_and_encode_like_eddsa(nonce_point, p);
+ curve448_point_destroy(p);
+ curve448_scalar_destroy(nonce_scalar_2);
+ }
+
+ {
+ uint8_t challenge[2 * EDDSA_448_PRIVATE_BYTES];
+
+ /* Compute the challenge */
+ if (!hash_init_with_dom(hashctx, prehashed, 0, context, context_len)
+ || !EVP_DigestUpdate(hashctx, nonce_point, sizeof(nonce_point))
+ || !EVP_DigestUpdate(hashctx, pubkey, EDDSA_448_PUBLIC_BYTES)
+ || !EVP_DigestUpdate(hashctx, message, message_len)
+ || !EVP_DigestFinalXOF(hashctx, challenge, sizeof(challenge)))
+ goto err;
+
+ curve448_scalar_decode_long(challenge_scalar, challenge,
+ sizeof(challenge));
+ OPENSSL_cleanse(challenge, sizeof(challenge));
+ }
+
+ curve448_scalar_mul(challenge_scalar, challenge_scalar, secret_scalar);
+ curve448_scalar_add(challenge_scalar, challenge_scalar, nonce_scalar);
+
+ OPENSSL_cleanse(signature, EDDSA_448_SIGNATURE_BYTES);
+ memcpy(signature, nonce_point, sizeof(nonce_point));
+ curve448_scalar_encode(&signature[EDDSA_448_PUBLIC_BYTES],
+ challenge_scalar);
+
+ curve448_scalar_destroy(secret_scalar);
+ curve448_scalar_destroy(nonce_scalar);
+ curve448_scalar_destroy(challenge_scalar);
+
+ ret = C448_SUCCESS;
+ err:
+ EVP_MD_CTX_free(hashctx);
+ return ret;
+}
+
+c448_error_t c448_ed448_sign_prehash(
+ uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t privkey[EDDSA_448_PRIVATE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t hash[64], const uint8_t *context,
+ size_t context_len)
+{
+ return c448_ed448_sign(signature, privkey, pubkey, hash, 64, 1, context,
+ context_len);
+}
+
+c448_error_t c448_ed448_verify(
+ const uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t *message, size_t message_len,
+ uint8_t prehashed, const uint8_t *context,
+ uint8_t context_len)
+{
+ curve448_point_t pk_point, r_point;
+ c448_error_t error =
+ curve448_point_decode_like_eddsa_and_mul_by_ratio(pk_point, pubkey);
+ curve448_scalar_t challenge_scalar;
+ curve448_scalar_t response_scalar;
+
+ if (C448_SUCCESS != error)
+ return error;
+
+ error =
+ curve448_point_decode_like_eddsa_and_mul_by_ratio(r_point, signature);
+ if (C448_SUCCESS != error)
+ return error;
+
+ {
+ /* Compute the challenge */
+ EVP_MD_CTX *hashctx = EVP_MD_CTX_new();
+ uint8_t challenge[2 * EDDSA_448_PRIVATE_BYTES];
+
+ if (hashctx == NULL
+ || !hash_init_with_dom(hashctx, prehashed, 0, context,
+ context_len)
+ || !EVP_DigestUpdate(hashctx, signature, EDDSA_448_PUBLIC_BYTES)
+ || !EVP_DigestUpdate(hashctx, pubkey, EDDSA_448_PUBLIC_BYTES)
+ || !EVP_DigestUpdate(hashctx, message, message_len)
+ || !EVP_DigestFinalXOF(hashctx, challenge, sizeof(challenge))) {
+ EVP_MD_CTX_free(hashctx);
+ return C448_FAILURE;
+ }
+
+ EVP_MD_CTX_free(hashctx);
+ curve448_scalar_decode_long(challenge_scalar, challenge,
+ sizeof(challenge));
+ OPENSSL_cleanse(challenge, sizeof(challenge));
+ }
+ curve448_scalar_sub(challenge_scalar, curve448_scalar_zero,
+ challenge_scalar);
+
+ curve448_scalar_decode_long(response_scalar,
+ &signature[EDDSA_448_PUBLIC_BYTES],
+ EDDSA_448_PRIVATE_BYTES);
+
+ /* pk_point = -c(x(P)) + (cx + k)G = kG */
+ curve448_base_double_scalarmul_non_secret(pk_point,
+ response_scalar,
+ pk_point, challenge_scalar);
+ return c448_succeed_if(curve448_point_eq(pk_point, r_point));
+}
+
+c448_error_t c448_ed448_verify_prehash(
+ const uint8_t signature[EDDSA_448_SIGNATURE_BYTES],
+ const uint8_t pubkey[EDDSA_448_PUBLIC_BYTES],
+ const uint8_t hash[64], const uint8_t *context,
+ uint8_t context_len)
+{
+ return c448_ed448_verify(signature, pubkey, hash, 64, 1, context,
+ context_len);
+}
+
+int ED448_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
+ const uint8_t public_key[57], const uint8_t private_key[57],
+ const uint8_t *context, size_t context_len)
+{
+ return c448_ed448_sign(out_sig, private_key, public_key, message,
+ message_len, 0, context, context_len)
+ == C448_SUCCESS;
+}
+
+int ED448_verify(const uint8_t *message, size_t message_len,
+ const uint8_t signature[114], const uint8_t public_key[57],
+ const uint8_t *context, size_t context_len)
+{
+ return c448_ed448_verify(signature, public_key, message, message_len, 0,
+ context, (uint8_t)context_len) == C448_SUCCESS;
+}
+
+int ED448ph_sign(uint8_t *out_sig, const uint8_t hash[64],
+ const uint8_t public_key[57], const uint8_t private_key[57],
+ const uint8_t *context, size_t context_len)
+{
+ return c448_ed448_sign_prehash(out_sig, private_key, public_key, hash,
+ context, context_len) == C448_SUCCESS;
+
+}
+
+int ED448ph_verify(const uint8_t hash[64], const uint8_t signature[114],
+ const uint8_t public_key[57], const uint8_t *context,
+ size_t context_len)
+{
+ return c448_ed448_verify_prehash(signature, public_key, hash, context,
+ (uint8_t)context_len) == C448_SUCCESS;
+}
+
+int ED448_public_from_private(uint8_t out_public_key[57],
+ const uint8_t private_key[57])
+{
+ return c448_ed448_derive_public_key(out_public_key, private_key)
+ == C448_SUCCESS;
+}
diff --git a/crypto/ec/curve448/f_generic.c b/crypto/ec/curve448/f_generic.c
new file mode 100644
index 000000000000..ed8f36d868f9
--- /dev/null
+++ b/crypto/ec/curve448/f_generic.c
@@ -0,0 +1,204 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+#include "field.h"
+
+static const gf MODULUS = {
+ FIELD_LITERAL(0xffffffffffffff, 0xffffffffffffff, 0xffffffffffffff,
+ 0xffffffffffffff, 0xfffffffffffffe, 0xffffffffffffff,
+ 0xffffffffffffff, 0xffffffffffffff)
+};
+
+/* Serialize to wire format. */
+void gf_serialize(uint8_t serial[SER_BYTES], const gf x, int with_hibit)
+{
+ unsigned int j = 0, fill = 0;
+ dword_t buffer = 0;
+ int i;
+ gf red;
+
+ gf_copy(red, x);
+ gf_strong_reduce(red);
+ if (!with_hibit)
+ assert(gf_hibit(red) == 0);
+
+ for (i = 0; i < (with_hibit ? X_SER_BYTES : SER_BYTES); i++) {
+ if (fill < 8 && j < NLIMBS) {
+ buffer |= ((dword_t) red->limb[LIMBPERM(j)]) << fill;
+ fill += LIMB_PLACE_VALUE(LIMBPERM(j));
+ j++;
+ }
+ serial[i] = (uint8_t)buffer;
+ fill -= 8;
+ buffer >>= 8;
+ }
+}
+
+/* Return high bit of x = low bit of 2x mod p */
+mask_t gf_hibit(const gf x)
+{
+ gf y;
+
+ gf_add(y, x, x);
+ gf_strong_reduce(y);
+ return 0 - (y->limb[0] & 1);
+}
+
+/* Return high bit of x = low bit of 2x mod p */
+mask_t gf_lobit(const gf x)
+{
+ gf y;
+
+ gf_copy(y, x);
+ gf_strong_reduce(y);
+ return 0 - (y->limb[0] & 1);
+}
+
+/* Deserialize from wire format; return -1 on success and 0 on failure. */
+mask_t gf_deserialize(gf x, const uint8_t serial[SER_BYTES], int with_hibit,
+ uint8_t hi_nmask)
+{
+ unsigned int j = 0, fill = 0;
+ dword_t buffer = 0;
+ dsword_t scarry = 0;
+ const unsigned nbytes = with_hibit ? X_SER_BYTES : SER_BYTES;
+ unsigned int i;
+ mask_t succ;
+
+ for (i = 0; i < NLIMBS; i++) {
+ while (fill < LIMB_PLACE_VALUE(LIMBPERM(i)) && j < nbytes) {
+ uint8_t sj;
+
+ sj = serial[j];
+ if (j == nbytes - 1)
+ sj &= ~hi_nmask;
+ buffer |= ((dword_t) sj) << fill;
+ fill += 8;
+ j++;
+ }
+ x->limb[LIMBPERM(i)] = (word_t)
+ ((i < NLIMBS - 1) ? buffer & LIMB_MASK(LIMBPERM(i)) : buffer);
+ fill -= LIMB_PLACE_VALUE(LIMBPERM(i));
+ buffer >>= LIMB_PLACE_VALUE(LIMBPERM(i));
+ scarry =
+ (scarry + x->limb[LIMBPERM(i)] -
+ MODULUS->limb[LIMBPERM(i)]) >> (8 * sizeof(word_t));
+ }
+ succ = with_hibit ? 0 - (mask_t) 1 : ~gf_hibit(x);
+ return succ & word_is_zero((word_t)buffer) & ~word_is_zero((word_t)scarry);
+}
+
+/* Reduce to canonical form. */
+void gf_strong_reduce(gf a)
+{
+ dsword_t scarry;
+ word_t scarry_0;
+ dword_t carry = 0;
+ unsigned int i;
+
+ /* first, clear high */
+ gf_weak_reduce(a); /* Determined to have negligible perf impact. */
+
+ /* now the total is less than 2p */
+
+ /* compute total_value - p. No need to reduce mod p. */
+ scarry = 0;
+ for (i = 0; i < NLIMBS; i++) {
+ scarry = scarry + a->limb[LIMBPERM(i)] - MODULUS->limb[LIMBPERM(i)];
+ a->limb[LIMBPERM(i)] = scarry & LIMB_MASK(LIMBPERM(i));
+ scarry >>= LIMB_PLACE_VALUE(LIMBPERM(i));
+ }
+
+ /*
+ * uncommon case: it was >= p, so now scarry = 0 and this = x common case:
+ * it was < p, so now scarry = -1 and this = x - p + 2^255 so let's add
+ * back in p. will carry back off the top for 2^255.
+ */
+ assert(scarry == 0 || scarry == -1);
+
+ scarry_0 = (word_t)scarry;
+
+ /* add it back */
+ for (i = 0; i < NLIMBS; i++) {
+ carry =
+ carry + a->limb[LIMBPERM(i)] +
+ (scarry_0 & MODULUS->limb[LIMBPERM(i)]);
+ a->limb[LIMBPERM(i)] = carry & LIMB_MASK(LIMBPERM(i));
+ carry >>= LIMB_PLACE_VALUE(LIMBPERM(i));
+ }
+
+ assert(carry < 2 && ((word_t)carry + scarry_0) == 0);
+}
+
+/* Subtract two gf elements d=a-b */
+void gf_sub(gf d, const gf a, const gf b)
+{
+ gf_sub_RAW(d, a, b);
+ gf_bias(d, 2);
+ gf_weak_reduce(d);
+}
+
+/* Add two field elements d = a+b */
+void gf_add(gf d, const gf a, const gf b)
+{
+ gf_add_RAW(d, a, b);
+ gf_weak_reduce(d);
+}
+
+/* Compare a==b */
+mask_t gf_eq(const gf a, const gf b)
+{
+ gf c;
+ mask_t ret = 0;
+ unsigned int i;
+
+ gf_sub(c, a, b);
+ gf_strong_reduce(c);
+
+ for (i = 0; i < NLIMBS; i++)
+ ret |= c->limb[LIMBPERM(i)];
+
+ return word_is_zero(ret);
+}
+
+mask_t gf_isr(gf a, const gf x)
+{
+ gf L0, L1, L2;
+
+ gf_sqr(L1, x);
+ gf_mul(L2, x, L1);
+ gf_sqr(L1, L2);
+ gf_mul(L2, x, L1);
+ gf_sqrn(L1, L2, 3);
+ gf_mul(L0, L2, L1);
+ gf_sqrn(L1, L0, 3);
+ gf_mul(L0, L2, L1);
+ gf_sqrn(L2, L0, 9);
+ gf_mul(L1, L0, L2);
+ gf_sqr(L0, L1);
+ gf_mul(L2, x, L0);
+ gf_sqrn(L0, L2, 18);
+ gf_mul(L2, L1, L0);
+ gf_sqrn(L0, L2, 37);
+ gf_mul(L1, L2, L0);
+ gf_sqrn(L0, L1, 37);
+ gf_mul(L1, L2, L0);
+ gf_sqrn(L0, L1, 111);
+ gf_mul(L2, L1, L0);
+ gf_sqr(L0, L2);
+ gf_mul(L1, x, L0);
+ gf_sqrn(L0, L1, 223);
+ gf_mul(L1, L2, L0);
+ gf_sqr(L2, L1);
+ gf_mul(L0, L2, x);
+ gf_copy(a, L1);
+ return gf_eq(L0, ONE);
+}
diff --git a/crypto/ec/curve448/field.h b/crypto/ec/curve448/field.h
new file mode 100644
index 000000000000..d96d4c023d44
--- /dev/null
+++ b/crypto/ec/curve448/field.h
@@ -0,0 +1,168 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2014 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_FIELD_H
+# define HEADER_FIELD_H
+
+# include "internal/constant_time_locl.h"
+# include <string.h>
+# include <assert.h>
+# include "word.h"
+
+# define NLIMBS (64/sizeof(word_t))
+# define X_SER_BYTES 56
+# define SER_BYTES 56
+
+# if defined(__GNUC__) || defined(__clang__)
+# define INLINE_UNUSED __inline__ __attribute__((__unused__,__always_inline__))
+# define RESTRICT __restrict__
+# define ALIGNED __attribute__((__aligned__(16)))
+# else
+# define INLINE_UNUSED ossl_inline
+# define RESTRICT
+# define ALIGNED
+# endif
+
+typedef struct gf_s {
+ word_t limb[NLIMBS];
+} ALIGNED gf_s, gf[1];
+
+/* RFC 7748 support */
+# define X_PUBLIC_BYTES X_SER_BYTES
+# define X_PRIVATE_BYTES X_PUBLIC_BYTES
+# define X_PRIVATE_BITS 448
+
+static INLINE_UNUSED void gf_copy(gf out, const gf a)
+{
+ *out = *a;
+}
+
+static INLINE_UNUSED void gf_add_RAW(gf out, const gf a, const gf b);
+static INLINE_UNUSED void gf_sub_RAW(gf out, const gf a, const gf b);
+static INLINE_UNUSED void gf_bias(gf inout, int amount);
+static INLINE_UNUSED void gf_weak_reduce(gf inout);
+
+void gf_strong_reduce(gf inout);
+void gf_add(gf out, const gf a, const gf b);
+void gf_sub(gf out, const gf a, const gf b);
+void gf_mul(gf_s * RESTRICT out, const gf a, const gf b);
+void gf_mulw_unsigned(gf_s * RESTRICT out, const gf a, uint32_t b);
+void gf_sqr(gf_s * RESTRICT out, const gf a);
+mask_t gf_isr(gf a, const gf x); /** a^2 x = 1, QNR, or 0 if x=0. Return true if successful */
+mask_t gf_eq(const gf x, const gf y);
+mask_t gf_lobit(const gf x);
+mask_t gf_hibit(const gf x);
+
+void gf_serialize(uint8_t *serial, const gf x, int with_highbit);
+mask_t gf_deserialize(gf x, const uint8_t serial[SER_BYTES], int with_hibit,
+ uint8_t hi_nmask);
+
+# include "f_impl.h" /* Bring in the inline implementations */
+
+# define LIMBPERM(i) (i)
+# define LIMB_MASK(i) (((1)<<LIMB_PLACE_VALUE(i))-1)
+
+static const gf ZERO = {{{0}}}, ONE = {{{1}}};
+
+/* Square x, n times. */
+static ossl_inline void gf_sqrn(gf_s * RESTRICT y, const gf x, int n)
+{
+ gf tmp;
+
+ assert(n > 0);
+ if (n & 1) {
+ gf_sqr(y, x);
+ n--;
+ } else {
+ gf_sqr(tmp, x);
+ gf_sqr(y, tmp);
+ n -= 2;
+ }
+ for (; n; n -= 2) {
+ gf_sqr(tmp, y);
+ gf_sqr(y, tmp);
+ }
+}
+
+# define gf_add_nr gf_add_RAW
+
+/* Subtract mod p. Bias by 2 and don't reduce */
+static ossl_inline void gf_sub_nr(gf c, const gf a, const gf b)
+{
+ gf_sub_RAW(c, a, b);
+ gf_bias(c, 2);
+ if (GF_HEADROOM < 3)
+ gf_weak_reduce(c);
+}
+
+/* Subtract mod p. Bias by amt but don't reduce. */
+static ossl_inline void gf_subx_nr(gf c, const gf a, const gf b, int amt)
+{
+ gf_sub_RAW(c, a, b);
+ gf_bias(c, amt);
+ if (GF_HEADROOM < amt + 1)
+ gf_weak_reduce(c);
+}
+
+/* Mul by signed int. Not constant-time WRT the sign of that int. */
+static ossl_inline void gf_mulw(gf c, const gf a, int32_t w)
+{
+ if (w > 0) {
+ gf_mulw_unsigned(c, a, w);
+ } else {
+ gf_mulw_unsigned(c, a, -w);
+ gf_sub(c, ZERO, c);
+ }
+}
+
+/* Constant time, x = is_z ? z : y */
+static ossl_inline void gf_cond_sel(gf x, const gf y, const gf z, mask_t is_z)
+{
+ size_t i;
+
+ for (i = 0; i < NLIMBS; i++) {
+#if ARCH_WORD_BITS == 32
+ x[0].limb[i] = constant_time_select_32(is_z, z[0].limb[i],
+ y[0].limb[i]);
+#else
+ /* Must be 64 bit */
+ x[0].limb[i] = constant_time_select_64(is_z, z[0].limb[i],
+ y[0].limb[i]);
+#endif
+ }
+}
+
+/* Constant time, if (neg) x=-x; */
+static ossl_inline void gf_cond_neg(gf x, mask_t neg)
+{
+ gf y;
+
+ gf_sub(y, ZERO, x);
+ gf_cond_sel(x, x, y, neg);
+}
+
+/* Constant time, if (swap) (x,y) = (y,x); */
+static ossl_inline void gf_cond_swap(gf x, gf_s * RESTRICT y, mask_t swap)
+{
+ size_t i;
+
+ for (i = 0; i < NLIMBS; i++) {
+#if ARCH_WORD_BITS == 32
+ constant_time_cond_swap_32(swap, &(x[0].limb[i]), &(y->limb[i]));
+#else
+ /* Must be 64 bit */
+ constant_time_cond_swap_64(swap, &(x[0].limb[i]), &(y->limb[i]));
+#endif
+ }
+}
+
+#endif /* HEADER_FIELD_H */
diff --git a/crypto/ec/curve448/point_448.h b/crypto/ec/curve448/point_448.h
new file mode 100644
index 000000000000..0ef3b8714e26
--- /dev/null
+++ b/crypto/ec/curve448/point_448.h
@@ -0,0 +1,301 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_POINT_448_H
+# define HEADER_POINT_448_H
+
+# include "curve448utils.h"
+# include "field.h"
+
+/* Comb config: number of combs, n, t, s. */
+#define COMBS_N 5
+#define COMBS_T 5
+#define COMBS_S 18
+
+/* Projective Niels coordinates */
+typedef struct {
+ gf a, b, c;
+} niels_s, niels_t[1];
+typedef struct {
+ niels_t n;
+ gf z;
+} pniels_t[1];
+
+/* Precomputed base */
+struct curve448_precomputed_s {
+ niels_t table[COMBS_N << (COMBS_T - 1)];
+};
+
+# define C448_SCALAR_LIMBS ((446-1)/C448_WORD_BITS+1)
+
+/* The number of bits in a scalar */
+# define C448_SCALAR_BITS 446
+
+/* Number of bytes in a serialized scalar. */
+# define C448_SCALAR_BYTES 56
+
+/* X448 encoding ratio. */
+# define X448_ENCODE_RATIO 2
+
+/* Number of bytes in an x448 public key */
+# define X448_PUBLIC_BYTES 56
+
+/* Number of bytes in an x448 private key */
+# define X448_PRIVATE_BYTES 56
+
+/* Twisted Edwards extended homogeneous coordinates */
+typedef struct curve448_point_s {
+ gf x, y, z, t;
+} curve448_point_t[1];
+
+/* Precomputed table based on a point. Can be trivial implementation. */
+struct curve448_precomputed_s;
+
+/* Precomputed table based on a point. Can be trivial implementation. */
+typedef struct curve448_precomputed_s curve448_precomputed_s;
+
+/* Scalar is stored packed, because we don't need the speed. */
+typedef struct curve448_scalar_s {
+ c448_word_t limb[C448_SCALAR_LIMBS];
+} curve448_scalar_t[1];
+
+/* A scalar equal to 1. */
+extern const curve448_scalar_t curve448_scalar_one;
+
+/* A scalar equal to 0. */
+extern const curve448_scalar_t curve448_scalar_zero;
+
+/* The identity point on the curve. */
+extern const curve448_point_t curve448_point_identity;
+
+/* Precomputed table for the base point on the curve. */
+extern const struct curve448_precomputed_s *curve448_precomputed_base;
+extern const niels_t *curve448_wnaf_base;
+
+/*
+ * Read a scalar from wire format or from bytes.
+ *
+ * ser (in): Serialized form of a scalar.
+ * out (out): Deserialized form.
+ *
+ * Returns:
+ * C448_SUCCESS: The scalar was correctly encoded.
+ * C448_FAILURE: The scalar was greater than the modulus, and has been reduced
+ * modulo that modulus.
+ */
+c448_error_t curve448_scalar_decode(curve448_scalar_t out,
+ const unsigned char ser[C448_SCALAR_BYTES]);
+
+/*
+ * Read a scalar from wire format or from bytes. Reduces mod scalar prime.
+ *
+ * ser (in): Serialized form of a scalar.
+ * ser_len (in): Length of serialized form.
+ * out (out): Deserialized form.
+ */
+void curve448_scalar_decode_long(curve448_scalar_t out,
+ const unsigned char *ser, size_t ser_len);
+
+/*
+ * Serialize a scalar to wire format.
+ *
+ * ser (out): Serialized form of a scalar.
+ * s (in): Deserialized scalar.
+ */
+void curve448_scalar_encode(unsigned char ser[C448_SCALAR_BYTES],
+ const curve448_scalar_t s);
+
+/*
+ * Add two scalars. |a|, |b| and |out| may alias each other.
+ *
+ * a (in): One scalar.
+ * b (in): Another scalar.
+ * out (out): a+b.
+ */
+void curve448_scalar_add(curve448_scalar_t out,
+ const curve448_scalar_t a, const curve448_scalar_t b);
+
+/*
+ * Subtract two scalars. |a|, |b| and |out| may alias each other.
+ * a (in): One scalar.
+ * b (in): Another scalar.
+ * out (out): a-b.
+ */
+void curve448_scalar_sub(curve448_scalar_t out,
+ const curve448_scalar_t a, const curve448_scalar_t b);
+
+/*
+ * Multiply two scalars. |a|, |b| and |out| may alias each other.
+ *
+ * a (in): One scalar.
+ * b (in): Another scalar.
+ * out (out): a*b.
+ */
+void curve448_scalar_mul(curve448_scalar_t out,
+ const curve448_scalar_t a, const curve448_scalar_t b);
+
+/*
+* Halve a scalar. |a| and |out| may alias each other.
+*
+* a (in): A scalar.
+* out (out): a/2.
+*/
+void curve448_scalar_halve(curve448_scalar_t out, const curve448_scalar_t a);
+
+/*
+ * Copy a scalar. The scalars may alias each other, in which case this
+ * function does nothing.
+ *
+ * a (in): A scalar.
+ * out (out): Will become a copy of a.
+ */
+static ossl_inline void curve448_scalar_copy(curve448_scalar_t out,
+ const curve448_scalar_t a)
+{
+ *out = *a;
+}
+
+/*
+ * Copy a point. The input and output may alias, in which case this function
+ * does nothing.
+ *
+ * a (out): A copy of the point.
+ * b (in): Any point.
+ */
+static ossl_inline void curve448_point_copy(curve448_point_t a,
+ const curve448_point_t b)
+{
+ *a = *b;
+}
+
+/*
+ * Test whether two points are equal. If yes, return C448_TRUE, else return
+ * C448_FALSE.
+ *
+ * a (in): A point.
+ * b (in): Another point.
+ *
+ * Returns:
+ * C448_TRUE: The points are equal.
+ * C448_FALSE: The points are not equal.
+ */
+__owur c448_bool_t curve448_point_eq(const curve448_point_t a,
+ const curve448_point_t b);
+
+/*
+ * Double a point. Equivalent to curve448_point_add(two_a,a,a), but potentially
+ * faster.
+ *
+ * two_a (out): The sum a+a.
+ * a (in): A point.
+ */
+void curve448_point_double(curve448_point_t two_a, const curve448_point_t a);
+
+/*
+ * RFC 7748 Diffie-Hellman scalarmul. This function uses a different
+ * (non-Decaf) encoding.
+ *
+ * out (out): The scaled point base*scalar
+ * base (in): The point to be scaled.
+ * scalar (in): The scalar to multiply by.
+ *
+ * Returns:
+ * C448_SUCCESS: The scalarmul succeeded.
+ * C448_FAILURE: The scalarmul didn't succeed, because the base point is in a
+ * small subgroup.
+ */
+__owur c448_error_t x448_int(uint8_t out[X448_PUBLIC_BYTES],
+ const uint8_t base[X448_PUBLIC_BYTES],
+ const uint8_t scalar[X448_PRIVATE_BYTES]);
+
+/*
+ * Multiply a point by X448_ENCODE_RATIO, then encode it like RFC 7748.
+ *
+ * This function is mainly used internally, but is exported in case
+ * it will be useful.
+ *
+ * The ratio is necessary because the internal representation doesn't
+ * track the cofactor information, so on output we must clear the cofactor.
+ * This would multiply by the cofactor, but in fact internally points are always
+ * even, so it multiplies by half the cofactor instead.
+ *
+ * As it happens, this aligns with the base point definitions; that is,
+ * if you pass the Decaf/Ristretto base point to this function, the result
+ * will be X448_ENCODE_RATIO times the X448
+ * base point.
+ *
+ * out (out): The scaled and encoded point.
+ * p (in): The point to be scaled and encoded.
+ */
+void curve448_point_mul_by_ratio_and_encode_like_x448(
+ uint8_t out[X448_PUBLIC_BYTES],
+ const curve448_point_t p);
+
+/*
+ * RFC 7748 Diffie-Hellman base point scalarmul. This function uses a different
+ * (non-Decaf) encoding.
+ *
+ * out (out): The scaled point base*scalar
+ * scalar (in): The scalar to multiply by.
+ */
+void x448_derive_public_key(uint8_t out[X448_PUBLIC_BYTES],
+ const uint8_t scalar[X448_PRIVATE_BYTES]);
+
+/*
+ * Multiply a precomputed base point by a scalar: out = scalar*base.
+ *
+ * scaled (out): The scaled point base*scalar
+ * base (in): The point to be scaled.
+ * scalar (in): The scalar to multiply by.
+ */
+void curve448_precomputed_scalarmul(curve448_point_t scaled,
+ const curve448_precomputed_s * base,
+ const curve448_scalar_t scalar);
+
+/*
+ * Multiply two base points by two scalars:
+ * combo = scalar1*curve448_point_base + scalar2*base2.
+ *
+ * Otherwise equivalent to curve448_point_double_scalarmul, but may be
+ * faster at the expense of being variable time.
+ *
+ * combo (out): The linear combination scalar1*base + scalar2*base2.
+ * scalar1 (in): A first scalar to multiply by.
+ * base2 (in): A second point to be scaled.
+ * scalar2 (in) A second scalar to multiply by.
+ *
+ * Warning: This function takes variable time, and may leak the scalars used.
+ * It is designed for signature verification.
+ */
+void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
+ const curve448_scalar_t scalar1,
+ const curve448_point_t base2,
+ const curve448_scalar_t scalar2);
+
+/*
+ * Test that a point is valid, for debugging purposes.
+ *
+ * to_test (in): The point to test.
+ *
+ * Returns:
+ * C448_TRUE The point is valid.
+ * C448_FALSE The point is invalid.
+ */
+__owur c448_bool_t curve448_point_valid(const curve448_point_t to_test);
+
+/* Overwrite scalar with zeros. */
+void curve448_scalar_destroy(curve448_scalar_t scalar);
+
+/* Overwrite point with zeros. */
+void curve448_point_destroy(curve448_point_t point);
+
+#endif /* HEADER_POINT_448_H */
diff --git a/crypto/ec/curve448/scalar.c b/crypto/ec/curve448/scalar.c
new file mode 100644
index 000000000000..b5702c025570
--- /dev/null
+++ b/crypto/ec/curve448/scalar.c
@@ -0,0 +1,235 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2015-2016 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+#include <openssl/crypto.h>
+
+#include "word.h"
+#include "point_448.h"
+
+static const c448_word_t MONTGOMERY_FACTOR = (c448_word_t) 0x3bd440fae918bc5;
+static const curve448_scalar_t sc_p = {
+ {
+ {
+ SC_LIMB(0x2378c292ab5844f3), SC_LIMB(0x216cc2728dc58f55),
+ SC_LIMB(0xc44edb49aed63690), SC_LIMB(0xffffffff7cca23e9),
+ SC_LIMB(0xffffffffffffffff), SC_LIMB(0xffffffffffffffff),
+ SC_LIMB(0x3fffffffffffffff)
+ }
+ }
+}, sc_r2 = {
+ {
+ {
+
+ SC_LIMB(0xe3539257049b9b60), SC_LIMB(0x7af32c4bc1b195d9),
+ SC_LIMB(0x0d66de2388ea1859), SC_LIMB(0xae17cf725ee4d838),
+ SC_LIMB(0x1a9cc14ba3c47c44), SC_LIMB(0x2052bcb7e4d070af),
+ SC_LIMB(0x3402a939f823b729)
+ }
+ }
+};
+
+#define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
+
+const curve448_scalar_t curve448_scalar_one = {{{1}}};
+const curve448_scalar_t curve448_scalar_zero = {{{0}}};
+
+/*
+ * {extra,accum} - sub +? p
+ * Must have extra <= 1
+ */
+static void sc_subx(curve448_scalar_t out,
+ const c448_word_t accum[C448_SCALAR_LIMBS],
+ const curve448_scalar_t sub,
+ const curve448_scalar_t p, c448_word_t extra)
+{
+ c448_dsword_t chain = 0;
+ unsigned int i;
+ c448_word_t borrow;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ chain = (chain + accum[i]) - sub->limb[i];
+ out->limb[i] = (c448_word_t)chain;
+ chain >>= WBITS;
+ }
+ borrow = (c448_word_t)chain + extra; /* = 0 or -1 */
+
+ chain = 0;
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ chain = (chain + out->limb[i]) + (p->limb[i] & borrow);
+ out->limb[i] = (c448_word_t)chain;
+ chain >>= WBITS;
+ }
+}
+
+static void sc_montmul(curve448_scalar_t out, const curve448_scalar_t a,
+ const curve448_scalar_t b)
+{
+ unsigned int i, j;
+ c448_word_t accum[C448_SCALAR_LIMBS + 1] = { 0 };
+ c448_word_t hi_carry = 0;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ c448_word_t mand = a->limb[i];
+ const c448_word_t *mier = b->limb;
+
+ c448_dword_t chain = 0;
+ for (j = 0; j < C448_SCALAR_LIMBS; j++) {
+ chain += ((c448_dword_t) mand) * mier[j] + accum[j];
+ accum[j] = (c448_word_t)chain;
+ chain >>= WBITS;
+ }
+ accum[j] = (c448_word_t)chain;
+
+ mand = accum[0] * MONTGOMERY_FACTOR;
+ chain = 0;
+ mier = sc_p->limb;
+ for (j = 0; j < C448_SCALAR_LIMBS; j++) {
+ chain += (c448_dword_t) mand *mier[j] + accum[j];
+ if (j)
+ accum[j - 1] = (c448_word_t)chain;
+ chain >>= WBITS;
+ }
+ chain += accum[j];
+ chain += hi_carry;
+ accum[j - 1] = (c448_word_t)chain;
+ hi_carry = chain >> WBITS;
+ }
+
+ sc_subx(out, accum, sc_p, sc_p, hi_carry);
+}
+
+void curve448_scalar_mul(curve448_scalar_t out, const curve448_scalar_t a,
+ const curve448_scalar_t b)
+{
+ sc_montmul(out, a, b);
+ sc_montmul(out, out, sc_r2);
+}
+
+void curve448_scalar_sub(curve448_scalar_t out, const curve448_scalar_t a,
+ const curve448_scalar_t b)
+{
+ sc_subx(out, a->limb, b, sc_p, 0);
+}
+
+void curve448_scalar_add(curve448_scalar_t out, const curve448_scalar_t a,
+ const curve448_scalar_t b)
+{
+ c448_dword_t chain = 0;
+ unsigned int i;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ chain = (chain + a->limb[i]) + b->limb[i];
+ out->limb[i] = (c448_word_t)chain;
+ chain >>= WBITS;
+ }
+ sc_subx(out, out->limb, sc_p, sc_p, (c448_word_t)chain);
+}
+
+static ossl_inline void scalar_decode_short(curve448_scalar_t s,
+ const unsigned char *ser,
+ size_t nbytes)
+{
+ size_t i, j, k = 0;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ c448_word_t out = 0;
+
+ for (j = 0; j < sizeof(c448_word_t) && k < nbytes; j++, k++)
+ out |= ((c448_word_t) ser[k]) << (8 * j);
+ s->limb[i] = out;
+ }
+}
+
+c448_error_t curve448_scalar_decode(
+ curve448_scalar_t s,
+ const unsigned char ser[C448_SCALAR_BYTES])
+{
+ unsigned int i;
+ c448_dsword_t accum = 0;
+
+ scalar_decode_short(s, ser, C448_SCALAR_BYTES);
+ for (i = 0; i < C448_SCALAR_LIMBS; i++)
+ accum = (accum + s->limb[i] - sc_p->limb[i]) >> WBITS;
+ /* Here accum == 0 or -1 */
+
+ curve448_scalar_mul(s, s, curve448_scalar_one); /* ham-handed reduce */
+
+ return c448_succeed_if(~word_is_zero((uint32_t)accum));
+}
+
+void curve448_scalar_destroy(curve448_scalar_t scalar)
+{
+ OPENSSL_cleanse(scalar, sizeof(curve448_scalar_t));
+}
+
+void curve448_scalar_decode_long(curve448_scalar_t s,
+ const unsigned char *ser, size_t ser_len)
+{
+ size_t i;
+ curve448_scalar_t t1, t2;
+
+ if (ser_len == 0) {
+ curve448_scalar_copy(s, curve448_scalar_zero);
+ return;
+ }
+
+ i = ser_len - (ser_len % C448_SCALAR_BYTES);
+ if (i == ser_len)
+ i -= C448_SCALAR_BYTES;
+
+ scalar_decode_short(t1, &ser[i], ser_len - i);
+
+ if (ser_len == sizeof(curve448_scalar_t)) {
+ assert(i == 0);
+ /* ham-handed reduce */
+ curve448_scalar_mul(s, t1, curve448_scalar_one);
+ curve448_scalar_destroy(t1);
+ return;
+ }
+
+ while (i) {
+ i -= C448_SCALAR_BYTES;
+ sc_montmul(t1, t1, sc_r2);
+ (void)curve448_scalar_decode(t2, ser + i);
+ curve448_scalar_add(t1, t1, t2);
+ }
+
+ curve448_scalar_copy(s, t1);
+ curve448_scalar_destroy(t1);
+ curve448_scalar_destroy(t2);
+}
+
+void curve448_scalar_encode(unsigned char ser[C448_SCALAR_BYTES],
+ const curve448_scalar_t s)
+{
+ unsigned int i, j, k = 0;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ for (j = 0; j < sizeof(c448_word_t); j++, k++)
+ ser[k] = s->limb[i] >> (8 * j);
+ }
+}
+
+void curve448_scalar_halve(curve448_scalar_t out, const curve448_scalar_t a)
+{
+ c448_word_t mask = 0 - (a->limb[0] & 1);
+ c448_dword_t chain = 0;
+ unsigned int i;
+
+ for (i = 0; i < C448_SCALAR_LIMBS; i++) {
+ chain = (chain + a->limb[i]) + (sc_p->limb[i] & mask);
+ out->limb[i] = (c448_word_t)chain;
+ chain >>= C448_WORD_BITS;
+ }
+ for (i = 0; i < C448_SCALAR_LIMBS - 1; i++)
+ out->limb[i] = out->limb[i] >> 1 | out->limb[i + 1] << (WBITS - 1);
+ out->limb[i] = out->limb[i] >> 1 | (c448_word_t)(chain << (WBITS - 1));
+}
diff --git a/crypto/ec/curve448/word.h b/crypto/ec/curve448/word.h
new file mode 100644
index 000000000000..a48b9e053a5c
--- /dev/null
+++ b/crypto/ec/curve448/word.h
@@ -0,0 +1,81 @@
+/*
+ * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 2014 Cryptography Research, Inc.
+ *
+ * Licensed under the OpenSSL license (the "License"). You may not use
+ * this file except in compliance with the License. You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ *
+ * Originally written by Mike Hamburg
+ */
+
+#ifndef HEADER_WORD_H
+# define HEADER_WORD_H
+
+# include <string.h>
+# include <assert.h>
+# include <stdlib.h>
+# include <openssl/e_os2.h>
+# include "arch_intrinsics.h"
+# include "curve448utils.h"
+
+# if (ARCH_WORD_BITS == 64)
+typedef uint64_t word_t, mask_t;
+typedef __uint128_t dword_t;
+typedef int32_t hsword_t;
+typedef int64_t sword_t;
+typedef __int128_t dsword_t;
+# elif (ARCH_WORD_BITS == 32)
+typedef uint32_t word_t, mask_t;
+typedef uint64_t dword_t;
+typedef int16_t hsword_t;
+typedef int32_t sword_t;
+typedef int64_t dsword_t;
+# else
+# error "For now, we only support 32- and 64-bit architectures."
+# endif
+
+/*
+ * Scalar limbs are keyed off of the API word size instead of the arch word
+ * size.
+ */
+# if C448_WORD_BITS == 64
+# define SC_LIMB(x) (x)
+# elif C448_WORD_BITS == 32
+# define SC_LIMB(x) ((uint32_t)(x)),((x) >> 32)
+# else
+# error "For now we only support 32- and 64-bit architectures."
+# endif
+
+/*
+ * The plan on booleans: The external interface uses c448_bool_t, but this
+ * might be a different size than our particular arch's word_t (and thus
+ * mask_t). Also, the caller isn't guaranteed to pass it as nonzero. So
+ * bool_to_mask converts word sizes and checks nonzero. On the flip side,
+ * mask_t is always -1 or 0, but it might be a different size than
+ * c448_bool_t. On the third hand, we have success vs boolean types, but
+ * that's handled in common.h: it converts between c448_bool_t and
+ * c448_error_t.
+ */
+static ossl_inline c448_bool_t mask_to_bool(mask_t m)
+{
+ return (c448_sword_t)(sword_t)m;
+}
+
+static ossl_inline mask_t bool_to_mask(c448_bool_t m)
+{
+ /* On most arches this will be optimized to a simple cast. */
+ mask_t ret = 0;
+ unsigned int i;
+ unsigned int limit = sizeof(c448_bool_t) / sizeof(mask_t);
+
+ if (limit < 1)
+ limit = 1;
+ for (i = 0; i < limit; i++)
+ ret |= ~word_is_zero(m >> (i * 8 * sizeof(word_t)));
+
+ return ret;
+}
+
+#endif /* HEADER_WORD_H */