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authorNagi5Yeq <nagi5yeq@gmail.com>2019-07-04 23:55:21 +0800
committerNagi5Yeq <nagi5yeq@gmail.com>2019-07-04 23:55:21 +0800
commit28bc016fb0dc11bf2a4c3e157daac24b2d9482f6 (patch)
tree187edb3f152852940610e0cd09a5b2d16f83cc99
parent22aca24950b45892a70aef77a5f197e521fc11b5 (diff)
add ge448 and data
-rw-r--r--fe448.c61
-rw-r--r--fe448.h2
-rw-r--r--ge448.c272
-rw-r--r--ge448.h34
-rw-r--r--ge448_base.data1490
-rw-r--r--ge448_gen.c99
-rw-r--r--sc448.c131
-rw-r--r--sc448.h13
8 files changed, 2006 insertions, 96 deletions
diff --git a/fe448.c b/fe448.c
index 961e944..b0c413d 100644
--- a/fe448.c
+++ b/fe448.c
@@ -342,4 +342,65 @@ void fe448_invert(fe448 *r, const fe448 *x)
/* pow(2**446-2**222-1) */
void fe448_pow446(fe448 *r, const fe448 *x)
{
+ fe448 t0;
+ fe448 t1;
+ fe448 t2;
+ int i;
+
+ fe448_square(&t0, x);
+ fe448_mul(&t0, &t0, x); /* 2^2 - 1 */
+ fe448_square(&t0, &t0); /* 2^3 - 2 */
+ fe448_mul(&t1, &t0, x); /* 2^3 - 1 */
+ fe448_square(&t0, &t1); /* 2^4 - 2^1 */
+ for (i = 0; i < 2; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^6 - 2^3 */
+ fe448_mul(&t1, &t1, &t0); /* 2^6 - 1 */
+ fe448_square(&t0, &t1); /* 2^7 - 2^1 */
+ for (i = 0; i < 5; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^12 - 2^6 */
+ fe448_mul(&t1, &t1, &t0); /* 2^12 - 1 */
+ fe448_square(&t1, &t1); /* 2^13 - 2^1 */
+ fe448_mul(&t1, &t1, x); /* 2^13 - 1 */
+ fe448_square(&t0, &t1); /* 2^14 - 2^1 */
+ for (i = 0; i < 12; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^26 - 2^13 */
+ fe448_mul(&t1, &t1, &t0); /* 2^26 - 1 */
+ fe448_square(&t1, &t1); /* 2^26 - 2^1 */
+ fe448_mul(&t1, &t1, x); /* 2^27 - 1 */
+ fe448_square(&t0, &t1); /* 2^28 - 2^1 */
+ for (i = 0; i < 26; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^54 - 2^27 */
+ fe448_mul(&t1, &t1, &t0); /* 2^54 - 1 */
+ fe448_square(&t1, &t1); /* 2^55 - 2^1 */
+ fe448_mul(&t1, &t1, x); /* 2^55 - 1 */
+ fe448_square(&t0, &t1); /* 2^56 - 2^1 */
+ for (i = 0; i < 54; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^110 - 2^55 */
+ fe448_mul(&t1, &t1, &t0); /* 2^110 - 1 */
+ fe448_square(&t1, &t1); /* 2^111 - 2^1 */
+ fe448_mul(&t1, &t1, x); /* 2^111 - 1 */
+ fe448_square(&t0, &t1); /* 2^112 - 2^1 */
+ for (i = 0; i < 110; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^222 - 2^111 */
+ fe448_mul(&t2, &t1, &t0); /* 2^222 - 1 */
+ fe448_square(&t1, &t2); /* 2^223 - 2^1 */
+ fe448_mul(&t1, &t1, x); /* 2^223 - 1 */
+ fe448_square(&t0, &t1); /* 2^224 - 2^1 */
+ for (i = 0; i < 222; i++)
+ {
+ fe448_square(&t0, &t0);
+ } /* 2^446 - 2^223 */
+ fe448_mul(r, &t2, &t0); /* 2^446 - 2^222 - 1 */
}
diff --git a/fe448.h b/fe448.h
index 6eda9d1..535eeb5 100644
--- a/fe448.h
+++ b/fe448.h
@@ -19,7 +19,7 @@
#define fe448_mul crypto_sign_ed448_ref_fe448_mul
#define fe448_square crypto_sign_ed448_ref_fe448_square
#define fe448_invert crypto_sign_ed448_ref_fe448_invert
-#define fe448_pow2523 crypto_sign_ed448_ref_fe448_pow2523
+#define fe448_pow446 crypto_sign_ed448_ref_fe448_pow446
typedef struct
{
diff --git a/ge448.c b/ge448.c
new file mode 100644
index 0000000..5916e42
--- /dev/null
+++ b/ge448.c
@@ -0,0 +1,272 @@
+#include "fe448.h"
+#include "sc448.h"
+#include "ge448.h"
+
+/*
+ * Ed448-Goldilocks: x^2 + y^2 = 1 + dx^2y^2
+ * with d = -39081
+ */
+
+/* d */
+static const fe448 ge448_ecd = {{0x56, 0x67, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00}};
+
+#define ge448_p3 ge448 /* use projective coordinates only */
+
+typedef struct
+{
+ fe448 x;
+ fe448 y;
+} ge448_aff;
+
+/* Packed coordinates of the base point */
+const ge448 ge448_base = {{{0x5E, 0xC0, 0x0C, 0xC7, 0x2B, 0xA8, 0x26, 0x26, 0x8E, 0x93, 0x00, 0x8B, 0xE1, 0x80, 0x3B, 0x43,
+ 0x11, 0x65, 0xB6, 0x2A, 0xF7, 0x1A, 0xAE, 0x12, 0x64, 0xA4, 0xD3, 0xA3, 0x24, 0xE3, 0x6D, 0xEA,
+ 0x67, 0x17, 0x0F, 0x47, 0x70, 0x65, 0x14, 0x9E, 0xDA, 0x36, 0xBF, 0x22, 0xA6, 0x15, 0x1D, 0x22,
+ 0xED, 0x0D, 0xED, 0x6B, 0xC6, 0x70, 0x19, 0x4F, 0x00}},
+ {{0x14, 0xFA, 0x30, 0xF2, 0x5B, 0x79, 0x08, 0x98, 0xAD, 0xC8, 0xD7, 0x4E, 0x2C, 0x13, 0xBD, 0xFD,
+ 0xC4, 0x39, 0x7C, 0xE6, 0x1C, 0xFF, 0xD3, 0x3A, 0xD7, 0xC2, 0xA0, 0x05, 0x1E, 0x9C, 0x78, 0x87,
+ 0x40, 0x98, 0xA3, 0x6C, 0x73, 0x73, 0xEA, 0x4B, 0x62, 0xC7, 0xC9, 0x56, 0x37, 0x20, 0x76, 0x88,
+ 0x24, 0xBC, 0xB6, 0x6E, 0x71, 0x46, 0x3F, 0x69, 0x00}},
+ {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}};
+
+/* Multiples of the base point in affine representation */
+static const ge448_aff ge448_base_multiples_affine[745] = {
+#include "ge448_base.data"
+};
+
+/* point addition from rfc8032 */
+static void ge25519_mixadd2(ge448_p3 *r, const ge448_aff *q)
+{
+ fe448 a, b, c, d, e, f, g, h;
+ a = r->z; /* A=Z1*Z2 (Z2=1) */
+ fe448_square(&b, &a); /* B=A^2 */
+ fe448_mul(&c, &r->x, &q->x); /* C=X1*X2 */
+ fe448_mul(&d, &r->y, &q->y); /* D=Y1*Y2 */
+ fe448_mul(&e, &c, &d);
+ fe448_mul(&e, &e, &ge448_ecd); /* E=d*C*D */
+ fe448_sub(&f, &b, &e); /* F=B-E */
+ fe448_add(&g, &b, &e); /* G=B+E */
+ fe448_add(&b, &r->x, &r->y);
+ fe448_add(&h, &q->x, &q->y);
+ fe448_mul(&h, &b, &h); /* H=(X1+Y1)*(X2+Y2) */
+ fe448_sub(&h, &h, &c);
+ fe448_sub(&h, &h, &d);
+ fe448_mul(&h, &h, &f);
+ fe448_mul(&r->x, &a, &h); /* X3=A*F*(H-C-D) */
+ fe448_sub(&d, &d, &c);
+ fe448_mul(&d, &d, &g);
+ fe448_mul(&r->y, &a, &d); /* Y3=A*G*(D-C) */
+ fe448_mul(&r->z, &f, &g); /* Z3=F*G */
+}
+
+static void add_p3(ge448_p3 *r, const ge448_p3 *p, const ge448_p3 *q)
+{
+ fe448 a, b, c, d, e, f, g, h;
+ fe448_mul(&a, &p->z, &q->z); /* A=Z1*Z2 */
+ fe448_square(&b, &a); /* B=A^2 */
+ fe448_mul(&c, &p->x, &q->x); /* C=X1*X2 */
+ fe448_mul(&d, &p->y, &q->y); /* D=Y1*Y2 */
+ fe448_mul(&e, &c, &d);
+ fe448_mul(&e, &e, &ge448_ecd); /* E=d*C*D */
+ fe448_sub(&f, &b, &e); /* F=B-E */
+ fe448_add(&g, &b, &e); /* G=B+E */
+ fe448_add(&b, &p->x, &p->y);
+ fe448_add(&h, &q->x, &q->y);
+ fe448_mul(&h, &b, &h); /* H=(X1+Y1)*(X2+Y2) */
+ fe448_sub(&h, &h, &c);
+ fe448_sub(&h, &h, &d);
+ fe448_mul(&h, &h, &f);
+ fe448_mul(&r->x, &a, &h); /* X3=A*F*(H-C-D) */
+ fe448_sub(&d, &d, &c);
+ fe448_mul(&d, &d, &g);
+ fe448_mul(&r->y, &a, &d); /* Y3=A*G*(D-C) */
+ fe448_mul(&r->z, &f, &g); /* Z3=F*G */
+}
+
+/* point doubling from rfc8032 */
+static void dbl_p3(ge448_p3 *r, const ge448_p3 *p)
+{
+ fe448 b, c, d, e, h, j;
+ fe448_add(&b, &p->x, &p->y);
+ fe448_square(&b, &b); /* B=(X1+Y1)^2 */
+ fe448_square(&c, &p->x); /* C=X1^2 */
+ fe448_square(&d, &p->y); /* D=Y1^2 */
+ fe448_add(&e, &c, &d); /* E=C+D */
+ fe448_square(&h, &p->z); /* H=Z1^2 */
+ fe448_sub(&j, &e, &h);
+ fe448_sub(&j, &j, &h); /* J=E-2*H */
+ fe448_sub(&h, &b, &e);
+ fe448_mul(&r->x, &h, &j); /* X3=(B-E)*J */
+ fe448_sub(&h, &c, &d);
+ fe448_mul(&r->y, &e, &h); /* Y3=E*(C-D) */
+ fe448_mul(&r->z, &e, &j); /* Z3=E*J */
+}
+
+/* Constant-time version of: if(b) r = p */
+static void cmov_aff(ge448_aff *r, const ge448_aff *p, unsigned char b)
+{
+ fe448_cmov(&r->x, &p->x, b);
+ fe448_cmov(&r->y, &p->y, b);
+}
+
+static unsigned char equal(signed char b, signed char c)
+{
+ unsigned char ub = b;
+ unsigned char uc = c;
+ unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
+ crypto_uint32 y = x; /* 0: yes; 1..255: no */
+ y -= 1; /* 4294967295: yes; 0..254: no */
+ y >>= 31; /* 1: yes; 0: no */
+ return y;
+}
+
+static unsigned char negative(signed char b)
+{
+ unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
+ x >>= 63; /* 1: yes; 0: no */
+ return x;
+}
+
+static void choose_t(ge448_aff *t, unsigned long long pos, signed char b)
+{
+ /* constant time */
+ fe448 v;
+ *t = ge448_base_multiples_affine[5 * pos + 0];
+ cmov_aff(t, &ge448_base_multiples_affine[5 * pos + 1], equal(b, 1) | equal(b, -1));
+ cmov_aff(t, &ge448_base_multiples_affine[5 * pos + 2], equal(b, 2) | equal(b, -2));
+ cmov_aff(t, &ge448_base_multiples_affine[5 * pos + 3], equal(b, 3) | equal(b, -3));
+ cmov_aff(t, &ge448_base_multiples_affine[5 * pos + 4], equal(b, -4));
+ fe448_neg(&v, &t->x);
+ fe448_cmov(&t->x, &v, negative(b));
+}
+
+static void setneutral(ge448 *r)
+{
+ fe448_setzero(&r->x);
+ fe448_setone(&r->y);
+ fe448_setone(&r->z);
+}
+
+/* ********************************************************************
+ * EXPORTED FUNCTIONS
+ ******************************************************************** */
+
+/* return 0 on success, -1 otherwise */
+int ge448_unpackneg_vartime(ge448_p3 *r, const unsigned char p[57])
+{
+ unsigned char par;
+ fe448 t, chk, num, num2, den, den2, n3d;
+ fe448_setone(&r->z);
+ par = p[56] >> 7;
+ fe448_unpack(&r->y, p);
+ fe448_square(&num, &r->y); /* x = y^2 */
+ fe448_mul(&den, &num, &ge448_ecd); /* den = dy^2 */
+ fe448_sub(&num, &num, &r->z); /* x = y^2-1 */
+ fe448_sub(&den, &r->z, &den); /* den = dy^2-1 */
+
+ /* Computation of sqrt(num/den) */
+ /* 1.: computation of num^3 * den */
+ fe448_square(&num2, &num);
+ fe448_mul(&n3d, &num2, &num);
+ fe448_mul(&n3d, &n3d, &den);
+
+ /* 2. computation of r->x = n3d * (num^5*den^3)^((p-3)/4) */
+ fe448_square(&den2, &den);
+ fe448_mul(&t, &n3d, &den2);
+ fe448_mul(&t, &t, &num2);
+ fe448_pow446(&t, &t);
+ fe448_mul(&r->x, &t, &n3d);
+
+ /* 3. Check whether square root exists */
+ fe448_square(&chk, &r->x);
+ fe448_mul(&chk, &chk, &den);
+ if (!fe448_iseq_vartime(&chk, &num))
+ return -1;
+
+ /* 4. Choose the desired square root according to parity: */
+ if (fe448_getparity(&r->x) != (1 - par))
+ fe448_neg(&r->x, &r->x);
+ return 0;
+}
+
+void ge448_pack(unsigned char r[57], const ge448_p3 *p)
+{
+ fe448 tx, ty, zi;
+ fe448_invert(&zi, &p->z);
+ fe448_mul(&tx, &p->x, &zi);
+ fe448_mul(&ty, &p->y, &zi);
+ fe448_pack(r, &ty);
+ r[56] ^= fe448_getparity(&tx) << 7;
+}
+
+int ge448_isneutral_vartime(const ge448_p3 *p)
+{
+ int ret = 1;
+ if (!fe448_iszero(&p->x))
+ ret = 0;
+ if (!fe448_iseq_vartime(&p->y, &p->z))
+ ret = 0;
+ return ret;
+}
+
+/* computes [s1]p1 + [s2]p2 */
+void ge448_double_scalarmult_vartime(ge448_p3 *r, const ge448_p3 *p1, const sc448 *s1, const ge448_p3 *p2, const sc448 *s2)
+{
+ ge448_p3 pre[16];
+ unsigned char b[223];
+ int i;
+
+ /* precomputation s2 s1 */
+ setneutral(pre); /* 00 00 */
+ pre[1] = *p1; /* 00 01 */
+ dbl_p3(&pre[2], p1); /* 00 10 */
+ add_p3(&pre[3], &pre[1], &pre[2]); /* 00 11 */
+ pre[4] = *p2; /* 01 00 */
+ add_p3(&pre[5], &pre[1], &pre[4]); /* 01 01 */
+ add_p3(&pre[6], &pre[2], &pre[4]); /* 01 10 */
+ add_p3(&pre[7], &pre[3], &pre[4]); /* 01 11 */
+ dbl_p3(&pre[8], p2); /* 10 00 */
+ add_p3(&pre[9], &pre[1], &pre[8]); /* 10 01 */
+ dbl_p3(&pre[10], &pre[5]); /* 10 10 */
+ add_p3(&pre[11], &pre[3], &pre[8]); /* 10 11 */
+ add_p3(&pre[12], &pre[4], &pre[8]); /* 11 00 */
+ add_p3(&pre[13], &pre[1], &pre[12]); /* 11 01 */
+ add_p3(&pre[14], &pre[2], &pre[12]); /* 11 10 */
+ add_p3(&pre[15], &pre[3], &pre[12]); /* 11 11 */
+
+ sc448_2interleave2(b, s1, s2);
+
+ /* scalar multiplication */
+ *r = pre[b[222]];
+ for (i = 125; i >= 0; i--)
+ {
+ dbl_p3(r, r);
+ dbl_p3(r, r);
+ if (b[i] != 0)
+ {
+ add_p3(r, r, &pre[b[i]]);
+ }
+ }
+}
+
+void ge448_scalarmult_base(ge448_p3 *r, const sc448 *s)
+{
+ signed char b[149];
+ int i;
+ ge448_aff t;
+ sc448_window3(b, s);
+
+ choose_t((ge448_aff *)r, 0, b[0]);
+ fe448_setone(&r->z);
+ for (i = 1; i < 149; i++)
+ {
+ choose_t(&t, (unsigned long long)i, b[i]);
+ ge25519_mixadd2(r, &t);
+ }
+}
diff --git a/ge448.h b/ge448.h
new file mode 100644
index 0000000..dbdfccf
--- /dev/null
+++ b/ge448.h
@@ -0,0 +1,34 @@
+#ifndef GE448_H
+#define GE448_H
+
+#include "fe448.h"
+#include "sc448.h"
+
+#define ge448 crypto_sign_ed448_ref_ge448
+#define ge448_base crypto_sign_ed448_ref_ge448_base
+#define ge448_unpackneg_vartime crypto_sign_ed448_ref_unpackneg_vartime
+#define ge448_pack crypto_sign_ed448_ref_pack
+#define ge448_isneutral_vartime crypto_sign_ed448_ref_isneutral_vartime
+#define ge448_double_scalarmult_vartime crypto_sign_ed448_ref_double_scalarmult_vartime
+#define ge448_scalarmult_base crypto_sign_ed448_ref_scalarmult_base
+
+typedef struct
+{
+ fe448 x;
+ fe448 y;
+ fe448 z;
+} ge448;
+
+extern const ge448 ge448_base;
+
+int ge448_unpackneg_vartime(ge448 *r, const unsigned char p[57]);
+
+void ge448_pack(unsigned char r[57], const ge448 *p);
+
+int ge448_isneutral_vartime(const ge448 *p);
+
+void ge448_double_scalarmult_vartime(ge448 *r, const ge448 *p1, const sc448 *s1, const ge448 *p2, const sc448 *s2);
+
+void ge448_scalarmult_base(ge448 *r, const sc448 *s);
+
+#endif
diff --git a/ge448_base.data b/ge448_base.data
new file mode 100644
index 0000000..9afd629
--- /dev/null
+++ b/ge448_base.data
@@ -0,0 +1,1490 @@
+{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
+ {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
+{{{0x5e, 0xc0, 0x0c, 0xc7, 0x2b, 0xa8, 0x26, 0x26, 0x8e, 0x93, 0x00, 0x8b, 0xe1, 0x80, 0x3b, 0x43, 0x11, 0x65, 0xb6, 0x2a, 0xf7, 0x1a, 0xae, 0x12, 0x64, 0xa4, 0xd3, 0xa3, 0x24, 0xe3, 0x6d, 0xea, 0x67, 0x17, 0x0f, 0x47, 0x70, 0x65, 0x14, 0x9e, 0xda, 0x36, 0xbf, 0x22, 0xa6, 0x15, 0x1d, 0x22, 0xed, 0x0d, 0xed, 0x6b, 0xc6, 0x70, 0x19, 0x4f, 0x00}},
+ {{0x14, 0xfa, 0x30, 0xf2, 0x5b, 0x79, 0x08, 0x98, 0xad, 0xc8, 0xd7, 0x4e, 0x2c, 0x13, 0xbd, 0xfd, 0xc4, 0x39, 0x7c, 0xe6, 0x1c, 0xff, 0xd3, 0x3a, 0xd7, 0xc2, 0xa0, 0x05, 0x1e, 0x9c, 0x78, 0x87, 0x40, 0x98, 0xa3, 0x6c, 0x73, 0x73, 0xea, 0x4b, 0x62, 0xc7, 0xc9, 0x56, 0x37, 0x20, 0x76, 0x88, 0x24, 0xbc, 0xb6, 0x6e, 0x71, 0x46, 0x3f, 0x69, 0x00}}},
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+ {{0xcb, 0x85, 0x91, 0x65, 0x42, 0x2e, 0x8b, 0x56, 0x10, 0x0e, 0xf8, 0xb9, 0x95, 0xf0, 0x56, 0x4e, 0x17, 0x97, 0x6b, 0xb6, 0x6c, 0x43, 0x94, 0xed, 0x09, 0x87, 0xf5, 0x48, 0x29, 0x6b, 0xe1, 0x36, 0x23, 0x95, 0x7b, 0x7b, 0x54, 0x85, 0xc0, 0x5d, 0xf3, 0x0a, 0xcf, 0x4d, 0x23, 0x16, 0x66, 0x70, 0xaf, 0x76, 0xfa, 0x6f, 0x33, 0x56, 0xae, 0x17, 0x00}}},
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+ {{0x01, 0xc6, 0x1e, 0xa0, 0xaa, 0xe5, 0xda, 0x91, 0x33, 0x8c, 0x59, 0xc2, 0xc5, 0x9d, 0xf2, 0x08, 0x22, 0x32, 0x08, 0x86, 0xc0, 0xa3, 0x22, 0x00, 0x76, 0x6e, 0xa5, 0xb2, 0x3c, 0x0c, 0x4d, 0x4b, 0x4f, 0x78, 0x41, 0x22, 0xf4, 0x93, 0x71, 0x3b, 0xb5, 0x19, 0xa4, 0xb2, 0x09, 0x06, 0xaf, 0x0d, 0xda, 0xf6, 0x0a, 0xff, 0xd4, 0x48, 0x7d, 0x11, 0x00}}},
+{{{0x54, 0xd4, 0xd2, 0xea, 0x75, 0xf3, 0x5c, 0x28, 0x5d, 0x70, 0x68, 0x86, 0x4e, 0x09, 0x6f, 0x0b, 0xc8, 0xc4, 0x42, 0xa7, 0x57, 0x69, 0x22, 0x58, 0xa5, 0xa8, 0x5c, 0x68, 0x55, 0xc9, 0x89, 0x10, 0x6e, 0x1b, 0xbc, 0x6b, 0xb0, 0xe0, 0x1c, 0x2b, 0xde, 0x0f, 0x7d, 0x55, 0x1d, 0xc7, 0x8e, 0xd4, 0xbe, 0xfc, 0xd2, 0x97, 0x69, 0x5a, 0xb8, 0x2d, 0x00}},
+ {{0xc8, 0xb6, 0xeb, 0xa6, 0x13, 0x29, 0xc0, 0x3a, 0x39, 0x65, 0x75, 0x40, 0xa0, 0xa7, 0x16, 0xdc, 0xea, 0x7e, 0xab, 0xf3, 0x7e, 0xc9, 0xbd, 0xd6, 0xf3, 0x1c, 0xce, 0x5b, 0x75, 0x16, 0x61, 0xbb, 0xae, 0xf0, 0x43, 0x4b, 0x72, 0x3a, 0x6a, 0x25, 0xdc, 0xe5, 0xbe, 0x8f, 0x73, 0x31, 0x1b, 0xdc, 0x52, 0xca, 0x43, 0xf7, 0x2f, 0xe4, 0x9f, 0x54, 0x00}}},
+{{{0x56, 0x8a, 0x7e, 0xae, 0x05, 0x16, 0x9e, 0x10, 0x2e, 0xcd, 0xad, 0x60, 0x50, 0xdb, 0xdd, 0x66, 0x9f, 0x23, 0x9e, 0xf5, 0x13, 0x11, 0x7d, 0xe1, 0x81, 0x1f, 0xf7, 0x2f, 0x50, 0x8c, 0xd6, 0x98, 0x01, 0xa6, 0x44, 0x22, 0x63, 0x3f, 0x17, 0x67, 0xa2, 0x1d, 0x94, 0x81, 0x71, 0x99, 0xbd, 0x32, 0x9a, 0x0c, 0x24, 0x5a, 0xc0, 0x52, 0x19, 0xcd, 0x00}},
+ {{0x25, 0x2d, 0x64, 0x6f, 0x61, 0x78, 0xa4, 0xac, 0xb9, 0x3b, 0xf3, 0xe0, 0x5b, 0xd1, 0x8d, 0xfe, 0x74, 0xda, 0x5d, 0x83, 0x60, 0xfa, 0x6b, 0xf6, 0xd8, 0x1e, 0xf0, 0xa6, 0xed, 0xa8, 0x6c, 0x41, 0x81, 0x6e, 0xaa, 0x9b, 0xdf, 0x5d, 0x72, 0x25, 0x73, 0x70, 0xc5, 0xf0, 0xa8, 0x65, 0x57, 0xb4, 0x63, 0x7f, 0xf6, 0x78, 0x59, 0x3e, 0xea, 0xc8, 0x00}}},
+{{{0xa9, 0x27, 0xd4, 0x00, 0x79, 0xf1, 0x2e, 0x4e, 0x04, 0x02, 0xea, 0xa1, 0x47, 0xd9, 0xdc, 0xc2, 0x16, 0x15, 0xb9, 0x2d, 0x50, 0x77, 0x1a, 0xd6, 0x36, 0x60, 0xa9, 0x4c, 0xb4, 0x0a, 0x95, 0x23, 0xbe, 0xd6, 0x35, 0xf2, 0xdf, 0xa3, 0x52, 0x43, 0x46, 0xf0, 0x9b, 0x15, 0x44, 0xcb, 0xcc, 0x53, 0x9f, 0x9f, 0x59, 0x42, 0x90, 0x4d, 0x80, 0xbd, 0x00}},
+ {{0xf7, 0xdb, 0xbc, 0xcd, 0x01, 0x5b, 0x2c, 0xbe, 0x18, 0x96, 0x92, 0x18, 0xce, 0x46, 0x46, 0x34, 0x64, 0x9c, 0x02, 0x75, 0x7a, 0xe8, 0xf3, 0x9f, 0xac, 0x33, 0x73, 0xee, 0x07, 0x16, 0x0d, 0x5b, 0xa5, 0x22, 0xaf, 0x8d, 0x59, 0x9d, 0x9e, 0x58, 0xe2, 0x09, 0xb9, 0x7d, 0x0b, 0xf8, 0x78, 0x65, 0xf3, 0x2c, 0xa9, 0xe3, 0x59, 0xf6, 0x5c, 0xab, 0x00}}},
+{{{0xfd, 0x1f, 0xa8, 0x15, 0x17, 0x1c, 0x16, 0x83, 0xf1, 0x6b, 0x5a, 0x29, 0x9c, 0x32, 0xad, 0x37, 0xe6, 0x1d, 0xb4, 0xad, 0x41, 0xdb, 0xd7, 0x47, 0x63, 0xdf, 0x9e, 0xd5, 0xfa, 0xcd, 0xdc, 0x54, 0x3b, 0x12, 0xde, 0x0f, 0x9a, 0x05, 0xdd, 0x67, 0x54, 0xb9, 0x6c, 0x54, 0x43, 0xc1, 0xd8, 0xa5, 0xaa, 0x66, 0xa5, 0x43, 0xbf, 0x7f, 0x6b, 0x3f, 0x00}},
+ {{0xc2, 0x15, 0x1f, 0x3f, 0xc3, 0xa3, 0xc3, 0xe6, 0x10, 0x03, 0x66, 0xe1, 0x21, 0x4f, 0x72, 0x52, 0x5c, 0x7f, 0x65, 0x03, 0x4f, 0x5a, 0x02, 0xe3, 0x86, 0x8d, 0xf6, 0x8b, 0xfe, 0x6d, 0x45, 0xed, 0x16, 0x77, 0x86, 0x4d, 0x27, 0x23, 0xb6, 0x08, 0x85, 0x4a, 0x48, 0x1d, 0x31, 0xd3, 0x88, 0xf6, 0x4e, 0x47, 0xd6, 0x1e, 0xa8, 0x39, 0x42, 0xfa, 0x00}}},
+{{{0xe5, 0x9b, 0x3e, 0x65, 0x41, 0xbd, 0x49, 0x57, 0x6f, 0x64, 0x26, 0xc0, 0x3a, 0xee, 0x61, 0xf6, 0x30, 0xb6, 0x09, 0xbf, 0x0d, 0xb4, 0xdb, 0x16, 0xc3, 0xf5, 0xd8, 0x4c, 0x93, 0x3e, 0x78, 0x3a, 0x8a, 0xdf, 0xfb, 0x9f, 0xb1, 0xba, 0x7b, 0x10, 0x0d, 0x58, 0xfd, 0xbd, 0x10, 0xdf, 0xe7, 0x5f, 0x9d, 0x09, 0x3a, 0xbd, 0xa3, 0x26, 0x24, 0x29, 0x00}},
+ {{0x5b, 0xc7, 0xc2, 0x85, 0x7c, 0xfe, 0xe1, 0xac, 0x43, 0xf8, 0x4c, 0xf7, 0x01, 0xb6, 0xba, 0x91, 0x40, 0xe0, 0xdc, 0xf4, 0x01, 0xb5, 0xdb, 0x8d, 0x58, 0xd4, 0x01, 0x97, 0x62, 0xd9, 0xeb, 0x2f, 0xaa, 0x10, 0x5c, 0x0d, 0x8f, 0x83, 0x02, 0xfa, 0x24, 0x64, 0x7f, 0xec, 0x50, 0xa0, 0x80, 0xac, 0x3e, 0xd0, 0xa8, 0xc5, 0xaa, 0x66, 0xf3, 0x5e, 0x00}}},
+{{{0xb6, 0xd0, 0x23, 0xa7, 0xc0, 0x05, 0x98, 0xa0, 0x0a, 0xe0, 0xdb, 0x7d, 0x19, 0xe1, 0x02, 0x18, 0x65, 0x2b, 0xdb, 0x0f, 0x39, 0x30, 0x60, 0x4d, 0x5f, 0xd9, 0x80, 0x05, 0x18, 0xec, 0xf0, 0xad, 0x99, 0x92, 0x87, 0xa6, 0x39, 0x7e, 0x5f, 0xc6, 0xf3, 0xb6, 0xd7, 0x9d, 0x6a, 0x58, 0xc3, 0x3a, 0x24, 0x85, 0xde, 0x10, 0x04, 0x71, 0x4e, 0x5f, 0x00}},
+ {{0x14, 0x66, 0x08, 0x1b, 0xee, 0xa8, 0x22, 0xc0, 0x01, 0xf7, 0xc0, 0x97, 0x28, 0xa9, 0x48, 0x68, 0xe3, 0x08, 0x9f, 0xb4, 0x7d, 0x2d, 0xbb, 0xf7, 0xeb, 0xee, 0x7b, 0xca, 0x0f, 0x98, 0x0a, 0xb2, 0x36, 0x9b, 0xe2, 0xbc, 0xa3, 0x75, 0xba, 0x05, 0x3d, 0x35, 0x2b, 0xe9, 0x39, 0x73, 0xf4, 0x95, 0x1d, 0x19, 0x8b, 0x27, 0x1b, 0x93, 0x3c, 0xa5, 0x00}}},
+{{{0x65, 0x0f, 0xf8, 0x91, 0x6e, 0x91, 0x9c, 0x6b, 0xcf, 0xb7, 0x67, 0xbd, 0x97, 0xb0, 0x79, 0x3e, 0xbd, 0xab, 0x72, 0x9c, 0x9f, 0x72, 0xe5, 0x03, 0x47, 0x0e, 0x8b, 0x72, 0x59, 0x0e, 0x43, 0x4a, 0x4a, 0x51, 0x1e, 0x58, 0x80, 0x01, 0x6b, 0x30, 0x14, 0xa1, 0xc8, 0xd1, 0xe1, 0x33, 0x71, 0xd0, 0xbf, 0x2c, 0x89, 0x48, 0x10, 0xa8, 0x0e, 0x2a, 0x00}},
+ {{0x15, 0xcb, 0x27, 0xc1, 0xe2, 0x6f, 0x09, 0x58, 0xf1, 0xc5, 0x76, 0x8c, 0x5a, 0xd9, 0x32, 0xe8, 0x15, 0x83, 0xc8, 0x65, 0xce, 0x61, 0x78, 0x96, 0xfc, 0x82, 0x90, 0xb5, 0x39, 0x7e, 0xef, 0x7c, 0xe7, 0x07, 0x89, 0xf7, 0x6f, 0x67, 0xac, 0x3e, 0xdb, 0x81, 0x91, 0xff, 0x23, 0x75, 0x7f, 0x32, 0x3d, 0xf9, 0x6a, 0x39, 0x2c, 0xd4, 0x47, 0x6b, 0x00}}},
+{{{0x75, 0x2c, 0x8d, 0x19, 0xd4, 0x97, 0x40, 0xbd, 0x95, 0x44, 0x4c, 0xe8, 0x99, 0x21, 0x5c, 0x17, 0x0d, 0x5f, 0x46, 0x9d, 0x02, 0x7d, 0x5a, 0x9c, 0x92, 0xb2, 0xce, 0xd4, 0x84, 0xa8, 0xd5, 0x6c, 0x61, 0xf7, 0x48, 0xc1, 0xe5, 0xfe, 0xd2, 0x67, 0x58, 0x17, 0xcf, 0x3c, 0xb5, 0xe5, 0x1a, 0x70, 0x60, 0x44, 0xb2, 0x75, 0x8d, 0x4d, 0x4e, 0x26, 0x00}},
+ {{0x28, 0x8e, 0xf5, 0x5b, 0xee, 0x73, 0x6b, 0xd7, 0x28, 0xd1, 0x81, 0x3c, 0x67, 0xcd, 0x4f, 0xd6, 0x46, 0xd2, 0x07, 0x41, 0xfd, 0xa4, 0x2b, 0xe4, 0xb9, 0xfb, 0xcb, 0x9e, 0xa7, 0x7f, 0xe9, 0x42, 0xa1, 0x82, 0x8a, 0x7b, 0x70, 0x5e, 0xc4, 0x11, 0x30, 0x8b, 0x40, 0xbe, 0x16, 0x00, 0x3d, 0x6f, 0xd7, 0x14, 0x64, 0xa1, 0xc0, 0x0f, 0xda, 0x74, 0x00}}},
+{{{0x48, 0x2d, 0x0f, 0x00, 0xc5, 0x65, 0x10, 0x94, 0x79, 0x79, 0x1b, 0x48, 0x32, 0x9a, 0xa2, 0x44, 0x68, 0xed, 0xcf, 0xde, 0x33, 0xfb, 0xaf, 0xa2, 0xfc, 0x4a, 0x29, 0x40, 0xd5, 0x36, 0x31, 0x43, 0xf4, 0x57, 0xae, 0x87, 0xcc, 0x7d, 0x3c, 0xc2, 0xd7, 0x44, 0xc7, 0x44, 0xe1, 0x42, 0x52, 0x88, 0x82, 0x78, 0xb7, 0x65, 0x17, 0xb5, 0xc7, 0xd7, 0x00}},
+ {{0x55, 0xe4, 0x74, 0xb7, 0xb0, 0x5e, 0x26, 0x7c, 0xd9, 0x30, 0x60, 0x68, 0xa3, 0xfc, 0xc0, 0xe5, 0x02, 0xf1, 0x57, 0x9f, 0x6a, 0x4d, 0x74, 0xa3, 0xc4, 0xa7, 0x20, 0x15, 0x50, 0x90, 0x2f, 0xc8, 0x43, 0x86, 0x2f, 0x77, 0x9f, 0xca, 0xd7, 0x19, 0xa9, 0xe8, 0xe8, 0x26, 0x7a, 0x8f, 0xa1, 0xb6, 0xc2, 0x2d, 0x46, 0x3c, 0x69, 0xfd, 0xe8, 0x31, 0x00}}},
+{{{0x73, 0x80, 0xa7, 0xca, 0xf3, 0x4a, 0x87, 0x8d, 0x8c, 0xa8, 0x8b, 0x32, 0x4d, 0x3e, 0x0d, 0x70, 0xf1, 0x96, 0x96, 0x8e, 0x25, 0x8c, 0xd4, 0xcb, 0x17, 0xa0, 0xa2, 0xce, 0x69, 0x43, 0xe3, 0x36, 0x35, 0xf8, 0x0e, 0x5e, 0x4a, 0x24, 0x32, 0xf3, 0x0e, 0x39, 0x2f, 0x08, 0x4c, 0x5d, 0x04, 0xca, 0x50, 0x91, 0x59, 0x7c, 0x9c, 0xd9, 0x7b, 0x01, 0x00}},
+ {{0xdd, 0xe6, 0x9e, 0xae, 0xa2, 0x88, 0x7a, 0x73, 0x50, 0xed, 0x0c, 0xc3, 0x24, 0x31, 0x67, 0x80, 0x7f, 0x5b, 0xab, 0x22, 0x56, 0xa4, 0x11, 0x56, 0x6e, 0xfe, 0x69, 0x27, 0x51, 0x4b, 0x58, 0x13, 0xb6, 0xd2, 0x85, 0xf3, 0xbf, 0xf0, 0xc0, 0x9d, 0x32, 0xf3, 0x54, 0x05, 0x59, 0x53, 0xe5, 0xcc, 0xac, 0x50, 0xfc, 0x49, 0xf2, 0x74, 0x2d, 0x16, 0x00}}},
+{{{0x4b, 0x3d, 0x88, 0xc3, 0x92, 0x8a, 0x14, 0x69, 0xe9, 0x16, 0xdf, 0x38, 0xde, 0xb0, 0xe5, 0x92, 0xcf, 0x60, 0xab, 0xe2, 0x6b, 0xed, 0x9d, 0xf6, 0x51, 0x75, 0x5a, 0x79, 0x34, 0xa8, 0xac, 0x0a, 0xb3, 0x1d, 0xb3, 0xf5, 0xb3, 0x0d, 0x35, 0xa8, 0x66, 0x66, 0xfc, 0x6e, 0x8d, 0xec, 0x53, 0x22, 0x7a, 0xa0, 0x8a, 0x64, 0xa9, 0xa1, 0xd6, 0x17, 0x00}},
+ {{0x48, 0x41, 0xb5, 0xd7, 0x1c, 0x62, 0x99, 0x63, 0x8b, 0xef, 0x25, 0x44, 0x03, 0xa1, 0x43, 0xc0, 0xe5, 0xf2, 0x04, 0x90, 0x63, 0x3a, 0x0a, 0x6f, 0x09, 0x47, 0x12, 0x99, 0x5b, 0x04, 0xd6, 0xdd, 0xaa, 0xe8, 0x89, 0x32, 0x49, 0xdc, 0x9d, 0x91, 0x50, 0x3f, 0x64, 0xf3, 0xd2, 0xfd, 0xa5, 0x9c, 0xdb, 0xf1, 0xde, 0x9c, 0x94, 0x2d, 0xcf, 0xb9, 0x00}}},
+{{{0x81, 0x68, 0x4a, 0xbc, 0x3b, 0xd3, 0x48, 0x00, 0x61, 0xfb, 0x79, 0x0f, 0xf9, 0x8c, 0xc9, 0x7c, 0xe9, 0xf0, 0x67, 0x10, 0x0c, 0xe4, 0x6b, 0xa1, 0x78, 0x1c, 0x30, 0xf7, 0x0e, 0x8d, 0xe9, 0x45, 0xf4, 0xc6, 0x8c, 0x9b, 0x63, 0x8c, 0x6c, 0xf4, 0xbc, 0x3c, 0xec, 0xb9, 0xb9, 0x22, 0xbe, 0xad, 0x0d, 0xf1, 0x56, 0xbd, 0xcd, 0x6e, 0x9e, 0x4a, 0x00}},
+ {{0x2f, 0x79, 0xd7, 0xf4, 0x36, 0x8c, 0x14, 0x2e, 0x21, 0x97, 0x4c, 0x8e, 0x54, 0x28, 0x36, 0x27, 0xc2, 0xdb, 0x9f, 0x54, 0x6d, 0xee, 0x11, 0xf4, 0xeb, 0x2d, 0xa9, 0x5e, 0x75, 0x10, 0xbd, 0xdb, 0xaf, 0x6e, 0x01, 0x9d, 0xa2, 0xb2, 0x67, 0x46, 0x93, 0x96, 0x8e, 0x07, 0x98, 0xb4, 0x06, 0x2c, 0x08, 0x63, 0xe6, 0x2f, 0xb8, 0x62, 0x1f, 0x8f, 0x00}}},
+{{{0x6c, 0x29, 0x6c, 0x91, 0x2c, 0x65, 0x9f, 0xb1, 0xff, 0x91, 0x1a, 0x8e, 0x6a, 0x6b, 0xf9, 0x5b, 0x25, 0xf7, 0x34, 0x41, 0x87, 0x6c, 0xa2, 0x25, 0x10, 0x0d, 0x75, 0x7d, 0xd4, 0xd1, 0xdb, 0x82, 0x8b, 0xed, 0x77, 0x80, 0x15, 0xb7, 0x0f, 0xba, 0x7d, 0x9a, 0x69, 0x75, 0xae, 0xe7, 0xdc, 0xbd, 0x57, 0x8b, 0x94, 0xbf, 0xa6, 0x62, 0x3a, 0xa9, 0x00}},
+ {{0xc7, 0x36, 0xc1, 0x36, 0xf7, 0xd0, 0x66, 0x79, 0xaf, 0xbb, 0x21, 0xa3, 0xb4, 0x82, 0x58, 0x09, 0x28, 0x3c, 0x6c, 0x8c, 0xcb, 0xff, 0xcb, 0xd3, 0xd6, 0xd1, 0xcf, 0x63, 0x91, 0x4a, 0xb2, 0xbc, 0x4e, 0x3b, 0x0b, 0x97, 0x4f, 0xe0, 0xda, 0x30, 0x3a, 0x58, 0xb5, 0x56, 0x14, 0xad, 0x47, 0x04, 0x72, 0xda, 0xad, 0x64, 0x2e, 0xd9, 0xb4, 0x0d, 0x00}}},
+{{{0x01, 0xcb, 0xa2, 0x8f, 0xe9, 0x01, 0xe0, 0x19, 0xa5, 0xc3, 0xa3, 0x58, 0x84, 0x83, 0xe1, 0x58, 0x80, 0xe9, 0x1f, 0x06, 0xc4, 0x6b, 0xec, 0xf8, 0x02, 0x10, 0x67, 0xac, 0x05, 0x3e, 0x24, 0x83, 0xff, 0xd5, 0x45, 0x01, 0xf9, 0xab, 0x5f, 0xce, 0x55, 0x73, 0xac, 0x70, 0x5a, 0x96, 0x26, 0x93, 0xd9, 0xdb, 0x32, 0xd8, 0x0f, 0x9a, 0x92, 0x4a, 0x00}},
+ {{0x21, 0x16, 0x60, 0x99, 0xf1, 0x73, 0x76, 0x65, 0x55, 0x39, 0x59, 0x16, 0x83, 0x7c, 0x23, 0x2d, 0x61, 0x0c, 0x3f, 0x82, 0x09, 0xe9, 0x42, 0x9e, 0x7a, 0x0e, 0x4b, 0x41, 0xa2, 0xe0, 0x83, 0x5c, 0x77, 0x98, 0x74, 0x52, 0x01, 0xcd, 0x65, 0x34, 0xa2, 0x05, 0xc3, 0xbf, 0xcb, 0xf4, 0x07, 0x46, 0x65, 0x5d, 0xc7, 0xb1, 0xd1, 0x3e, 0xea, 0x4d, 0x00}}} \ No newline at end of file
diff --git a/ge448_gen.c b/ge448_gen.c
new file mode 100644
index 0000000..dfe5ed3
--- /dev/null
+++ b/ge448_gen.c
@@ -0,0 +1,99 @@
+#include "ge448.h"
+#include <stdio.h>
+#include <assert.h>
+
+const ge448 ge448_base = {{{0x5E, 0xC0, 0x0C, 0xC7, 0x2B, 0xA8, 0x26, 0x26, 0x8E, 0x93, 0x00, 0x8B, 0xE1, 0x80, 0x3B, 0x43,
+ 0x11, 0x65, 0xB6, 0x2A, 0xF7, 0x1A, 0xAE, 0x12, 0x64, 0xA4, 0xD3, 0xA3, 0x24, 0xE3, 0x6D, 0xEA,
+ 0x67, 0x17, 0x0F, 0x47, 0x70, 0x65, 0x14, 0x9E, 0xDA, 0x36, 0xBF, 0x22, 0xA6, 0x15, 0x1D, 0x22,
+ 0xED, 0x0D, 0xED, 0x6B, 0xC6, 0x70, 0x19, 0x4F, 0x00}},
+ {{0x14, 0xFA, 0x30, 0xF2, 0x5B, 0x79, 0x08, 0x98, 0xAD, 0xC8, 0xD7, 0x4E, 0x2C, 0x13, 0xBD, 0xFD,
+ 0xC4, 0x39, 0x7C, 0xE6, 0x1C, 0xFF, 0xD3, 0x3A, 0xD7, 0xC2, 0xA0, 0x05, 0x1E, 0x9C, 0x78, 0x87,
+ 0x40, 0x98, 0xA3, 0x6C, 0x73, 0x73, 0xEA, 0x4B, 0x62, 0xC7, 0xC9, 0x56, 0x37, 0x20, 0x76, 0x88,
+ 0x24, 0xBC, 0xB6, 0x6E, 0x71, 0x46, 0x3F, 0x69, 0x00}},
+ {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}};
+
+#define ge448_p3 ge448 /* use projective coordinates only */
+
+static const fe448 ge448_ecd = {{0x56, 0x67, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00}};
+
+static void add_p3(ge448_p3 *r, const ge448_p3 *p, const ge448_p3 *q)
+{
+ fe448 a, b, c, d, e, f, g, h;
+ fe448_mul(&a, &p->z, &q->z); /* A=Z1*Z2 */
+ fe448_square(&b, &a); /* B=A^2 */
+ fe448_mul(&c, &p->x, &q->x); /* C=X1*X2 */
+ fe448_mul(&d, &p->y, &q->y); /* D=Y1*Y2 */
+ fe448_mul(&e, &c, &d);
+ fe448_mul(&e, &e, &ge448_ecd); /* E=d*C*D */
+ fe448_sub(&f, &b, &e); /* F=B-E */
+ fe448_add(&g, &b, &e); /* G=B+E */
+ fe448_add(&b, &p->x, &p->y);
+ fe448_add(&h, &q->x, &q->y);
+ fe448_mul(&h, &b, &h); /* H=(X1+Y1)*(X2+Y2) */
+ fe448_sub(&h, &h, &c);
+ fe448_sub(&h, &h, &d);
+ fe448_mul(&h, &h, &f);
+ fe448_mul(&r->x, &a, &h); /* X3=A*F*(H-C-D) */
+ fe448_sub(&d, &d, &c);
+ fe448_mul(&d, &d, &g);
+ fe448_mul(&r->y, &a, &d); /* Y3=A*G*(D-C) */
+ fe448_mul(&r->z, &f, &g); /* Z3=F*G */
+}
+
+static void output(ge448_p3 *p)
+{
+ int i, printed;
+ fe448 tx, ty, zi;
+ unsigned char px[57], py[57];
+ char buffer[4000];
+ fe448_invert(&zi, &p->z);
+ fe448_mul(&tx, &p->x, &zi);
+ fe448_mul(&ty, &p->y, &zi);
+ fe448_pack(px, &tx);
+ fe448_pack(py, &ty);
+ printed = 0;
+ printed += sprintf(buffer + printed, "{{{0x%02x", px[0]);
+ for (i = 1; i < 57; i++)
+ {
+ printed += sprintf(buffer + printed, ", 0x%02x", px[i]);
+ }
+ printed += sprintf(buffer + printed, "}},\n");
+ printed += sprintf(buffer + printed, " {{0x%02x", py[0]);
+ for (i = 1; i < 57; i++)
+ {
+ printed += sprintf(buffer + printed, ", 0x%02x", py[i]);
+ }
+ printed += sprintf(buffer + printed, "}}}");
+ assert(fwrite(buffer, 1, printed, stdout) == printed);
+}
+
+int main()
+{
+ int i;
+ ge448 neutral = {{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
+ {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
+ {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}};
+ output(&neutral);
+ for (i = 1; i < 745; i++)
+ {
+ add_p3(&neutral, &neutral, &ge448_base);
+ puts(",");
+ output(&neutral);
+ }
+ return 0;
+} \ No newline at end of file
diff --git a/sc448.c b/sc448.c
index 75ea623..9a0f58c 100644
--- a/sc448.c
+++ b/sc448.c
@@ -198,94 +198,55 @@ void sc448_mul(sc448 *r, const sc448 *x, const sc448 *y)
barrett_reduce(r, t);
}
- /* not implemented yet */
-#if 0
-void sc25519_window3(signed char r[85], const sc25519 *s)
-{
- char carry;
- int i;
- for (i = 0; i < 10; i++)
- {
- r[8 * i + 0] = s->v[3 * i + 0] & 7;
- r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
- r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
- r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
- r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
- r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;
- r[8 * i + 5] = (s->v[3 * i + 1] >> 7) & 7;
- r[8 * i + 5] ^= (s->v[3 * i + 2] << 1) & 7;
- r[8 * i + 6] = (s->v[3 * i + 2] >> 2) & 7;
- r[8 * i + 7] = (s->v[3 * i + 2] >> 5) & 7;
- }
- r[8 * i + 0] = s->v[3 * i + 0] & 7;
- r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
- r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
- r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
- r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
- r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;
-
- /* Making it signed */
- carry = 0;
- for (i = 0; i < 84; i++)
- {
- r[i] += carry;
- r[i + 1] += r[i] >> 3;
- r[i] &= 7;
- carry = r[i] >> 2;
- r[i] -= carry << 3;
- }
- r[84] += carry;
-}
-void sc25519_window5(signed char r[51], const sc25519 *s)
+void sc448_window3(signed char r[149], const sc448 *s)
{
- char carry;
- int i;
- for (i = 0; i < 6; i++)
- {
- r[8 * i + 0] = s->v[5 * i + 0] & 31;
- r[8 * i + 1] = (s->v[5 * i + 0] >> 5) & 31;
- r[8 * i + 1] ^= (s->v[5 * i + 1] << 3) & 31;
- r[8 * i + 2] = (s->v[5 * i + 1] >> 2) & 31;
- r[8 * i + 3] = (s->v[5 * i + 1] >> 7) & 31;
- r[8 * i + 3] ^= (s->v[5 * i + 2] << 1) & 31;
- r[8 * i + 4] = (s->v[5 * i + 2] >> 4) & 31;
- r[8 * i + 4] ^= (s->v[5 * i + 3] << 4) & 31;
- r[8 * i + 5] = (s->v[5 * i + 3] >> 1) & 31;
- r[8 * i + 6] = (s->v[5 * i + 3] >> 6) & 31;
- r[8 * i + 6] ^= (s->v[5 * i + 4] << 2) & 31;
- r[8 * i + 7] = (s->v[5 * i + 4] >> 3) & 31;
- }
- r[8 * i + 0] = s->v[5 * i + 0] & 31;
- r[8 * i + 1] = (s->v[5 * i + 0] >> 5) & 31;
- r[8 * i + 1] ^= (s->v[5 * i + 1] << 3) & 31;
- r[8 * i + 2] = (s->v[5 * i + 1] >> 2) & 31;
-
- /* Making it signed */
- carry = 0;
- for (i = 0; i < 50; i++)
- {
- r[i] += carry;
- r[i + 1] += r[i] >> 5;
- r[i] &= 31;
- carry = r[i] >> 4;
- r[i] -= carry << 5;
- }
- r[50] += carry;
+ char carry;
+ int i;
+ for (i = 0; i < 18; i++)
+ {
+ r[8 * i + 0] = s->v[3 * i + 0] & 7;
+ r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
+ r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
+ r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
+ r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
+ r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;
+ r[8 * i + 5] = (s->v[3 * i + 1] >> 7) & 7;
+ r[8 * i + 5] ^= (s->v[3 * i + 2] << 1) & 7;
+ r[8 * i + 6] = (s->v[3 * i + 2] >> 2) & 7;
+ r[8 * i + 7] = (s->v[3 * i + 2] >> 5) & 7;
+ }
+ r[8 * i + 0] = s->v[3 * i + 0] & 7;
+ r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
+ r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
+ r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
+ r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
+ r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;
+
+ /* Making it signed */
+ carry = 0;
+ for (i = 0; i < 148; i++)
+ {
+ r[i] += carry;
+ r[i + 1] += r[i] >> 3;
+ r[i] &= 7;
+ carry = r[i] >> 2;
+ r[i] -= carry << 3;
+ }
+ r[148] += carry;
}
-void sc25519_2interleave2(unsigned char r[127], const sc25519 *s1, const sc25519 *s2)
+void sc448_2interleave2(unsigned char r[223], const sc448 *s1, const sc448 *s2)
{
- int i;
- for (i = 0; i < 31; i++)
- {
- r[4 * i] = (s1->v[i] & 3) ^ ((s2->v[i] & 3) << 2);
- r[4 * i + 1] = ((s1->v[i] >> 2) & 3) ^ (((s2->v[i] >> 2) & 3) << 2);
- r[4 * i + 2] = ((s1->v[i] >> 4) & 3) ^ (((s2->v[i] >> 4) & 3) << 2);
- r[4 * i + 3] = ((s1->v[i] >> 6) & 3) ^ (((s2->v[i] >> 6) & 3) << 2);
- }
- r[124] = (s1->v[31] & 3) ^ ((s2->v[31] & 3) << 2);
- r[125] = ((s1->v[31] >> 2) & 3) ^ (((s2->v[31] >> 2) & 3) << 2);
- r[126] = ((s1->v[31] >> 4) & 3) ^ (((s2->v[31] >> 4) & 3) << 2);
+ int i;
+ for (i = 0; i < 55; i++)
+ {
+ r[4 * i] = (s1->v[i] & 3) ^ ((s2->v[i] & 3) << 2);
+ r[4 * i + 1] = ((s1->v[i] >> 2) & 3) ^ (((s2->v[i] >> 2) & 3) << 2);
+ r[4 * i + 2] = ((s1->v[i] >> 4) & 3) ^ (((s2->v[i] >> 4) & 3) << 2);
+ r[4 * i + 3] = ((s1->v[i] >> 6) & 3) ^ (((s2->v[i] >> 6) & 3) << 2);
+ }
+ r[220] = (s1->v[55] & 3) ^ ((s2->v[55] & 3) << 2);
+ r[221] = ((s1->v[55] >> 2) & 3) ^ (((s2->v[55] >> 2) & 3) << 2);
+ r[222] = ((s1->v[55] >> 4) & 3) ^ (((s2->v[55] >> 4) & 3) << 2);
}
-#endif
diff --git a/sc448.h b/sc448.h
index 86d1c31..245763e 100644
--- a/sc448.h
+++ b/sc448.h
@@ -14,7 +14,6 @@
#define sc448_sub_nored crypto_sign_ed448_ref_sc448_sub_nored
#define sc448_mul crypto_sign_ed448_ref_sc448_mul
#define sc448_window3 crypto_sign_ed448_ref_sc448_window3
-#define sc448_window5 crypto_sign_ed448_ref_sc448_window5
#define sc448_2interleave2 crypto_sign_ed448_ref_sc448_2interleave2
typedef struct
@@ -41,17 +40,11 @@ void sc448_sub_nored(sc448 *r, const sc448 *x, const sc448 *y);
void sc448_mul(sc448 *r, const sc448 *x, const sc448 *y);
-#if 0
/* Convert s into a representation of the form \sum_{i=0}^{84}r[i]2^3
* with r[i] in {-4,...,3}
*/
-void sc448_window3(signed char r[85], const sc448 *s);
+void sc448_window3(signed char r[149], const sc448 *s);
-/* Convert s into a representation of the form \sum_{i=0}^{50}r[i]2^5
- * with r[i] in {-16,...,15}
- */
-void sc448_window5(signed char r[51], const sc448 *s);
+void sc448_2interleave2(unsigned char r[223], const sc448 *s1, const sc448 *s2);
-void sc448_2interleave2(unsigned char r[127], const sc448 *s1, const sc448 *s2);
-#endif
-#endif
+#endif \ No newline at end of file