6 * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
7 * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
8 * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
26 #define ge25519_p3 ge25519
28 /* Windowsize for fixed-window scalar multiplication */
29 #define WINDOWSIZE 2 /* Should be 1,2, or 4 */
30 #define WINDOWMASK ((1<<WINDOWSIZE)-1)
32 /* packed parameter d in the Edwards curve equation */
33 static const unsigned char ecd
[32] = {0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
34 0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52};
36 /* Packed coordinates of the base point */
37 static const unsigned char ge25519_base_x
[32] = {0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
38 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21};
39 static const unsigned char ge25519_base_y
[32] = {0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
40 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66};
41 static const unsigned char ge25519_base_z
[32] = {1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
42 static const unsigned char ge25519_base_t
[32] = {0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
43 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67};
45 /* Packed coordinates of the neutral element */
46 static const unsigned char ge25519_neutral_x
[32] = {0};
47 static const unsigned char ge25519_neutral_y
[32] = {1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
48 static const unsigned char ge25519_neutral_z
[32] = {1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
49 static const unsigned char ge25519_neutral_t
[32] = {0};
51 static void p1p1_to_p2(ge25519_p2
*r
, const ge25519_p1p1
*p
)
53 fe25519_mul(&r
->x
, &p
->x
, &p
->t
);
54 fe25519_mul(&r
->y
, &p
->y
, &p
->z
);
55 fe25519_mul(&r
->z
, &p
->z
, &p
->t
);
58 static void p1p1_to_p3(ge25519_p3
*r
, const ge25519_p1p1
*p
)
60 p1p1_to_p2((ge25519_p2
*)r
, p
);
61 fe25519_mul(&r
->t
, &p
->x
, &p
->y
);
64 /* Constant-time version of: if(b) r = p */
65 static void cmov_p3(ge25519_p3
*r
, const ge25519_p3
*p
, unsigned char b
)
67 fe25519_cmov(&r
->x
, &p
->x
, b
);
68 fe25519_cmov(&r
->y
, &p
->y
, b
);
69 fe25519_cmov(&r
->z
, &p
->z
, b
);
70 fe25519_cmov(&r
->t
, &p
->t
, b
);
73 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
74 static void dbl_p1p1(ge25519_p1p1
*r
, const ge25519_p2
*p
)
77 fe25519_square(&a
, &p
->x
);
78 fe25519_square(&b
, &p
->y
);
79 fe25519_square(&c
, &p
->z
);
80 fe25519_add(&c
, &c
, &c
);
83 fe25519_add(&r
->x
, &p
->x
, &p
->y
);
84 fe25519_square(&r
->x
, &r
->x
);
85 fe25519_sub(&r
->x
, &r
->x
, &a
);
86 fe25519_sub(&r
->x
, &r
->x
, &b
);
87 fe25519_add(&r
->z
, &d
, &b
);
88 fe25519_sub(&r
->t
, &r
->z
, &c
);
89 fe25519_sub(&r
->y
, &d
, &b
);
92 static void add_p1p1(ge25519_p1p1
*r
, const ge25519_p3
*p
, const ge25519_p3
*q
)
94 fe25519 a
, b
, c
, d
, t
, fd
;
95 fe25519_unpack(&fd
, ecd
);
97 fe25519_sub(&a
, &p
->y
, &p
->x
); // A = (Y1-X1)*(Y2-X2)
98 fe25519_sub(&t
, &q
->y
, &q
->x
);
99 fe25519_mul(&a
, &a
, &t
);
100 fe25519_add(&b
, &p
->x
, &p
->y
); // B = (Y1+X1)*(Y2+X2)
101 fe25519_add(&t
, &q
->x
, &q
->y
);
102 fe25519_mul(&b
, &b
, &t
);
103 fe25519_mul(&c
, &p
->t
, &q
->t
); //C = T1*k*T2
104 fe25519_mul(&c
, &c
, &fd
);
105 fe25519_add(&c
, &c
, &c
); //XXX: Can save this addition by precomputing 2*ecd
106 fe25519_mul(&d
, &p
->z
, &q
->z
); //D = Z1*2*Z2
107 fe25519_add(&d
, &d
, &d
);
108 fe25519_sub(&r
->x
, &b
, &a
); // E = B-A
109 fe25519_sub(&r
->t
, &d
, &c
); // F = D-C
110 fe25519_add(&r
->z
, &d
, &c
); // G = D+C
111 fe25519_add(&r
->y
, &b
, &a
); // H = B+A
114 /* ********************************************************************
116 ******************************************************************** */
118 /* return 0 on success, -1 otherwise */
119 int ge25519_unpack_vartime(ge25519_p3
*r
, const unsigned char p
[32])
123 fe25519_setone(&r
->z
);
124 fe25519_unpack(&fd
, ecd
);
125 unsigned char par
= p
[31] >> 7;
126 fe25519_unpack(&r
->y
, p
);
127 fe25519_square(&r
->x
, &r
->y
);
128 fe25519_mul(&t
, &r
->x
, &fd
);
129 fe25519_sub(&r
->x
, &r
->x
, &r
->z
);
130 fe25519_add(&t
, &r
->z
, &t
);
131 fe25519_invert(&t
, &t
);
132 fe25519_mul(&r
->x
, &r
->x
, &t
);
133 ret
= fe25519_sqrt_vartime(&r
->x
, &r
->x
, par
);
134 fe25519_mul(&r
->t
, &r
->x
, &r
->y
);
138 void ge25519_pack(unsigned char r
[32], const ge25519_p3
*p
)
141 fe25519_invert(&zi
, &p
->z
);
142 fe25519_mul(&tx
, &p
->x
, &zi
);
143 fe25519_mul(&ty
, &p
->y
, &zi
);
144 fe25519_pack(r
, &ty
);
145 r
[31] ^= fe25519_getparity(&tx
) << 7;
148 void ge25519_add(ge25519_p3
*r
, const ge25519_p3
*p
, const ge25519_p3
*q
)
151 add_p1p1(&grp1p1
, p
, q
);
152 p1p1_to_p3(r
, &grp1p1
);
155 void ge25519_double(ge25519_p3
*r
, const ge25519_p3
*p
)
158 dbl_p1p1(&grp1p1
, (ge25519_p2
*)p
);
159 p1p1_to_p3(r
, &grp1p1
);
162 void ge25519_scalarmult(ge25519_p3
*r
, const ge25519_p3
*p
, const sc25519
*s
)
166 fe25519_unpack(&g
.x
, ge25519_neutral_x
);
167 fe25519_unpack(&g
.y
, ge25519_neutral_y
);
168 fe25519_unpack(&g
.z
, ge25519_neutral_z
);
169 fe25519_unpack(&g
.t
, ge25519_neutral_t
);
171 ge25519_p3 pre
[(1 << WINDOWSIZE
)];
175 unsigned char sb
[32];
176 sc25519_to32bytes(sb
, s
);
181 for(i
=2;i
<(1<<WINDOWSIZE
);i
+=2)
183 dbl_p1p1(&tp1p1
, (ge25519_p2
*)(pre
+i
/2));
184 p1p1_to_p3(pre
+i
, &tp1p1
);
185 add_p1p1(&tp1p1
, pre
+i
, pre
+1);
186 p1p1_to_p3(pre
+i
+1, &tp1p1
);
189 // Fixed-window scalar multiplication
192 for(j
=8-WINDOWSIZE
;j
>=0;j
-=WINDOWSIZE
)
194 for(k
=0;k
<WINDOWSIZE
-1;k
++)
196 dbl_p1p1(&tp1p1
, (ge25519_p2
*)&g
);
197 p1p1_to_p2((ge25519_p2
*)&g
, &tp1p1
);
199 dbl_p1p1(&tp1p1
, (ge25519_p2
*)&g
);
200 p1p1_to_p3(&g
, &tp1p1
);
201 // Cache-timing resistant loading of precomputed value:
202 w
= (sb
[i
-1]>>j
) & WINDOWMASK
;
204 for(k
=1;k
<(1<<WINDOWSIZE
);k
++)
205 cmov_p3(&t
, &pre
[k
], k
==w
);
207 add_p1p1(&tp1p1
, &g
, &t
);
208 if(j
!= 0) p1p1_to_p2((ge25519_p2
*)&g
, &tp1p1
);
209 else p1p1_to_p3(&g
, &tp1p1
); /* convert to p3 representation at the end */
218 void ge25519_scalarmult_base(ge25519_p3
*r
, const sc25519
*s
)
220 /* XXX: Better algorithm for known-base-point scalar multiplication */
222 fe25519_unpack(&t
.x
, ge25519_base_x
);
223 fe25519_unpack(&t
.y
, ge25519_base_y
);
224 fe25519_unpack(&t
.z
, ge25519_base_z
);
225 fe25519_unpack(&t
.t
, ge25519_base_t
);
226 ge25519_scalarmult(r
, &t
, s
);