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);
12 static const fe25519 ge25519_ecd
= {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
13 0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
15 static const fe25519 ge25519_ec2d
= {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
16 0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
18 static const fe25519 ge25519_sqrtm1
= {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
19 0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};
21 #define ge25519_p3 ge25519
45 /* Packed coordinates of the base point */
46 const ge25519 ge25519_base
= {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
47 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
48 {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
49 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
50 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
51 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
52 {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
53 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};
55 /* Multiples of the base point in affine representation */
56 static const ge25519_aff ge25519_base_multiples_affine
[425] = {
57 #include "ge25519_base.data"
60 static void p1p1_to_p2(ge25519_p2
*r
, const ge25519_p1p1
*p
)
62 fe25519_mul(&r
->x
, &p
->x
, &p
->t
);
63 fe25519_mul(&r
->y
, &p
->y
, &p
->z
);
64 fe25519_mul(&r
->z
, &p
->z
, &p
->t
);
67 static void p1p1_to_p3(ge25519_p3
*r
, const ge25519_p1p1
*p
)
69 p1p1_to_p2((ge25519_p2
*)r
, p
);
70 fe25519_mul(&r
->t
, &p
->x
, &p
->y
);
73 static void ge25519_mixadd2(ge25519_p3
*r
, const ge25519_aff
*q
)
75 fe25519 a
,b
,t1
,t2
,c
,d
,e
,f
,g
,h
,qt
;
76 fe25519_mul(&qt
, &q
->x
, &q
->y
);
77 fe25519_sub(&a
, &r
->y
, &r
->x
); /* A = (Y1-X1)*(Y2-X2) */
78 fe25519_add(&b
, &r
->y
, &r
->x
); /* B = (Y1+X1)*(Y2+X2) */
79 fe25519_sub(&t1
, &q
->y
, &q
->x
);
80 fe25519_add(&t2
, &q
->y
, &q
->x
);
81 fe25519_mul(&a
, &a
, &t1
);
82 fe25519_mul(&b
, &b
, &t2
);
83 fe25519_sub(&e
, &b
, &a
); /* E = B-A */
84 fe25519_add(&h
, &b
, &a
); /* H = B+A */
85 fe25519_mul(&c
, &r
->t
, &qt
); /* C = T1*k*T2 */
86 fe25519_mul(&c
, &c
, &ge25519_ec2d
);
87 fe25519_add(&d
, &r
->z
, &r
->z
); /* D = Z1*2 */
88 fe25519_sub(&f
, &d
, &c
); /* F = D-C */
89 fe25519_add(&g
, &d
, &c
); /* G = D+C */
90 fe25519_mul(&r
->x
, &e
, &f
);
91 fe25519_mul(&r
->y
, &h
, &g
);
92 fe25519_mul(&r
->z
, &g
, &f
);
93 fe25519_mul(&r
->t
, &e
, &h
);
96 static void add_p1p1(ge25519_p1p1
*r
, const ge25519_p3
*p
, const ge25519_p3
*q
)
98 fe25519 a
, b
, c
, d
, t
;
100 fe25519_sub(&a
, &p
->y
, &p
->x
); /* A = (Y1-X1)*(Y2-X2) */
101 fe25519_sub(&t
, &q
->y
, &q
->x
);
102 fe25519_mul(&a
, &a
, &t
);
103 fe25519_add(&b
, &p
->x
, &p
->y
); /* B = (Y1+X1)*(Y2+X2) */
104 fe25519_add(&t
, &q
->x
, &q
->y
);
105 fe25519_mul(&b
, &b
, &t
);
106 fe25519_mul(&c
, &p
->t
, &q
->t
); /* C = T1*k*T2 */
107 fe25519_mul(&c
, &c
, &ge25519_ec2d
);
108 fe25519_mul(&d
, &p
->z
, &q
->z
); /* D = Z1*2*Z2 */
109 fe25519_add(&d
, &d
, &d
);
110 fe25519_sub(&r
->x
, &b
, &a
); /* E = B-A */
111 fe25519_sub(&r
->t
, &d
, &c
); /* F = D-C */
112 fe25519_add(&r
->z
, &d
, &c
); /* G = D+C */
113 fe25519_add(&r
->y
, &b
, &a
); /* H = B+A */
116 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
117 static void dbl_p1p1(ge25519_p1p1
*r
, const ge25519_p2
*p
)
120 fe25519_square(&a
, &p
->x
);
121 fe25519_square(&b
, &p
->y
);
122 fe25519_square(&c
, &p
->z
);
123 fe25519_add(&c
, &c
, &c
);
126 fe25519_add(&r
->x
, &p
->x
, &p
->y
);
127 fe25519_square(&r
->x
, &r
->x
);
128 fe25519_sub(&r
->x
, &r
->x
, &a
);
129 fe25519_sub(&r
->x
, &r
->x
, &b
);
130 fe25519_add(&r
->z
, &d
, &b
);
131 fe25519_sub(&r
->t
, &r
->z
, &c
);
132 fe25519_sub(&r
->y
, &d
, &b
);
135 /* Constant-time version of: if(b) r = p */
136 static void cmov_aff(ge25519_aff
*r
, const ge25519_aff
*p
, unsigned char b
)
138 fe25519_cmov(&r
->x
, &p
->x
, b
);
139 fe25519_cmov(&r
->y
, &p
->y
, b
);
142 static unsigned char equal(signed char b
,signed char c
)
144 unsigned char ub
= b
;
145 unsigned char uc
= c
;
146 unsigned char x
= ub
^ uc
; /* 0: yes; 1..255: no */
147 crypto_uint32 y
= x
; /* 0: yes; 1..255: no */
148 y
-= 1; /* 4294967295: yes; 0..254: no */
149 y
>>= 31; /* 1: yes; 0: no */
153 static unsigned char negative(signed char b
)
155 unsigned long long x
= b
; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
156 x
>>= 63; /* 1: yes; 0: no */
160 static void choose_t(ge25519_aff
*t
, unsigned long long pos
, signed char b
)
164 *t
= ge25519_base_multiples_affine
[5*pos
+0];
165 cmov_aff(t
, &ge25519_base_multiples_affine
[5*pos
+1],equal(b
,1) | equal(b
,-1));
166 cmov_aff(t
, &ge25519_base_multiples_affine
[5*pos
+2],equal(b
,2) | equal(b
,-2));
167 cmov_aff(t
, &ge25519_base_multiples_affine
[5*pos
+3],equal(b
,3) | equal(b
,-3));
168 cmov_aff(t
, &ge25519_base_multiples_affine
[5*pos
+4],equal(b
,-4));
169 fe25519_neg(&v
, &t
->x
);
170 fe25519_cmov(&t
->x
, &v
, negative(b
));
173 static void setneutral(ge25519
*r
)
175 fe25519_setzero(&r
->x
);
176 fe25519_setone(&r
->y
);
177 fe25519_setone(&r
->z
);
178 fe25519_setzero(&r
->t
);
181 /* ********************************************************************
183 ******************************************************************** */
185 /* return 0 on success, -1 otherwise */
186 int ge25519_unpackneg_vartime(ge25519_p3
*r
, const unsigned char p
[32])
188 fe25519 t
, chk
, num
, den
, den2
, den4
, den6
;
189 fe25519_setone(&r
->z
);
190 unsigned char par
= p
[31] >> 7;
191 fe25519_unpack(&r
->y
, p
);
192 fe25519_square(&num
, &r
->y
); /* x = y^2 */
193 fe25519_mul(&den
, &num
, &ge25519_ecd
); /* den = dy^2 */
194 fe25519_sub(&num
, &num
, &r
->z
); /* x = y^2-1 */
195 fe25519_add(&den
, &r
->z
, &den
); /* den = dy^2+1 */
197 /* Computation of sqrt(num/den) */
198 /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
199 fe25519_square(&den2
, &den
);
200 fe25519_square(&den4
, &den2
);
201 fe25519_mul(&den6
, &den4
, &den2
);
202 fe25519_mul(&t
, &den6
, &num
);
203 fe25519_mul(&t
, &t
, &den
);
205 fe25519_pow2523(&t
, &t
);
206 /* 2. computation of r->x = t * num * den^3 */
207 fe25519_mul(&t
, &t
, &num
);
208 fe25519_mul(&t
, &t
, &den
);
209 fe25519_mul(&t
, &t
, &den
);
210 fe25519_mul(&r
->x
, &t
, &den
);
212 /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
213 fe25519_square(&chk
, &r
->x
);
214 fe25519_mul(&chk
, &chk
, &den
);
215 if (!fe25519_iseq_vartime(&chk
, &num
))
216 fe25519_mul(&r
->x
, &r
->x
, &ge25519_sqrtm1
);
218 /* 4. Now we have one of the two square roots, except if input was not a square */
219 fe25519_square(&chk
, &r
->x
);
220 fe25519_mul(&chk
, &chk
, &den
);
221 if (!fe25519_iseq_vartime(&chk
, &num
))
224 /* 5. Choose the desired square root according to parity: */
225 if(fe25519_getparity(&r
->x
) != (1-par
))
226 fe25519_neg(&r
->x
, &r
->x
);
228 fe25519_mul(&r
->t
, &r
->x
, &r
->y
);
232 void ge25519_pack(unsigned char r
[32], const ge25519_p3
*p
)
235 fe25519_invert(&zi
, &p
->z
);
236 fe25519_mul(&tx
, &p
->x
, &zi
);
237 fe25519_mul(&ty
, &p
->y
, &zi
);
238 fe25519_pack(r
, &ty
);
239 r
[31] ^= fe25519_getparity(&tx
) << 7;
242 int ge25519_isneutral_vartime(const ge25519_p3
*p
)
245 if(!fe25519_iszero(&p
->x
)) ret
= 0;
246 if(!fe25519_iseq_vartime(&p
->y
, &p
->z
)) ret
= 0;
250 /* computes [s1]p1 + [s2]p2 */
251 void ge25519_double_scalarmult_vartime(ge25519_p3
*r
, const ge25519_p3
*p1
, const sc25519
*s1
, const ge25519_p3
*p2
, const sc25519
*s2
)
255 unsigned char b
[127];
257 /* precomputation s2 s1 */
258 setneutral(pre
); /* 00 00 */
259 pre
[1] = *p1
; /* 00 01 */
260 dbl_p1p1(&tp1p1
,(ge25519_p2
*)p1
); p1p1_to_p3( &pre
[2], &tp1p1
); /* 00 10 */
261 add_p1p1(&tp1p1
,&pre
[1], &pre
[2]); p1p1_to_p3( &pre
[3], &tp1p1
); /* 00 11 */
262 pre
[4] = *p2
; /* 01 00 */
263 add_p1p1(&tp1p1
,&pre
[1], &pre
[4]); p1p1_to_p3( &pre
[5], &tp1p1
); /* 01 01 */
264 add_p1p1(&tp1p1
,&pre
[2], &pre
[4]); p1p1_to_p3( &pre
[6], &tp1p1
); /* 01 10 */
265 add_p1p1(&tp1p1
,&pre
[3], &pre
[4]); p1p1_to_p3( &pre
[7], &tp1p1
); /* 01 11 */
266 dbl_p1p1(&tp1p1
,(ge25519_p2
*)p2
); p1p1_to_p3( &pre
[8], &tp1p1
); /* 10 00 */
267 add_p1p1(&tp1p1
,&pre
[1], &pre
[8]); p1p1_to_p3( &pre
[9], &tp1p1
); /* 10 01 */
268 dbl_p1p1(&tp1p1
,(ge25519_p2
*)&pre
[5]); p1p1_to_p3(&pre
[10], &tp1p1
); /* 10 10 */
269 add_p1p1(&tp1p1
,&pre
[3], &pre
[8]); p1p1_to_p3(&pre
[11], &tp1p1
); /* 10 11 */
270 add_p1p1(&tp1p1
,&pre
[4], &pre
[8]); p1p1_to_p3(&pre
[12], &tp1p1
); /* 11 00 */
271 add_p1p1(&tp1p1
,&pre
[1],&pre
[12]); p1p1_to_p3(&pre
[13], &tp1p1
); /* 11 01 */
272 add_p1p1(&tp1p1
,&pre
[2],&pre
[12]); p1p1_to_p3(&pre
[14], &tp1p1
); /* 11 10 */
273 add_p1p1(&tp1p1
,&pre
[3],&pre
[12]); p1p1_to_p3(&pre
[15], &tp1p1
); /* 11 11 */
275 sc25519_2interleave2(b
,s1
,s2
);
277 /* scalar multiplication */
282 dbl_p1p1(&tp1p1
, (ge25519_p2
*)r
);
283 p1p1_to_p2((ge25519_p2
*) r
, &tp1p1
);
284 dbl_p1p1(&tp1p1
, (ge25519_p2
*)r
);
287 p1p1_to_p3(r
, &tp1p1
);
288 add_p1p1(&tp1p1
, r
, &pre
[b
[i
]]);
290 if(i
!= 0) p1p1_to_p2((ge25519_p2
*)r
, &tp1p1
);
291 else p1p1_to_p3(r
, &tp1p1
);
295 void ge25519_scalarmult_base(ge25519_p3
*r
, const sc25519
*s
)
300 sc25519_window3(b
,s
);
302 choose_t((ge25519_aff
*)r
, 0, b
[0]);
303 fe25519_setone(&r
->z
);
304 fe25519_mul(&r
->t
, &r
->x
, &r
->y
);
307 choose_t(&t
, (unsigned long long) i
, b
[i
]);
308 ge25519_mixadd2(r
, &t
);