1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
#include "fe.h"
#include "crypto_int64.h"
#include "crypto_uint64.h"
/*
h = f * g
Can overlap h with f or g.
Preconditions:
|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
|g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
Postconditions:
|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
*/
/*
Notes on implementation strategy:
Using schoolbook multiplication.
Karatsuba would save a little in some cost models.
Most multiplications by 2 and 19 are 32-bit precomputations;
cheaper than 64-bit postcomputations.
There is one remaining multiplication by 19 in the carry chain;
one *19 precomputation can be merged into this,
but the resulting data flow is considerably less clean.
There are 12 carries below.
10 of them are 2-way parallelizable and vectorizable.
Can get away with 11 carries, but then data flow is much deeper.
With tighter constraints on inputs can squeeze carries into int32.
*/
void fe_mul(fe h,const fe f,const fe g)
{
crypto_int32 f0 = f[0];
crypto_int32 f1 = f[1];
crypto_int32 f2 = f[2];
crypto_int32 f3 = f[3];
crypto_int32 f4 = f[4];
crypto_int32 f5 = f[5];
crypto_int32 f6 = f[6];
crypto_int32 f7 = f[7];
crypto_int32 f8 = f[8];
crypto_int32 f9 = f[9];
crypto_int32 g0 = g[0];
crypto_int32 g1 = g[1];
crypto_int32 g2 = g[2];
crypto_int32 g3 = g[3];
crypto_int32 g4 = g[4];
crypto_int32 g5 = g[5];
crypto_int32 g6 = g[6];
crypto_int32 g7 = g[7];
crypto_int32 g8 = g[8];
crypto_int32 g9 = g[9];
crypto_int32 g1_19 = 19 * g1; /* 1.959375*2^29 */
crypto_int32 g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
crypto_int32 g3_19 = 19 * g3;
crypto_int32 g4_19 = 19 * g4;
crypto_int32 g5_19 = 19 * g5;
crypto_int32 g6_19 = 19 * g6;
crypto_int32 g7_19 = 19 * g7;
crypto_int32 g8_19 = 19 * g8;
crypto_int32 g9_19 = 19 * g9;
crypto_int32 f1_2 = 2 * f1;
crypto_int32 f3_2 = 2 * f3;
crypto_int32 f5_2 = 2 * f5;
crypto_int32 f7_2 = 2 * f7;
crypto_int32 f9_2 = 2 * f9;
crypto_int64 f0g0 = f0 * (crypto_int64) g0;
crypto_int64 f0g1 = f0 * (crypto_int64) g1;
crypto_int64 f0g2 = f0 * (crypto_int64) g2;
crypto_int64 f0g3 = f0 * (crypto_int64) g3;
crypto_int64 f0g4 = f0 * (crypto_int64) g4;
crypto_int64 f0g5 = f0 * (crypto_int64) g5;
crypto_int64 f0g6 = f0 * (crypto_int64) g6;
crypto_int64 f0g7 = f0 * (crypto_int64) g7;
crypto_int64 f0g8 = f0 * (crypto_int64) g8;
crypto_int64 f0g9 = f0 * (crypto_int64) g9;
crypto_int64 f1g0 = f1 * (crypto_int64) g0;
crypto_int64 f1g1_2 = f1_2 * (crypto_int64) g1;
crypto_int64 f1g2 = f1 * (crypto_int64) g2;
crypto_int64 f1g3_2 = f1_2 * (crypto_int64) g3;
crypto_int64 f1g4 = f1 * (crypto_int64) g4;
crypto_int64 f1g5_2 = f1_2 * (crypto_int64) g5;
crypto_int64 f1g6 = f1 * (crypto_int64) g6;
crypto_int64 f1g7_2 = f1_2 * (crypto_int64) g7;
crypto_int64 f1g8 = f1 * (crypto_int64) g8;
crypto_int64 f1g9_38 = f1_2 * (crypto_int64) g9_19;
crypto_int64 f2g0 = f2 * (crypto_int64) g0;
crypto_int64 f2g1 = f2 * (crypto_int64) g1;
crypto_int64 f2g2 = f2 * (crypto_int64) g2;
crypto_int64 f2g3 = f2 * (crypto_int64) g3;
crypto_int64 f2g4 = f2 * (crypto_int64) g4;
crypto_int64 f2g5 = f2 * (crypto_int64) g5;
crypto_int64 f2g6 = f2 * (crypto_int64) g6;
crypto_int64 f2g7 = f2 * (crypto_int64) g7;
crypto_int64 f2g8_19 = f2 * (crypto_int64) g8_19;
crypto_int64 f2g9_19 = f2 * (crypto_int64) g9_19;
crypto_int64 f3g0 = f3 * (crypto_int64) g0;
crypto_int64 f3g1_2 = f3_2 * (crypto_int64) g1;
crypto_int64 f3g2 = f3 * (crypto_int64) g2;
crypto_int64 f3g3_2 = f3_2 * (crypto_int64) g3;
crypto_int64 f3g4 = f3 * (crypto_int64) g4;
crypto_int64 f3g5_2 = f3_2 * (crypto_int64) g5;
crypto_int64 f3g6 = f3 * (crypto_int64) g6;
crypto_int64 f3g7_38 = f3_2 * (crypto_int64) g7_19;
crypto_int64 f3g8_19 = f3 * (crypto_int64) g8_19;
crypto_int64 f3g9_38 = f3_2 * (crypto_int64) g9_19;
crypto_int64 f4g0 = f4 * (crypto_int64) g0;
crypto_int64 f4g1 = f4 * (crypto_int64) g1;
crypto_int64 f4g2 = f4 * (crypto_int64) g2;
crypto_int64 f4g3 = f4 * (crypto_int64) g3;
crypto_int64 f4g4 = f4 * (crypto_int64) g4;
crypto_int64 f4g5 = f4 * (crypto_int64) g5;
crypto_int64 f4g6_19 = f4 * (crypto_int64) g6_19;
crypto_int64 f4g7_19 = f4 * (crypto_int64) g7_19;
crypto_int64 f4g8_19 = f4 * (crypto_int64) g8_19;
crypto_int64 f4g9_19 = f4 * (crypto_int64) g9_19;
crypto_int64 f5g0 = f5 * (crypto_int64) g0;
crypto_int64 f5g1_2 = f5_2 * (crypto_int64) g1;
crypto_int64 f5g2 = f5 * (crypto_int64) g2;
crypto_int64 f5g3_2 = f5_2 * (crypto_int64) g3;
crypto_int64 f5g4 = f5 * (crypto_int64) g4;
crypto_int64 f5g5_38 = f5_2 * (crypto_int64) g5_19;
crypto_int64 f5g6_19 = f5 * (crypto_int64) g6_19;
crypto_int64 f5g7_38 = f5_2 * (crypto_int64) g7_19;
crypto_int64 f5g8_19 = f5 * (crypto_int64) g8_19;
crypto_int64 f5g9_38 = f5_2 * (crypto_int64) g9_19;
crypto_int64 f6g0 = f6 * (crypto_int64) g0;
crypto_int64 f6g1 = f6 * (crypto_int64) g1;
crypto_int64 f6g2 = f6 * (crypto_int64) g2;
crypto_int64 f6g3 = f6 * (crypto_int64) g3;
crypto_int64 f6g4_19 = f6 * (crypto_int64) g4_19;
crypto_int64 f6g5_19 = f6 * (crypto_int64) g5_19;
crypto_int64 f6g6_19 = f6 * (crypto_int64) g6_19;
crypto_int64 f6g7_19 = f6 * (crypto_int64) g7_19;
crypto_int64 f6g8_19 = f6 * (crypto_int64) g8_19;
crypto_int64 f6g9_19 = f6 * (crypto_int64) g9_19;
crypto_int64 f7g0 = f7 * (crypto_int64) g0;
crypto_int64 f7g1_2 = f7_2 * (crypto_int64) g1;
crypto_int64 f7g2 = f7 * (crypto_int64) g2;
crypto_int64 f7g3_38 = f7_2 * (crypto_int64) g3_19;
crypto_int64 f7g4_19 = f7 * (crypto_int64) g4_19;
crypto_int64 f7g5_38 = f7_2 * (crypto_int64) g5_19;
crypto_int64 f7g6_19 = f7 * (crypto_int64) g6_19;
crypto_int64 f7g7_38 = f7_2 * (crypto_int64) g7_19;
crypto_int64 f7g8_19 = f7 * (crypto_int64) g8_19;
crypto_int64 f7g9_38 = f7_2 * (crypto_int64) g9_19;
crypto_int64 f8g0 = f8 * (crypto_int64) g0;
crypto_int64 f8g1 = f8 * (crypto_int64) g1;
crypto_int64 f8g2_19 = f8 * (crypto_int64) g2_19;
crypto_int64 f8g3_19 = f8 * (crypto_int64) g3_19;
crypto_int64 f8g4_19 = f8 * (crypto_int64) g4_19;
crypto_int64 f8g5_19 = f8 * (crypto_int64) g5_19;
crypto_int64 f8g6_19 = f8 * (crypto_int64) g6_19;
crypto_int64 f8g7_19 = f8 * (crypto_int64) g7_19;
crypto_int64 f8g8_19 = f8 * (crypto_int64) g8_19;
crypto_int64 f8g9_19 = f8 * (crypto_int64) g9_19;
crypto_int64 f9g0 = f9 * (crypto_int64) g0;
crypto_int64 f9g1_38 = f9_2 * (crypto_int64) g1_19;
crypto_int64 f9g2_19 = f9 * (crypto_int64) g2_19;
crypto_int64 f9g3_38 = f9_2 * (crypto_int64) g3_19;
crypto_int64 f9g4_19 = f9 * (crypto_int64) g4_19;
crypto_int64 f9g5_38 = f9_2 * (crypto_int64) g5_19;
crypto_int64 f9g6_19 = f9 * (crypto_int64) g6_19;
crypto_int64 f9g7_38 = f9_2 * (crypto_int64) g7_19;
crypto_int64 f9g8_19 = f9 * (crypto_int64) g8_19;
crypto_int64 f9g9_38 = f9_2 * (crypto_int64) g9_19;
crypto_int64 h0 = f0g0+f1g9_38+f2g8_19+f3g7_38+f4g6_19+f5g5_38+f6g4_19+f7g3_38+f8g2_19+f9g1_38;
crypto_int64 h1 = f0g1+f1g0 +f2g9_19+f3g8_19+f4g7_19+f5g6_19+f6g5_19+f7g4_19+f8g3_19+f9g2_19;
crypto_int64 h2 = f0g2+f1g1_2 +f2g0 +f3g9_38+f4g8_19+f5g7_38+f6g6_19+f7g5_38+f8g4_19+f9g3_38;
crypto_int64 h3 = f0g3+f1g2 +f2g1 +f3g0 +f4g9_19+f5g8_19+f6g7_19+f7g6_19+f8g5_19+f9g4_19;
crypto_int64 h4 = f0g4+f1g3_2 +f2g2 +f3g1_2 +f4g0 +f5g9_38+f6g8_19+f7g7_38+f8g6_19+f9g5_38;
crypto_int64 h5 = f0g5+f1g4 +f2g3 +f3g2 +f4g1 +f5g0 +f6g9_19+f7g8_19+f8g7_19+f9g6_19;
crypto_int64 h6 = f0g6+f1g5_2 +f2g4 +f3g3_2 +f4g2 +f5g1_2 +f6g0 +f7g9_38+f8g8_19+f9g7_38;
crypto_int64 h7 = f0g7+f1g6 +f2g5 +f3g4 +f4g3 +f5g2 +f6g1 +f7g0 +f8g9_19+f9g8_19;
crypto_int64 h8 = f0g8+f1g7_2 +f2g6 +f3g5_2 +f4g4 +f5g3_2 +f6g2 +f7g1_2 +f8g0 +f9g9_38;
crypto_int64 h9 = f0g9+f1g8 +f2g7 +f3g6 +f4g5 +f5g4 +f6g3 +f7g2 +f8g1 +f9g0 ;
crypto_uint64 carry0;
crypto_uint64 carry1;
crypto_uint64 carry2;
crypto_uint64 carry3;
crypto_uint64 carry4;
crypto_uint64 carry5;
crypto_uint64 carry6;
crypto_uint64 carry7;
crypto_uint64 carry8;
crypto_uint64 carry9;
/*
|h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
|h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9
*/
carry0 = (h0 + (crypto_int64) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
carry4 = (h4 + (crypto_int64) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
/* |h0| <= 2^25 */
/* |h4| <= 2^25 */
/* |h1| <= 1.71*2^59 */
/* |h5| <= 1.71*2^59 */
carry1 = (h1 + (crypto_int64) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
carry5 = (h5 + (crypto_int64) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
/* |h1| <= 2^24; from now on fits into int32 */
/* |h5| <= 2^24; from now on fits into int32 */
/* |h2| <= 1.41*2^60 */
/* |h6| <= 1.41*2^60 */
carry2 = (h2 + (crypto_int64) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
carry6 = (h6 + (crypto_int64) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
/* |h2| <= 2^25; from now on fits into int32 unchanged */
/* |h6| <= 2^25; from now on fits into int32 unchanged */
/* |h3| <= 1.71*2^59 */
/* |h7| <= 1.71*2^59 */
carry3 = (h3 + (crypto_int64) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
carry7 = (h7 + (crypto_int64) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
/* |h3| <= 2^24; from now on fits into int32 unchanged */
/* |h7| <= 2^24; from now on fits into int32 unchanged */
/* |h4| <= 1.72*2^34 */
/* |h8| <= 1.41*2^60 */
carry4 = (h4 + (crypto_int64) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry8 = (h8 + (crypto_int64) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
/* |h4| <= 2^25; from now on fits into int32 unchanged */
/* |h8| <= 2^25; from now on fits into int32 unchanged */
/* |h5| <= 1.01*2^24 */
/* |h9| <= 1.71*2^59 */
carry9 = (h9 + (crypto_int64) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
/* |h9| <= 2^24; from now on fits into int32 unchanged */
/* |h0| <= 1.1*2^39 */
carry0 = (h0 + (crypto_int64) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
/* |h0| <= 2^25; from now on fits into int32 unchanged */
/* |h1| <= 1.01*2^24 */
h[0] = h0;
h[1] = h1;
h[2] = h2;
h[3] = h3;
h[4] = h4;
h[5] = h5;
h[6] = h6;
h[7] = h7;
h[8] = h8;
h[9] = h9;
}
|