1#include "fe.h"
2#include "crypto_int64.h"
3
4/*
5h = f * g
6Can overlap h with f or g.
7
8Preconditions:
9 |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
10 |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
11
12Postconditions:
13 |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
14*/
15
16/*
17Notes on implementation strategy:
18
19Using schoolbook multiplication.
20Karatsuba would save a little in some cost models.
21
22Most multiplications by 2 and 19 are 32-bit precomputations;
23cheaper than 64-bit postcomputations.
24
25There is one remaining multiplication by 19 in the carry chain;
26one *19 precomputation can be merged into this,
27but the resulting data flow is considerably less clean.
28
29There are 12 carries below.
3010 of them are 2-way parallelizable and vectorizable.
31Can get away with 11 carries, but then data flow is much deeper.
32
33With tighter constraints on inputs can squeeze carries into int32.
34*/
35
36void fe_mul(fe h,const fe f,const fe g)
37{
38 crypto_int32 f0 = f[0];
39 crypto_int32 f1 = f[1];
40 crypto_int32 f2 = f[2];
41 crypto_int32 f3 = f[3];
42 crypto_int32 f4 = f[4];
43 crypto_int32 f5 = f[5];
44 crypto_int32 f6 = f[6];
45 crypto_int32 f7 = f[7];
46 crypto_int32 f8 = f[8];
47 crypto_int32 f9 = f[9];
48 crypto_int32 g0 = g[0];
49 crypto_int32 g1 = g[1];
50 crypto_int32 g2 = g[2];
51 crypto_int32 g3 = g[3];
52 crypto_int32 g4 = g[4];
53 crypto_int32 g5 = g[5];
54 crypto_int32 g6 = g[6];
55 crypto_int32 g7 = g[7];
56 crypto_int32 g8 = g[8];
57 crypto_int32 g9 = g[9];
58 crypto_int32 g1_19 = 19 * g1; /* 1.959375*2^29 */
59 crypto_int32 g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
60 crypto_int32 g3_19 = 19 * g3;
61 crypto_int32 g4_19 = 19 * g4;
62 crypto_int32 g5_19 = 19 * g5;
63 crypto_int32 g6_19 = 19 * g6;
64 crypto_int32 g7_19 = 19 * g7;
65 crypto_int32 g8_19 = 19 * g8;
66 crypto_int32 g9_19 = 19 * g9;
67 crypto_int32 f1_2 = 2 * f1;
68 crypto_int32 f3_2 = 2 * f3;
69 crypto_int32 f5_2 = 2 * f5;
70 crypto_int32 f7_2 = 2 * f7;
71 crypto_int32 f9_2 = 2 * f9;
72 crypto_int64 f0g0 = f0 * (crypto_int64) g0;
73 crypto_int64 f0g1 = f0 * (crypto_int64) g1;
74 crypto_int64 f0g2 = f0 * (crypto_int64) g2;
75 crypto_int64 f0g3 = f0 * (crypto_int64) g3;
76 crypto_int64 f0g4 = f0 * (crypto_int64) g4;
77 crypto_int64 f0g5 = f0 * (crypto_int64) g5;
78 crypto_int64 f0g6 = f0 * (crypto_int64) g6;
79 crypto_int64 f0g7 = f0 * (crypto_int64) g7;
80 crypto_int64 f0g8 = f0 * (crypto_int64) g8;
81 crypto_int64 f0g9 = f0 * (crypto_int64) g9;
82 crypto_int64 f1g0 = f1 * (crypto_int64) g0;
83 crypto_int64 f1g1_2 = f1_2 * (crypto_int64) g1;
84 crypto_int64 f1g2 = f1 * (crypto_int64) g2;
85 crypto_int64 f1g3_2 = f1_2 * (crypto_int64) g3;
86 crypto_int64 f1g4 = f1 * (crypto_int64) g4;
87 crypto_int64 f1g5_2 = f1_2 * (crypto_int64) g5;
88 crypto_int64 f1g6 = f1 * (crypto_int64) g6;
89 crypto_int64 f1g7_2 = f1_2 * (crypto_int64) g7;
90 crypto_int64 f1g8 = f1 * (crypto_int64) g8;
91 crypto_int64 f1g9_38 = f1_2 * (crypto_int64) g9_19;
92 crypto_int64 f2g0 = f2 * (crypto_int64) g0;
93 crypto_int64 f2g1 = f2 * (crypto_int64) g1;
94 crypto_int64 f2g2 = f2 * (crypto_int64) g2;
95 crypto_int64 f2g3 = f2 * (crypto_int64) g3;
96 crypto_int64 f2g4 = f2 * (crypto_int64) g4;
97 crypto_int64 f2g5 = f2 * (crypto_int64) g5;
98 crypto_int64 f2g6 = f2 * (crypto_int64) g6;
99 crypto_int64 f2g7 = f2 * (crypto_int64) g7;
100 crypto_int64 f2g8_19 = f2 * (crypto_int64) g8_19;
101 crypto_int64 f2g9_19 = f2 * (crypto_int64) g9_19;
102 crypto_int64 f3g0 = f3 * (crypto_int64) g0;
103 crypto_int64 f3g1_2 = f3_2 * (crypto_int64) g1;
104 crypto_int64 f3g2 = f3 * (crypto_int64) g2;
105 crypto_int64 f3g3_2 = f3_2 * (crypto_int64) g3;
106 crypto_int64 f3g4 = f3 * (crypto_int64) g4;
107 crypto_int64 f3g5_2 = f3_2 * (crypto_int64) g5;
108 crypto_int64 f3g6 = f3 * (crypto_int64) g6;
109 crypto_int64 f3g7_38 = f3_2 * (crypto_int64) g7_19;
110 crypto_int64 f3g8_19 = f3 * (crypto_int64) g8_19;
111 crypto_int64 f3g9_38 = f3_2 * (crypto_int64) g9_19;
112 crypto_int64 f4g0 = f4 * (crypto_int64) g0;
113 crypto_int64 f4g1 = f4 * (crypto_int64) g1;
114 crypto_int64 f4g2 = f4 * (crypto_int64) g2;
115 crypto_int64 f4g3 = f4 * (crypto_int64) g3;
116 crypto_int64 f4g4 = f4 * (crypto_int64) g4;
117 crypto_int64 f4g5 = f4 * (crypto_int64) g5;
118 crypto_int64 f4g6_19 = f4 * (crypto_int64) g6_19;
119 crypto_int64 f4g7_19 = f4 * (crypto_int64) g7_19;
120 crypto_int64 f4g8_19 = f4 * (crypto_int64) g8_19;
121 crypto_int64 f4g9_19 = f4 * (crypto_int64) g9_19;
122 crypto_int64 f5g0 = f5 * (crypto_int64) g0;
123 crypto_int64 f5g1_2 = f5_2 * (crypto_int64) g1;
124 crypto_int64 f5g2 = f5 * (crypto_int64) g2;
125 crypto_int64 f5g3_2 = f5_2 * (crypto_int64) g3;
126 crypto_int64 f5g4 = f5 * (crypto_int64) g4;
127 crypto_int64 f5g5_38 = f5_2 * (crypto_int64) g5_19;
128 crypto_int64 f5g6_19 = f5 * (crypto_int64) g6_19;
129 crypto_int64 f5g7_38 = f5_2 * (crypto_int64) g7_19;
130 crypto_int64 f5g8_19 = f5 * (crypto_int64) g8_19;
131 crypto_int64 f5g9_38 = f5_2 * (crypto_int64) g9_19;
132 crypto_int64 f6g0 = f6 * (crypto_int64) g0;
133 crypto_int64 f6g1 = f6 * (crypto_int64) g1;
134 crypto_int64 f6g2 = f6 * (crypto_int64) g2;
135 crypto_int64 f6g3 = f6 * (crypto_int64) g3;
136 crypto_int64 f6g4_19 = f6 * (crypto_int64) g4_19;
137 crypto_int64 f6g5_19 = f6 * (crypto_int64) g5_19;
138 crypto_int64 f6g6_19 = f6 * (crypto_int64) g6_19;
139 crypto_int64 f6g7_19 = f6 * (crypto_int64) g7_19;
140 crypto_int64 f6g8_19 = f6 * (crypto_int64) g8_19;
141 crypto_int64 f6g9_19 = f6 * (crypto_int64) g9_19;
142 crypto_int64 f7g0 = f7 * (crypto_int64) g0;
143 crypto_int64 f7g1_2 = f7_2 * (crypto_int64) g1;
144 crypto_int64 f7g2 = f7 * (crypto_int64) g2;
145 crypto_int64 f7g3_38 = f7_2 * (crypto_int64) g3_19;
146 crypto_int64 f7g4_19 = f7 * (crypto_int64) g4_19;
147 crypto_int64 f7g5_38 = f7_2 * (crypto_int64) g5_19;
148 crypto_int64 f7g6_19 = f7 * (crypto_int64) g6_19;
149 crypto_int64 f7g7_38 = f7_2 * (crypto_int64) g7_19;
150 crypto_int64 f7g8_19 = f7 * (crypto_int64) g8_19;
151 crypto_int64 f7g9_38 = f7_2 * (crypto_int64) g9_19;
152 crypto_int64 f8g0 = f8 * (crypto_int64) g0;
153 crypto_int64 f8g1 = f8 * (crypto_int64) g1;
154 crypto_int64 f8g2_19 = f8 * (crypto_int64) g2_19;
155 crypto_int64 f8g3_19 = f8 * (crypto_int64) g3_19;
156 crypto_int64 f8g4_19 = f8 * (crypto_int64) g4_19;
157 crypto_int64 f8g5_19 = f8 * (crypto_int64) g5_19;
158 crypto_int64 f8g6_19 = f8 * (crypto_int64) g6_19;
159 crypto_int64 f8g7_19 = f8 * (crypto_int64) g7_19;
160 crypto_int64 f8g8_19 = f8 * (crypto_int64) g8_19;
161 crypto_int64 f8g9_19 = f8 * (crypto_int64) g9_19;
162 crypto_int64 f9g0 = f9 * (crypto_int64) g0;
163 crypto_int64 f9g1_38 = f9_2 * (crypto_int64) g1_19;
164 crypto_int64 f9g2_19 = f9 * (crypto_int64) g2_19;
165 crypto_int64 f9g3_38 = f9_2 * (crypto_int64) g3_19;
166 crypto_int64 f9g4_19 = f9 * (crypto_int64) g4_19;
167 crypto_int64 f9g5_38 = f9_2 * (crypto_int64) g5_19;
168 crypto_int64 f9g6_19 = f9 * (crypto_int64) g6_19;
169 crypto_int64 f9g7_38 = f9_2 * (crypto_int64) g7_19;
170 crypto_int64 f9g8_19 = f9 * (crypto_int64) g8_19;
171 crypto_int64 f9g9_38 = f9_2 * (crypto_int64) g9_19;
172 crypto_int64 h0 = f0g0+f1g9_38+f2g8_19+f3g7_38+f4g6_19+f5g5_38+f6g4_19+f7g3_38+f8g2_19+f9g1_38;
173 crypto_int64 h1 = f0g1+f1g0 +f2g9_19+f3g8_19+f4g7_19+f5g6_19+f6g5_19+f7g4_19+f8g3_19+f9g2_19;
174 crypto_int64 h2 = f0g2+f1g1_2 +f2g0 +f3g9_38+f4g8_19+f5g7_38+f6g6_19+f7g5_38+f8g4_19+f9g3_38;
175 crypto_int64 h3 = f0g3+f1g2 +f2g1 +f3g0 +f4g9_19+f5g8_19+f6g7_19+f7g6_19+f8g5_19+f9g4_19;
176 crypto_int64 h4 = f0g4+f1g3_2 +f2g2 +f3g1_2 +f4g0 +f5g9_38+f6g8_19+f7g7_38+f8g6_19+f9g5_38;
177 crypto_int64 h5 = f0g5+f1g4 +f2g3 +f3g2 +f4g1 +f5g0 +f6g9_19+f7g8_19+f8g7_19+f9g6_19;
178 crypto_int64 h6 = f0g6+f1g5_2 +f2g4 +f3g3_2 +f4g2 +f5g1_2 +f6g0 +f7g9_38+f8g8_19+f9g7_38;
179 crypto_int64 h7 = f0g7+f1g6 +f2g5 +f3g4 +f4g3 +f5g2 +f6g1 +f7g0 +f8g9_19+f9g8_19;
180 crypto_int64 h8 = f0g8+f1g7_2 +f2g6 +f3g5_2 +f4g4 +f5g3_2 +f6g2 +f7g1_2 +f8g0 +f9g9_38;
181 crypto_int64 h9 = f0g9+f1g8 +f2g7 +f3g6 +f4g5 +f5g4 +f6g3 +f7g2 +f8g1 +f9g0 ;
182 crypto_int64 carry0;
183 crypto_int64 carry1;
184 crypto_int64 carry2;
185 crypto_int64 carry3;
186 crypto_int64 carry4;
187 crypto_int64 carry5;
188 crypto_int64 carry6;
189 crypto_int64 carry7;
190 crypto_int64 carry8;
191 crypto_int64 carry9;
192
193 /*
194 |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
195 i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
196 |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
197 i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9
198 */
199
200 carry0 = (h0 + (crypto_int64) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
201 carry4 = (h4 + (crypto_int64) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
202 /* |h0| <= 2^25 */
203 /* |h4| <= 2^25 */
204 /* |h1| <= 1.71*2^59 */
205 /* |h5| <= 1.71*2^59 */
206
207 carry1 = (h1 + (crypto_int64) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
208 carry5 = (h5 + (crypto_int64) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
209 /* |h1| <= 2^24; from now on fits into int32 */
210 /* |h5| <= 2^24; from now on fits into int32 */
211 /* |h2| <= 1.41*2^60 */
212 /* |h6| <= 1.41*2^60 */
213
214 carry2 = (h2 + (crypto_int64) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
215 carry6 = (h6 + (crypto_int64) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
216 /* |h2| <= 2^25; from now on fits into int32 unchanged */
217 /* |h6| <= 2^25; from now on fits into int32 unchanged */
218 /* |h3| <= 1.71*2^59 */
219 /* |h7| <= 1.71*2^59 */
220
221 carry3 = (h3 + (crypto_int64) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
222 carry7 = (h7 + (crypto_int64) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
223 /* |h3| <= 2^24; from now on fits into int32 unchanged */
224 /* |h7| <= 2^24; from now on fits into int32 unchanged */
225 /* |h4| <= 1.72*2^34 */
226 /* |h8| <= 1.41*2^60 */
227
228 carry4 = (h4 + (crypto_int64) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
229 carry8 = (h8 + (crypto_int64) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
230 /* |h4| <= 2^25; from now on fits into int32 unchanged */
231 /* |h8| <= 2^25; from now on fits into int32 unchanged */
232 /* |h5| <= 1.01*2^24 */
233 /* |h9| <= 1.71*2^59 */
234
235 carry9 = (h9 + (crypto_int64) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
236 /* |h9| <= 2^24; from now on fits into int32 unchanged */
237 /* |h0| <= 1.1*2^39 */
238
239 carry0 = (h0 + (crypto_int64) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
240 /* |h0| <= 2^25; from now on fits into int32 unchanged */
241 /* |h1| <= 1.01*2^24 */
242
243 h[0] = h0;
244 h[1] = h1;
245 h[2] = h2;
246 h[3] = h3;
247 h[4] = h4;
248 h[5] = h5;
249 h[6] = h6;
250 h[7] = h7;
251 h[8] = h8;
252 h[9] = h9;
253}
254