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
|
// +build noasm !amd64,!arm64
package p751
import (
. "github.com/cloudflare/sidh/internal/arith"
. "github.com/cloudflare/sidh/internal/isogeny"
)
// Compute z = x + y (mod p).
func fp751AddReduced(z, x, y *FpElement) {
var carry uint64
// z=x+y % p751
for i := 0; i < NumWords; i++ {
z[i], carry = Addc64(carry, x[i], y[i])
}
// z = z - p751x2
carry = 0
for i := 0; i < NumWords; i++ {
z[i], carry = Subc64(carry, z[i], p751x2[i])
}
// z = z + p751x2
mask := uint64(0 - carry)
carry = 0
for i := 0; i < NumWords; i++ {
z[i], carry = Addc64(carry, z[i], p751x2[i]&mask)
}
}
// Compute z = x - y (mod p).
func fp751SubReduced(z, x, y *FpElement) {
var borrow uint64
for i := 0; i < NumWords; i++ {
z[i], borrow = Subc64(borrow, x[i], y[i])
}
mask := uint64(0 - borrow)
borrow = 0
for i := 0; i < NumWords; i++ {
z[i], borrow = Addc64(borrow, z[i], p751x2[i]&mask)
}
}
// Conditionally swaps bits in x and y in constant time.
// mask indicates bits to be swaped (set bits are swapped)
// For details see "Hackers Delight, 2.20"
//
// Implementation doesn't actually depend on a prime field.
func fp751ConditionalSwap(x, y *FpElement, mask uint8) {
var tmp, mask64 uint64
mask64 = 0 - uint64(mask)
for i := 0; i < len(x); i++ {
tmp = mask64 & (x[i] ^ y[i])
x[i] = tmp ^ x[i]
y[i] = tmp ^ y[i]
}
}
// Perform Montgomery reduction: set z = x R^{-1} (mod 2*p)
// with R=2^768. Destroys the input value.
func fp751MontgomeryReduce(z *FpElement, x *FpElementX2) {
var carry, t, u, v uint64
var uv Uint128
var count int
count = 5 // number of 0 digits in the least significat part of p751 + 1
for i := 0; i < NumWords; i++ {
for j := 0; j < i; j++ {
if j < (i - count + 1) {
uv = Mul64(z[j], p751p1[i-j])
v, carry = Addc64(0, uv.L, v)
u, carry = Addc64(carry, uv.H, u)
t += carry
}
}
v, carry = Addc64(0, v, x[i])
u, carry = Addc64(carry, u, 0)
t += carry
z[i] = v
v = u
u = t
t = 0
}
for i := NumWords; i < 2*NumWords-1; i++ {
if count > 0 {
count--
}
for j := i - NumWords + 1; j < NumWords; j++ {
if j < (NumWords - count) {
uv = Mul64(z[j], p751p1[i-j])
v, carry = Addc64(0, uv.L, v)
u, carry = Addc64(carry, uv.H, u)
t += carry
}
}
v, carry = Addc64(0, v, x[i])
u, carry = Addc64(carry, u, 0)
t += carry
z[i-NumWords] = v
v = u
u = t
t = 0
}
v, carry = Addc64(0, v, x[2*NumWords-1])
z[NumWords-1] = v
}
// Compute z = x * y.
func fp751Mul(z *FpElementX2, x, y *FpElement) {
var u, v, t uint64
var carry uint64
var uv Uint128
for i := uint64(0); i < NumWords; i++ {
for j := uint64(0); j <= i; j++ {
uv = Mul64(x[j], y[i-j])
v, carry = Addc64(0, uv.L, v)
u, carry = Addc64(carry, uv.H, u)
t += carry
}
z[i] = v
v = u
u = t
t = 0
}
for i := NumWords; i < (2*NumWords)-1; i++ {
for j := i - NumWords + 1; j < NumWords; j++ {
uv = Mul64(x[j], y[i-j])
v, carry = Addc64(0, uv.L, v)
u, carry = Addc64(carry, uv.H, u)
t += carry
}
z[i] = v
v = u
u = t
t = 0
}
z[2*NumWords-1] = v
}
// Compute z = x + y, without reducing mod p.
func fp751AddLazy(z, x, y *FpElement) {
var carry uint64
for i := 0; i < NumWords; i++ {
z[i], carry = Addc64(carry, x[i], y[i])
}
}
// Compute z = x + y, without reducing mod p.
func fp751X2AddLazy(z, x, y *FpElementX2) {
var carry uint64
for i := 0; i < 2*NumWords; i++ {
z[i], carry = Addc64(carry, x[i], y[i])
}
}
// Reduce a field element in [0, 2*p) to one in [0,p).
func fp751StrongReduce(x *FpElement) {
var borrow, mask uint64
for i := 0; i < NumWords; i++ {
x[i], borrow = Subc64(borrow, x[i], p751[i])
}
// Sets all bits if borrow = 1
mask = 0 - borrow
borrow = 0
for i := 0; i < NumWords; i++ {
x[i], borrow = Addc64(borrow, x[i], p751[i]&mask)
}
}
// Compute z = x - y, without reducing mod p.
func fp751X2SubLazy(z, x, y *FpElementX2) {
var borrow, mask uint64
for i := 0; i < len(z); i++ {
z[i], borrow = Subc64(borrow, x[i], y[i])
}
// Sets all bits if borrow = 1
mask = 0 - borrow
borrow = 0
for i := NumWords; i < len(z); i++ {
z[i], borrow = Addc64(borrow, z[i], p751[i-NumWords]&mask)
}
}
|
