p256 0.14.0

Pure Rust implementation of the NIST P-256 (a.k.a. secp256r1, prime256v1) elliptic curve as defined in SP 800-186, with support for ECDH, ECDSA signing/verification, and general purpose curve arithmetic
Documentation 
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

//! 32-bit secp256r1 field element algorithms.

use elliptic_curve::bigint::{Limb, U256, modular::ConstMontyParams};

const MODULUS: &[Limb; 8] = super::FieldParams::PARAMS.modulus().as_ref().as_limbs();

pub(super) const fn add(a: &U256, b: &U256) -> U256 {
    let a = a.as_limbs();
    let b = b.as_limbs();

    // Bit 256 of p is set, so addition can result in nine words.
    // let (w0, carry) = carrying_add(a[0], b[0], 0);
    let (w0, carry) = a[0].carrying_add(b[0], Limb::ZERO);
    let (w1, carry) = a[1].carrying_add(b[1], carry);
    let (w2, carry) = a[2].carrying_add(b[2], carry);
    let (w3, carry) = a[3].carrying_add(b[3], carry);
    let (w4, carry) = a[4].carrying_add(b[4], carry);
    let (w5, carry) = a[5].carrying_add(b[5], carry);
    let (w6, carry) = a[6].carrying_add(b[6], carry);
    let (w7, w8) = a[7].carrying_add(b[7], carry);

    // Attempt to subtract the modulus, to ensure the result is in the field.
    let (result, _) = sub_inner(
        [w0, w1, w2, w3, w4, w5, w6, w7, w8],
        [
            MODULUS[0],
            MODULUS[1],
            MODULUS[2],
            MODULUS[3],
            MODULUS[4],
            MODULUS[5],
            MODULUS[6],
            MODULUS[7],
            Limb::ZERO,
        ],
    );

    U256::new([
        result[0], result[1], result[2], result[3], result[4], result[5], result[6], result[7],
    ])
}

pub(super) const fn sub(a: &U256, b: &U256) -> U256 {
    let a = a.as_limbs();
    let b = b.as_limbs();

    let (result, _) = sub_inner(
        [a[0], a[1], a[2], a[3], a[4], a[5], a[6], a[7], Limb::ZERO],
        [b[0], b[1], b[2], b[3], b[4], b[5], b[6], b[7], Limb::ZERO],
    );

    U256::new([
        result[0], result[1], result[2], result[3], result[4], result[5], result[6], result[7],
    ])
}

#[inline]
pub(super) const fn to_canonical(a: &U256) -> U256 {
    montgomery_reduce(a, &U256::ZERO)
}

/// Montgomery Reduction
///
/// The general algorithm is:
/// ```text
/// A <- input (2n b-limbs)
/// for i in 0..n {
///     k <- A[i] p' mod b
///     A <- A + k p b^i
/// }
/// A <- A / b^n
/// if A >= p {
///     A <- A - p
/// }
/// ```
///
/// For secp256r1, with a 32-bit arithmetic, we have the following
/// simplifications:
///
/// - `p'` is 1, so our multiplicand is simply the first limb of the intermediate A.
///
/// - The first limb of p is 2^32 - 1; multiplications by this limb can be simplified
///   to a shift and subtraction:
///   ```text
///       a_i * (2^32 - 1) = a_i * 2^32 - a_i = (a_i << 32) - a_i
///   ```
///   However, because `p' = 1`, the first limb of p is multiplied by limb i of the
///   intermediate A and then immediately added to that same limb, so we simply
///   initialize the carry to limb i of the intermediate.
///
///   The same applies for the second and third limb.
///
/// - The fourth limb of p is zero, so we can ignore any multiplications by it and just
///   add the carry.
///
///   The same applies for the fifth and sixth limb.
///
/// - The seventh limb of p is one, so we can substitute a `mac` operation with a `carrying_add` one.
///
/// References:
/// - Handbook of Applied Cryptography, Chapter 14
///   Algorithm 14.32
///   http://cacr.uwaterloo.ca/hac/about/chap14.pdf
///
/// - Efficient and Secure Elliptic Curve Cryptography Implementation of Curve P-256
///   Algorithm 7) Montgomery Word-by-Word Reduction
///   https://csrc.nist.gov/csrc/media/events/workshop-on-elliptic-curve-cryptography-standards/documents/papers/session6-adalier-mehmet.pdf
#[inline]
#[allow(clippy::too_many_arguments)]
pub(super) const fn montgomery_reduce(lo: &U256, hi: &U256) -> U256 {
    let lo = lo.as_limbs();
    let hi = hi.as_limbs();

    let a0 = lo[0];
    let a1 = lo[1];
    let a2 = lo[2];
    let a3 = lo[3];
    let a4 = lo[4];
    let a5 = lo[5];
    let a6 = lo[6];
    let a7 = lo[7];
    let a8 = hi[0];
    let a9 = hi[1];
    let a10 = hi[2];
    let a11 = hi[3];
    let a12 = hi[4];
    let a13 = hi[5];
    let a14 = hi[6];
    let a15 = hi[7];

    let (a3, carry) = a3.carrying_add(Limb::ZERO, a0);
    let (a4, carry) = a4.carrying_add(Limb::ZERO, carry);
    let (a5, carry) = a5.carrying_add(Limb::ZERO, carry);
    let (a6, carry) = a6.carrying_add(a0, carry);

    // NOTE `MODULUS[7]` is 2^32 - 1, this could be optimized to `carrying_add` and `borrowing_sub`
    // but multiplication costs 1 clock-cycle on several architectures,
    // thanks to parallelization
    let (a7, carry) = a0.carrying_mul_add(MODULUS[7], a7, carry);

    // optimization with only carrying_add and borrowing_sub
    let (a8, carry2) = a8.carrying_add(Limb::ZERO, carry);

    let (a4, carry) = a4.carrying_add(Limb::ZERO, a1);
    let (a5, carry) = a5.carrying_add(Limb::ZERO, carry);
    let (a6, carry) = a6.carrying_add(Limb::ZERO, carry);
    let (a7, carry) = a7.carrying_add(a1, carry);
    let (a8, carry) = a1.carrying_mul_add(MODULUS[7], a8, carry);
    let (a9, carry2) = a9.carrying_add(carry2, carry);

    let (a5, carry) = a5.carrying_add(Limb::ZERO, a2);
    let (a6, carry) = a6.carrying_add(Limb::ZERO, carry);
    let (a7, carry) = a7.carrying_add(Limb::ZERO, carry);
    let (a8, carry) = a8.carrying_add(a2, carry);
    let (a9, carry) = a2.carrying_mul_add(MODULUS[7], a9, carry);
    let (a10, carry2) = a10.carrying_add(carry2, carry);

    let (a6, carry) = a6.carrying_add(Limb::ZERO, a3);
    let (a7, carry) = a7.carrying_add(Limb::ZERO, carry);
    let (a8, carry) = a8.carrying_add(Limb::ZERO, carry);
    let (a9, carry) = a9.carrying_add(a3, carry);
    let (a10, carry) = a3.carrying_mul_add(MODULUS[7], a10, carry);
    let (a11, carry2) = a11.carrying_add(carry2, carry);

    let (a7, carry) = a7.carrying_add(Limb::ZERO, a4);
    let (a8, carry) = a8.carrying_add(Limb::ZERO, carry);
    let (a9, carry) = a9.carrying_add(Limb::ZERO, carry);
    let (a10, carry) = a10.carrying_add(a4, carry);
    let (a11, carry) = a4.carrying_mul_add(MODULUS[7], a11, carry);
    let (a12, carry2) = a12.carrying_add(carry2, carry);

    let (a8, carry) = a8.carrying_add(Limb::ZERO, a5);
    let (a9, carry) = a9.carrying_add(Limb::ZERO, carry);
    let (a10, carry) = a10.carrying_add(Limb::ZERO, carry);
    let (a11, carry) = a11.carrying_add(a5, carry);
    let (a12, carry) = a5.carrying_mul_add(MODULUS[7], a12, carry);
    let (a13, carry2) = a13.carrying_add(carry2, carry);

    let (a9, carry) = a9.carrying_add(Limb::ZERO, a6);
    let (a10, carry) = a10.carrying_add(Limb::ZERO, carry);
    let (a11, carry) = a11.carrying_add(Limb::ZERO, carry);
    let (a12, carry) = a12.carrying_add(a6, carry);
    let (a13, carry) = a6.carrying_mul_add(MODULUS[7], a13, carry);
    let (a14, carry2) = a14.carrying_add(carry2, carry);

    let (a10, carry) = a10.carrying_add(Limb::ZERO, a7);
    let (a11, carry) = a11.carrying_add(Limb::ZERO, carry);
    let (a12, carry) = a12.carrying_add(Limb::ZERO, carry);
    let (a13, carry) = a13.carrying_add(a7, carry);
    let (a14, carry) = a7.carrying_mul_add(MODULUS[7], a14, carry);
    let (a15, a16) = a15.carrying_add(carry2, carry);

    // Result may be within MODULUS of the correct value
    let (result, _) = sub_inner(
        [a8, a9, a10, a11, a12, a13, a14, a15, a16],
        [
            MODULUS[0],
            MODULUS[1],
            MODULUS[2],
            MODULUS[3],
            MODULUS[4],
            MODULUS[5],
            MODULUS[6],
            MODULUS[7],
            Limb::ZERO,
        ],
    );

    U256::new([
        result[0], result[1], result[2], result[3], result[4], result[5], result[6], result[7],
    ])
}

#[inline]
#[allow(clippy::too_many_arguments)]
const fn sub_inner(l: [Limb; 9], r: [Limb; 9]) -> ([Limb; 8], Limb) {
    let (w0, borrow) = l[0].borrowing_sub(r[0], Limb::ZERO);
    let (w1, borrow) = l[1].borrowing_sub(r[1], borrow);
    let (w2, borrow) = l[2].borrowing_sub(r[2], borrow);
    let (w3, borrow) = l[3].borrowing_sub(r[3], borrow);
    let (w4, borrow) = l[4].borrowing_sub(r[4], borrow);
    let (w5, borrow) = l[5].borrowing_sub(r[5], borrow);
    let (w6, borrow) = l[6].borrowing_sub(r[6], borrow);
    let (w7, borrow) = l[7].borrowing_sub(r[7], borrow);
    let (_, borrow) = l[8].borrowing_sub(r[8], borrow);

    // If underflow occurred on the final limb, borrow = 0xfff...fff, otherwise
    // borrow = 0x000...000. Thus, we use it as a mask to conditionally add
    // the MODULUS.

    let (w0, carry) = w0.carrying_add(MODULUS[0].bitand(borrow), Limb::ZERO);
    let (w1, carry) = w1.carrying_add(MODULUS[1].bitand(borrow), carry);
    let (w2, carry) = w2.carrying_add(MODULUS[2].bitand(borrow), carry);
    let (w3, carry) = w3.carrying_add(MODULUS[3].bitand(borrow), carry);
    let (w4, carry) = w4.carrying_add(MODULUS[4].bitand(borrow), carry);
    let (w5, carry) = w5.carrying_add(MODULUS[5].bitand(borrow), carry);
    let (w6, carry) = w6.carrying_add(MODULUS[6].bitand(borrow), carry);
    let (w7, _) = w7.carrying_add(MODULUS[7].bitand(borrow), carry);

    ([w0, w1, w2, w3, w4, w5, w6, w7], borrow)
}