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
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2018 Isis Lovecruft, Henry de Valence
// See LICENSE for licensing information.
//
// Authors:
// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>

//! Field arithmetic modulo \\(p = 2\^{255} - 19\\).
//!
//! The `curve25519_dalek::field` module provides a type alias
//! `curve25519_dalek::field::FieldElement` to a field element type
//! defined in the `backend` module; either `FieldElement51` or
//! `FieldElement2625`.
//!
//! Field operations defined in terms of machine
//! operations, such as field multiplication or squaring, are defined in
//! the backend implementation.
//!
//! Field operations defined in terms of other field operations, such as
//! field inversion or square roots, are defined here.

use core::cmp::{Eq, PartialEq};

use subtle::ConditionallySelectable;
use subtle::ConditionallyNegatable;
use subtle::Choice;
use subtle::ConstantTimeEq;

use constants;
use backend;

#[cfg(feature = "u64_backend")]
pub use backend::serial::u64::field::*;
/// A `FieldElement` represents an element of the field
/// \\( \mathbb Z / (2\^{255} - 19)\\).
///
/// The `FieldElement` type is an alias for one of the platform-specific
/// implementations.
#[cfg(feature = "u64_backend")]
pub type FieldElement = backend::serial::u64::field::FieldElement51;

#[cfg(feature = "u32_backend")]
pub use backend::serial::u32::field::*;
/// A `FieldElement` represents an element of the field
/// \\( \mathbb Z / (2\^{255} - 19)\\).
///
/// The `FieldElement` type is an alias for one of the platform-specific
/// implementations.
#[cfg(feature = "u32_backend")]
pub type FieldElement = backend::serial::u32::field::FieldElement2625;

impl Eq for FieldElement {}

impl PartialEq for FieldElement {
    fn eq(&self, other: &FieldElement) -> bool {
        self.ct_eq(other).unwrap_u8() == 1u8
    }
}

impl ConstantTimeEq for FieldElement {
    /// Test equality between two `FieldElement`s.  Since the
    /// internal representation is not canonical, the field elements
    /// are normalized to wire format before comparison.
    fn ct_eq(&self, other: &FieldElement) -> Choice {
        self.to_bytes().ct_eq(&other.to_bytes())
    }
}

impl FieldElement {
    /// Determine if this `FieldElement` is negative, in the sense
    /// used in the ed25519 paper: `x` is negative if the low bit is
    /// set.
    ///
    /// # Return
    ///
    /// If negative, return `Choice(1)`.  Otherwise, return `Choice(0)`.
    pub fn is_negative(&self) -> Choice {
        let bytes = self.to_bytes();
        (bytes[0] & 1).into()
    }

    /// Determine if this `FieldElement` is zero.
    ///
    /// # Return
    ///
    /// If zero, return `Choice(1)`.  Otherwise, return `Choice(0)`.
    pub fn is_zero(&self) -> Choice {
        let zero = [0u8; 32];
        let bytes = self.to_bytes();

        bytes.ct_eq(&zero)
    }

    /// Compute (self^(2^250-1), self^11), used as a helper function
    /// within invert() and pow22523().
    fn pow22501(&self) -> (FieldElement, FieldElement) {
        // Instead of managing which temporary variables are used
        // for what, we define as many as we need and leave stack
        // allocation to the compiler
        //
        // Each temporary variable t_i is of the form (self)^e_i.
        // Squaring t_i corresponds to multiplying e_i by 2,
        // so the pow2k function shifts e_i left by k places.
        // Multiplying t_i and t_j corresponds to adding e_i + e_j.
        //
        // Temporary t_i                      Nonzero bits of e_i
        //
        let t0  = self.square();           // 1         e_0 = 2^1
        let t1  = t0.square().square();    // 3         e_1 = 2^3
        let t2  = self * &t1;              // 3,0       e_2 = 2^3 + 2^0
        let t3  = &t0 * &t2;               // 3,1,0
        let t4  = t3.square();             // 4,2,1
        let t5  = &t2 * &t4;               // 4,3,2,1,0
        let t6  = t5.pow2k(5);             // 9,8,7,6,5
        let t7  = &t6 * &t5;               // 9,8,7,6,5,4,3,2,1,0
        let t8  = t7.pow2k(10);            // 19..10
        let t9  = &t8 * &t7;               // 19..0
        let t10 = t9.pow2k(20);            // 39..20
        let t11 = &t10 * &t9;              // 39..0
        let t12 = t11.pow2k(10);           // 49..10
        let t13 = &t12 * &t7;              // 49..0
        let t14 = t13.pow2k(50);           // 99..50
        let t15 = &t14 * &t13;             // 99..0
        let t16 = t15.pow2k(100);          // 199..100
        let t17 = &t16 * &t15;             // 199..0
        let t18 = t17.pow2k(50);           // 249..50
        let t19 = &t18 * &t13;             // 249..0

        (t19, t3)
    }

    /// Given a slice of public `FieldElements`, replace each with its inverse.
    ///
    /// All input `FieldElements` **MUST** be nonzero.
    #[cfg(feature = "alloc")]
    pub fn batch_invert(inputs: &mut [FieldElement]) {
        // Montgomery’s Trick and Fast Implementation of Masked AES
        // Genelle, Prouff and Quisquater
        // Section 3.2

        let n = inputs.len();
        let mut scratch = vec![FieldElement::one(); n];

        // Keep an accumulator of all of the previous products
        let mut acc = FieldElement::one();

        // Pass through the input vector, recording the previous
        // products in the scratch space
        for (input, scratch) in inputs.iter().zip(scratch.iter_mut()) {
            *scratch = acc;
            acc = &acc * input;
        }

        // Compute the inverse of all products
        acc = acc.invert();

        // Pass through the vector backwards to compute the inverses
        // in place
        for (input, scratch) in inputs.iter_mut().rev().zip(scratch.into_iter().rev()) {
            let tmp = &acc * input;
            *input = &acc * &scratch;
            acc = tmp;
        }
    }

    /// Given a nonzero field element, compute its inverse.
    ///
    /// The inverse is computed as self^(p-2), since
    /// x^(p-2)x = x^(p-1) = 1 (mod p).
    ///
    /// This function returns zero on input zero.
    pub fn invert(&self) -> FieldElement {
        // The bits of p-2 = 2^255 -19 -2 are 11010111111...11.
        //
        //                                 nonzero bits of exponent
        let (t19, t3) = self.pow22501();   // t19: 249..0 ; t3: 3,1,0
        let t20 = t19.pow2k(5);            // 254..5
        let t21 = &t20 * &t3;              // 254..5,3,1,0

        t21
    }

    /// Raise this field element to the power (p-5)/8 = 2^252 -3.
    fn pow_p58(&self) -> FieldElement {
        // The bits of (p-5)/8 are 101111.....11.
        //
        //                                 nonzero bits of exponent
        let (t19, _) = self.pow22501();    // 249..0
        let t20 = t19.pow2k(2);            // 251..2
        let t21 = self * &t20;             // 251..2,0

        t21
    }

    /// Given `FieldElements` `u` and `v`, compute either `sqrt(u/v)`
    /// or `sqrt(i*u/v)` in constant time.
    ///
    /// This function always returns the nonnegative square root.
    ///
    /// # Return
    ///
    /// - `(Choice(1), +sqrt(u/v))  ` if `v` is nonzero and `u/v` is square;
    /// - `(Choice(1), zero)        ` if `u` is zero;
    /// - `(Choice(0), zero)        ` if `v` is zero and `u` is nonzero;
    /// - `(Choice(0), +sqrt(i*u/v))` if `u/v` is nonsquare (so `i*u/v` is square).
    ///
    pub fn sqrt_ratio_i(u: &FieldElement, v: &FieldElement) -> (Choice, FieldElement) {
        // Using the same trick as in ed25519 decoding, we merge the
        // inversion, the square root, and the square test as follows.
        //
        // To compute sqrt(α), we can compute β = α^((p+3)/8).
        // Then β^2 = ±α, so multiplying β by sqrt(-1) if necessary
        // gives sqrt(α).
        //
        // To compute 1/sqrt(α), we observe that
        //    1/β = α^(p-1 - (p+3)/8) = α^((7p-11)/8)
        //                            = α^3 * (α^7)^((p-5)/8).
        //
        // We can therefore compute sqrt(u/v) = sqrt(u)/sqrt(v)
        // by first computing
        //    r = u^((p+3)/8) v^(p-1-(p+3)/8)
        //      = u u^((p-5)/8) v^3 (v^7)^((p-5)/8)
        //      = (uv^3) (uv^7)^((p-5)/8).
        //
        // If v is nonzero and u/v is square, then r^2 = ±u/v,
        //                                     so vr^2 = ±u.
        // If vr^2 =  u, then sqrt(u/v) = r.
        // If vr^2 = -u, then sqrt(u/v) = r*sqrt(-1).
        //
        // If v is zero, r is also zero.

        let v3 = &v.square()  * v;
        let v7 = &v3.square() * v;
        let mut r = &(u * &v3) * &(u * &v7).pow_p58();
        let check = v * &r.square();

        let i = &constants::SQRT_M1;

        let correct_sign_sqrt   = check.ct_eq(        u);
        let flipped_sign_sqrt   = check.ct_eq(     &(-u));
        let flipped_sign_sqrt_i = check.ct_eq(&(&(-u)*i));

        let r_prime = &constants::SQRT_M1 * &r;
        r.conditional_assign(&r_prime, flipped_sign_sqrt | flipped_sign_sqrt_i);

        // Choose the nonnegative square root.
        let r_is_negative = r.is_negative();
        r.conditional_negate(r_is_negative);

        let was_nonzero_square = correct_sign_sqrt | flipped_sign_sqrt;

        (was_nonzero_square, r)
    }

    /// Attempt to compute `sqrt(1/self)` in constant time.
    ///
    /// Convenience wrapper around `sqrt_ratio_i`.
    ///
    /// This function always returns the nonnegative square root.
    ///
    /// # Return
    ///
    /// - `(Choice(1), +sqrt(1/self))  ` if `self` is a nonzero square;
    /// - `(Choice(0), zero)           ` if `self` is zero;
    /// - `(Choice(0), +sqrt(i/self))  ` if `self` is a nonzero nonsquare;
    ///
    pub fn invsqrt(&self) -> (Choice, FieldElement) {
        FieldElement::sqrt_ratio_i(&FieldElement::one(), self)
    }
}

#[cfg(test)]
mod test {
    use field::*;
    use subtle::ConditionallyNegatable;

    /// Random element a of GF(2^255-19), from Sage
    /// a = 1070314506888354081329385823235218444233221\
    ///     2228051251926706380353716438957572
    static A_BYTES: [u8; 32] =
        [ 0x04, 0xfe, 0xdf, 0x98, 0xa7, 0xfa, 0x0a, 0x68,
          0x84, 0x92, 0xbd, 0x59, 0x08, 0x07, 0xa7, 0x03,
          0x9e, 0xd1, 0xf6, 0xf2, 0xe1, 0xd9, 0xe2, 0xa4,
          0xa4, 0x51, 0x47, 0x36, 0xf3, 0xc3, 0xa9, 0x17];

    /// Byte representation of a**2
    static ASQ_BYTES: [u8; 32] =
        [ 0x75, 0x97, 0x24, 0x9e, 0xe6, 0x06, 0xfe, 0xab,
          0x24, 0x04, 0x56, 0x68, 0x07, 0x91, 0x2d, 0x5d,
          0x0b, 0x0f, 0x3f, 0x1c, 0xb2, 0x6e, 0xf2, 0xe2,
          0x63, 0x9c, 0x12, 0xba, 0x73, 0x0b, 0xe3, 0x62];

    /// Byte representation of 1/a
    static AINV_BYTES: [u8; 32] =
        [0x96, 0x1b, 0xcd, 0x8d, 0x4d, 0x5e, 0xa2, 0x3a,
         0xe9, 0x36, 0x37, 0x93, 0xdb, 0x7b, 0x4d, 0x70,
         0xb8, 0x0d, 0xc0, 0x55, 0xd0, 0x4c, 0x1d, 0x7b,
         0x90, 0x71, 0xd8, 0xe9, 0xb6, 0x18, 0xe6, 0x30];

    /// Byte representation of a^((p-5)/8)
    static AP58_BYTES: [u8; 32] =
        [0x6a, 0x4f, 0x24, 0x89, 0x1f, 0x57, 0x60, 0x36,
         0xd0, 0xbe, 0x12, 0x3c, 0x8f, 0xf5, 0xb1, 0x59,
         0xe0, 0xf0, 0xb8, 0x1b, 0x20, 0xd2, 0xb5, 0x1f,
         0x15, 0x21, 0xf9, 0xe3, 0xe1, 0x61, 0x21, 0x55];

    #[test]
    fn a_mul_a_vs_a_squared_constant() {
        let a = FieldElement::from_bytes(&A_BYTES);
        let asq = FieldElement::from_bytes(&ASQ_BYTES);
        assert_eq!(asq, &a * &a);
    }

    #[test]
    fn a_square_vs_a_squared_constant() {
        let a = FieldElement::from_bytes(&A_BYTES);
        let asq = FieldElement::from_bytes(&ASQ_BYTES);
        assert_eq!(asq, a.square());
    }

    #[test]
    fn a_square2_vs_a_squared_constant() {
        let a = FieldElement::from_bytes(&A_BYTES);
        let asq = FieldElement::from_bytes(&ASQ_BYTES);
        assert_eq!(a.square2(), &asq+&asq);
    }

    #[test]
    fn a_invert_vs_inverse_of_a_constant() {
        let a    = FieldElement::from_bytes(&A_BYTES);
        let ainv = FieldElement::from_bytes(&AINV_BYTES);
        let should_be_inverse = a.invert();
        assert_eq!(ainv, should_be_inverse);
        assert_eq!(FieldElement::one(), &a * &should_be_inverse);
    }

    #[test]
    fn batch_invert_a_matches_nonbatched() {
        let a    = FieldElement::from_bytes(&A_BYTES);
        let ap58 = FieldElement::from_bytes(&AP58_BYTES);
        let asq  = FieldElement::from_bytes(&ASQ_BYTES);
        let ainv = FieldElement::from_bytes(&AINV_BYTES);
        let a2   = &a + &a;
        let a_list = vec![a, ap58, asq, ainv, a2];
        let mut ainv_list = a_list.clone();
        FieldElement::batch_invert(&mut ainv_list[..]);
        for i in 0..5 {
            assert_eq!(a_list[i].invert(), ainv_list[i]);
        }
    }

    #[test]
    fn sqrt_ratio_behavior() {
        let zero = FieldElement::zero();
        let one = FieldElement::one();
        let i = constants::SQRT_M1;
        let two = &one + &one; // 2 is nonsquare mod p.
        let four = &two + &two; // 4 is square mod p.

        // 0/0 should return (1, 0) since u is 0
        let (choice, sqrt) = FieldElement::sqrt_ratio_i(&zero, &zero);
        assert_eq!(choice.unwrap_u8(), 1);
        assert_eq!(sqrt, zero);
        assert_eq!(sqrt.is_negative().unwrap_u8(), 0);

        // 1/0 should return (0, 0) since v is 0, u is nonzero
        let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &zero);
        assert_eq!(choice.unwrap_u8(), 0);
        assert_eq!(sqrt, zero);
        assert_eq!(sqrt.is_negative().unwrap_u8(), 0);

        // 2/1 is nonsquare, so we expect (0, sqrt(i*2))
        let (choice, sqrt) = FieldElement::sqrt_ratio_i(&two, &one);
        assert_eq!(choice.unwrap_u8(), 0);
        assert_eq!(sqrt.square(), &two * &i);
        assert_eq!(sqrt.is_negative().unwrap_u8(), 0);

        // 4/1 is square, so we expect (1, sqrt(4))
        let (choice, sqrt) = FieldElement::sqrt_ratio_i(&four, &one);
        assert_eq!(choice.unwrap_u8(), 1);
        assert_eq!(sqrt.square(), four);
        assert_eq!(sqrt.is_negative().unwrap_u8(), 0);

        // 1/4 is square, so we expect (1, 1/sqrt(4))
        let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &four);
        assert_eq!(choice.unwrap_u8(), 1);
        assert_eq!(&sqrt.square() * &four, one);
        assert_eq!(sqrt.is_negative().unwrap_u8(), 0);
    }

    #[test]
    fn a_p58_vs_ap58_constant() {
        let a    = FieldElement::from_bytes(&A_BYTES);
        let ap58 = FieldElement::from_bytes(&AP58_BYTES);
        assert_eq!(ap58, a.pow_p58());
    }

    #[test]
    fn equality() {
        let a    = FieldElement::from_bytes(&A_BYTES);
        let ainv = FieldElement::from_bytes(&AINV_BYTES);
        assert!(a == a);
        assert!(a != ainv);
    }

    /// Notice that the last element has the high bit set, which
    /// should be ignored
    static B_BYTES: [u8;32] =
        [113, 191, 169, 143,  91, 234, 121,  15,
         241, 131, 217,  36, 230, 101,  92, 234,
           8, 208, 170, 251,  97, 127,  70, 210,
          58,  23, 166,  87, 240, 169, 184, 178];

    #[test]
    fn from_bytes_highbit_is_ignored() {
        let mut cleared_bytes = B_BYTES;
        cleared_bytes[31] &= 127u8;
        let with_highbit_set    = FieldElement::from_bytes(&B_BYTES);
        let without_highbit_set = FieldElement::from_bytes(&cleared_bytes);
        assert_eq!(without_highbit_set, with_highbit_set);
    }

    #[test]
    fn conditional_negate() {
        let       one = FieldElement::one();
        let minus_one = FieldElement::minus_one();
        let mut x = one;
        x.conditional_negate(Choice::from(1));
        assert_eq!(x, minus_one);
        x.conditional_negate(Choice::from(0));
        assert_eq!(x, minus_one);
        x.conditional_negate(Choice::from(1));
        assert_eq!(x, one);
    }

    #[test]
    fn encoding_is_canonical() {
        // Encode 1 wrongly as 1 + (2^255 - 19) = 2^255 - 18
        let one_encoded_wrongly_bytes: [u8;32] = [0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f];
        // Decode to a field element
        let one = FieldElement::from_bytes(&one_encoded_wrongly_bytes);
        // .. then check that the encoding is correct
        let one_bytes = one.to_bytes();
        assert_eq!(one_bytes[0], 1);
        for i in 1..32 {
            assert_eq!(one_bytes[i], 0);
        }
    }

    #[test]
    fn batch_invert_empty() {
        FieldElement::batch_invert(&mut []);
    }
}