p256/arithmetic/
scalar.rs

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1//! Scalar field arithmetic modulo n = 115792089210356248762697446949407573529996955224135760342422259061068512044369
2
3#[cfg_attr(target_pointer_width = "32", path = "scalar/scalar32.rs")]
4#[cfg_attr(target_pointer_width = "64", path = "scalar/scalar64.rs")]
5mod scalar_impl;
6
7use self::scalar_impl::barrett_reduce;
8use crate::{FieldBytes, NistP256, SecretKey, ORDER_HEX};
9use core::{
10    fmt::{self, Debug},
11    iter::{Product, Sum},
12    ops::{Add, AddAssign, Mul, MulAssign, Neg, Shr, ShrAssign, Sub, SubAssign},
13};
14use elliptic_curve::{
15    bigint::{prelude::*, Limb, U256},
16    group::ff::{self, Field, PrimeField},
17    ops::{Invert, Reduce, ReduceNonZero},
18    rand_core::RngCore,
19    scalar::{FromUintUnchecked, IsHigh},
20    subtle::{
21        Choice, ConditionallySelectable, ConstantTimeEq, ConstantTimeGreater, ConstantTimeLess,
22        CtOption,
23    },
24    zeroize::DefaultIsZeroes,
25    Curve, ScalarPrimitive,
26};
27
28#[cfg(feature = "bits")]
29use {crate::ScalarBits, elliptic_curve::group::ff::PrimeFieldBits};
30
31#[cfg(feature = "serde")]
32use serdect::serde::{de, ser, Deserialize, Serialize};
33
34/// Constant representing the modulus
35/// n = FFFFFFFF 00000000 FFFFFFFF FFFFFFFF BCE6FAAD A7179E84 F3B9CAC2 FC632551
36pub(crate) const MODULUS: U256 = NistP256::ORDER;
37
38/// `MODULUS / 2`
39const FRAC_MODULUS_2: Scalar = Scalar(MODULUS.shr_vartime(1));
40
41/// MU = floor(2^512 / n)
42///    = 115792089264276142090721624801893421302707618245269942344307673200490803338238
43///    = 0x100000000fffffffffffffffeffffffff43190552df1a6c21012ffd85eedf9bfe
44pub const MU: [u64; 5] = [
45    0x012f_fd85_eedf_9bfe,
46    0x4319_0552_df1a_6c21,
47    0xffff_fffe_ffff_ffff,
48    0x0000_0000_ffff_ffff,
49    0x0000_0000_0000_0001,
50];
51
52/// Scalars are elements in the finite field modulo n.
53///
54/// # Trait impls
55///
56/// Much of the important functionality of scalars is provided by traits from
57/// the [`ff`](https://docs.rs/ff/) crate, which is re-exported as
58/// `p256::elliptic_curve::ff`:
59///
60/// - [`Field`](https://docs.rs/ff/latest/ff/trait.Field.html) -
61///   represents elements of finite fields and provides:
62///   - [`Field::random`](https://docs.rs/ff/latest/ff/trait.Field.html#tymethod.random) -
63///     generate a random scalar
64///   - `double`, `square`, and `invert` operations
65///   - Bounds for [`Add`], [`Sub`], [`Mul`], and [`Neg`] (as well as `*Assign` equivalents)
66///   - Bounds for [`ConditionallySelectable`] from the `subtle` crate
67/// - [`PrimeField`](https://docs.rs/ff/latest/ff/trait.PrimeField.html) -
68///   represents elements of prime fields and provides:
69///   - `from_repr`/`to_repr` for converting field elements from/to big integers.
70///   - `multiplicative_generator` and `root_of_unity` constants.
71/// - [`PrimeFieldBits`](https://docs.rs/ff/latest/ff/trait.PrimeFieldBits.html) -
72///   operations over field elements represented as bits (requires `bits` feature)
73///
74/// Please see the documentation for the relevant traits for more information.
75///
76/// # `serde` support
77///
78/// When the `serde` feature of this crate is enabled, the `Serialize` and
79/// `Deserialize` traits are impl'd for this type.
80///
81/// The serialization is a fixed-width big endian encoding. When used with
82/// textual formats, the binary data is encoded as hexadecimal.
83#[derive(Clone, Copy, Default)]
84pub struct Scalar(pub(crate) U256);
85
86impl Scalar {
87    /// Zero scalar.
88    pub const ZERO: Self = Self(U256::ZERO);
89
90    /// Multiplicative identity.
91    pub const ONE: Self = Self(U256::ONE);
92
93    /// Returns the SEC1 encoding of this scalar.
94    pub fn to_bytes(&self) -> FieldBytes {
95        self.0.to_be_byte_array()
96    }
97
98    /// Returns self + rhs mod n
99    pub const fn add(&self, rhs: &Self) -> Self {
100        Self(self.0.add_mod(&rhs.0, &NistP256::ORDER))
101    }
102
103    /// Returns 2*self.
104    pub const fn double(&self) -> Self {
105        self.add(self)
106    }
107
108    /// Returns self - rhs mod n.
109    pub const fn sub(&self, rhs: &Self) -> Self {
110        Self(self.0.sub_mod(&rhs.0, &NistP256::ORDER))
111    }
112
113    /// Returns self * rhs mod n
114    pub const fn multiply(&self, rhs: &Self) -> Self {
115        let (lo, hi) = self.0.mul_wide(&rhs.0);
116        Self(barrett_reduce(lo, hi))
117    }
118
119    /// Returns self * self mod p
120    pub const fn square(&self) -> Self {
121        // Schoolbook multiplication.
122        self.multiply(self)
123    }
124
125    /// Right shifts the scalar.
126    ///
127    /// Note: not constant-time with respect to the `shift` parameter.
128    pub const fn shr_vartime(&self, shift: usize) -> Scalar {
129        Self(self.0.shr_vartime(shift))
130    }
131
132    /// Returns the multiplicative inverse of self, if self is non-zero
133    pub fn invert(&self) -> CtOption<Self> {
134        CtOption::new(self.invert_unchecked(), !self.is_zero())
135    }
136
137    /// Returns the multiplicative inverse of self.
138    ///
139    /// Does not check that self is non-zero.
140    const fn invert_unchecked(&self) -> Self {
141        // We need to find b such that b * a ≡ 1 mod p. As we are in a prime
142        // field, we can apply Fermat's Little Theorem:
143        //
144        //    a^p         ≡ a mod p
145        //    a^(p-1)     ≡ 1 mod p
146        //    a^(p-2) * a ≡ 1 mod p
147        //
148        // Thus inversion can be implemented with a single exponentiation.
149        //
150        // This is `n - 2`, so the top right two digits are `4f` instead of `51`.
151        self.pow_vartime(&[
152            0xf3b9_cac2_fc63_254f,
153            0xbce6_faad_a717_9e84,
154            0xffff_ffff_ffff_ffff,
155            0xffff_ffff_0000_0000,
156        ])
157    }
158
159    /// Exponentiates `self` by `exp`, where `exp` is a little-endian order integer
160    /// exponent.
161    pub const fn pow_vartime(&self, exp: &[u64]) -> Self {
162        let mut res = Self::ONE;
163
164        let mut i = exp.len();
165        while i > 0 {
166            i -= 1;
167
168            let mut j = 64;
169            while j > 0 {
170                j -= 1;
171                res = res.square();
172
173                if ((exp[i] >> j) & 1) == 1 {
174                    res = res.multiply(self);
175                }
176            }
177        }
178
179        res
180    }
181
182    /// Is integer representing equivalence class odd?
183    pub fn is_odd(&self) -> Choice {
184        self.0.is_odd()
185    }
186
187    /// Is integer representing equivalence class even?
188    pub fn is_even(&self) -> Choice {
189        !self.is_odd()
190    }
191}
192
193impl AsRef<Scalar> for Scalar {
194    fn as_ref(&self) -> &Scalar {
195        self
196    }
197}
198
199impl Field for Scalar {
200    const ZERO: Self = Self::ZERO;
201    const ONE: Self = Self::ONE;
202
203    fn random(mut rng: impl RngCore) -> Self {
204        let mut bytes = FieldBytes::default();
205
206        // Generate a uniformly random scalar using rejection sampling,
207        // which produces a uniformly random distribution of scalars.
208        //
209        // This method is not constant time, but should be secure so long as
210        // rejected RNG outputs are unrelated to future ones (which is a
211        // necessary property of a `CryptoRng`).
212        //
213        // With an unbiased RNG, the probability of failing to complete after 4
214        // iterations is vanishingly small.
215        loop {
216            rng.fill_bytes(&mut bytes);
217            if let Some(scalar) = Scalar::from_repr(bytes).into() {
218                return scalar;
219            }
220        }
221    }
222
223    #[must_use]
224    fn square(&self) -> Self {
225        Scalar::square(self)
226    }
227
228    #[must_use]
229    fn double(&self) -> Self {
230        self.add(self)
231    }
232
233    fn invert(&self) -> CtOption<Self> {
234        Scalar::invert(self)
235    }
236
237    /// Tonelli-Shank's algorithm for q mod 16 = 1
238    /// <https://eprint.iacr.org/2012/685.pdf> (page 12, algorithm 5)
239    #[allow(clippy::many_single_char_names)]
240    fn sqrt(&self) -> CtOption<Self> {
241        // Note: `pow_vartime` is constant-time with respect to `self`
242        let w = self.pow_vartime(&[
243            0x279dce5617e3192a,
244            0xfde737d56d38bcf4,
245            0x07ffffffffffffff,
246            0x07fffffff8000000,
247        ]);
248
249        let mut v = Self::S;
250        let mut x = *self * w;
251        let mut b = x * w;
252        let mut z = Self::ROOT_OF_UNITY;
253
254        for max_v in (1..=Self::S).rev() {
255            let mut k = 1;
256            let mut tmp = b.square();
257            let mut j_less_than_v = Choice::from(1);
258
259            for j in 2..max_v {
260                let tmp_is_one = tmp.ct_eq(&Self::ONE);
261                let squared = Self::conditional_select(&tmp, &z, tmp_is_one).square();
262                tmp = Self::conditional_select(&squared, &tmp, tmp_is_one);
263                let new_z = Self::conditional_select(&z, &squared, tmp_is_one);
264                j_less_than_v &= !j.ct_eq(&v);
265                k = u32::conditional_select(&j, &k, tmp_is_one);
266                z = Self::conditional_select(&z, &new_z, j_less_than_v);
267            }
268
269            let result = x * z;
270            x = Self::conditional_select(&result, &x, b.ct_eq(&Self::ONE));
271            z = z.square();
272            b *= z;
273            v = k;
274        }
275
276        CtOption::new(x, x.square().ct_eq(self))
277    }
278
279    fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
280        ff::helpers::sqrt_ratio_generic(num, div)
281    }
282}
283
284impl PrimeField for Scalar {
285    type Repr = FieldBytes;
286
287    const MODULUS: &'static str = ORDER_HEX;
288    const NUM_BITS: u32 = 256;
289    const CAPACITY: u32 = 255;
290    const TWO_INV: Self = Self(U256::from_u8(2)).invert_unchecked();
291    const MULTIPLICATIVE_GENERATOR: Self = Self(U256::from_u8(7));
292    const S: u32 = 4;
293    const ROOT_OF_UNITY: Self = Self(U256::from_be_hex(
294        "ffc97f062a770992ba807ace842a3dfc1546cad004378daf0592d7fbb41e6602",
295    ));
296    const ROOT_OF_UNITY_INV: Self = Self::ROOT_OF_UNITY.invert_unchecked();
297    const DELTA: Self = Self(U256::from_u64(33232930569601));
298
299    /// Attempts to parse the given byte array as an SEC1-encoded scalar.
300    ///
301    /// Returns None if the byte array does not contain a big-endian integer in the range
302    /// [0, p).
303    fn from_repr(bytes: FieldBytes) -> CtOption<Self> {
304        let inner = U256::from_be_byte_array(bytes);
305        CtOption::new(Self(inner), inner.ct_lt(&NistP256::ORDER))
306    }
307
308    fn to_repr(&self) -> FieldBytes {
309        self.to_bytes()
310    }
311
312    fn is_odd(&self) -> Choice {
313        self.0.is_odd()
314    }
315}
316
317#[cfg(feature = "bits")]
318impl PrimeFieldBits for Scalar {
319    #[cfg(target_pointer_width = "32")]
320    type ReprBits = [u32; 8];
321
322    #[cfg(target_pointer_width = "64")]
323    type ReprBits = [u64; 4];
324
325    fn to_le_bits(&self) -> ScalarBits {
326        self.into()
327    }
328
329    fn char_le_bits() -> ScalarBits {
330        NistP256::ORDER.to_words().into()
331    }
332}
333
334impl DefaultIsZeroes for Scalar {}
335
336impl Eq for Scalar {}
337
338impl FromUintUnchecked for Scalar {
339    type Uint = U256;
340
341    fn from_uint_unchecked(uint: Self::Uint) -> Self {
342        Self(uint)
343    }
344}
345
346impl Invert for Scalar {
347    type Output = CtOption<Self>;
348
349    fn invert(&self) -> CtOption<Self> {
350        self.invert()
351    }
352
353    /// Fast variable-time inversion using Stein's algorithm.
354    ///
355    /// Returns none if the scalar is zero.
356    ///
357    /// <https://link.springer.com/article/10.1007/s13389-016-0135-4>
358    ///
359    /// ⚠️ WARNING!
360    ///
361    /// This method should not be used with (unblinded) secret scalars, as its
362    /// variable-time operation can potentially leak secrets through
363    /// sidechannels.
364    #[allow(non_snake_case)]
365    fn invert_vartime(&self) -> CtOption<Self> {
366        let mut u = *self;
367        let mut v = Self(MODULUS);
368        let mut A = Self::ONE;
369        let mut C = Self::ZERO;
370
371        while !bool::from(u.is_zero()) {
372            // u-loop
373            while bool::from(u.is_even()) {
374                u >>= 1;
375
376                let was_odd: bool = A.is_odd().into();
377                A >>= 1;
378
379                if was_odd {
380                    A += FRAC_MODULUS_2;
381                    A += Self::ONE;
382                }
383            }
384
385            // v-loop
386            while bool::from(v.is_even()) {
387                v >>= 1;
388
389                let was_odd: bool = C.is_odd().into();
390                C >>= 1;
391
392                if was_odd {
393                    C += FRAC_MODULUS_2;
394                    C += Self::ONE;
395                }
396            }
397
398            // sub-step
399            if u >= v {
400                u -= &v;
401                A -= &C;
402            } else {
403                v -= &u;
404                C -= &A;
405            }
406        }
407
408        CtOption::new(C, !self.is_zero())
409    }
410}
411
412impl IsHigh for Scalar {
413    fn is_high(&self) -> Choice {
414        self.0.ct_gt(&FRAC_MODULUS_2.0)
415    }
416}
417
418impl Shr<usize> for Scalar {
419    type Output = Self;
420
421    fn shr(self, rhs: usize) -> Self::Output {
422        self.shr_vartime(rhs)
423    }
424}
425
426impl Shr<usize> for &Scalar {
427    type Output = Scalar;
428
429    fn shr(self, rhs: usize) -> Self::Output {
430        self.shr_vartime(rhs)
431    }
432}
433
434impl ShrAssign<usize> for Scalar {
435    fn shr_assign(&mut self, rhs: usize) {
436        *self = *self >> rhs;
437    }
438}
439
440impl PartialEq for Scalar {
441    fn eq(&self, other: &Self) -> bool {
442        self.ct_eq(other).into()
443    }
444}
445
446impl PartialOrd for Scalar {
447    fn partial_cmp(&self, other: &Self) -> Option<core::cmp::Ordering> {
448        Some(self.cmp(other))
449    }
450}
451
452impl Ord for Scalar {
453    fn cmp(&self, other: &Self) -> core::cmp::Ordering {
454        self.0.cmp(&other.0)
455    }
456}
457
458impl From<u32> for Scalar {
459    fn from(k: u32) -> Self {
460        Scalar(k.into())
461    }
462}
463
464impl From<u64> for Scalar {
465    fn from(k: u64) -> Self {
466        Scalar(k.into())
467    }
468}
469
470impl From<u128> for Scalar {
471    fn from(k: u128) -> Self {
472        Scalar(k.into())
473    }
474}
475
476impl From<Scalar> for FieldBytes {
477    fn from(scalar: Scalar) -> Self {
478        scalar.to_bytes()
479    }
480}
481
482impl From<&Scalar> for FieldBytes {
483    fn from(scalar: &Scalar) -> Self {
484        scalar.to_bytes()
485    }
486}
487
488impl From<ScalarPrimitive<NistP256>> for Scalar {
489    fn from(scalar: ScalarPrimitive<NistP256>) -> Scalar {
490        Scalar(*scalar.as_uint())
491    }
492}
493
494impl From<&ScalarPrimitive<NistP256>> for Scalar {
495    fn from(scalar: &ScalarPrimitive<NistP256>) -> Scalar {
496        Scalar(*scalar.as_uint())
497    }
498}
499
500impl From<Scalar> for ScalarPrimitive<NistP256> {
501    fn from(scalar: Scalar) -> ScalarPrimitive<NistP256> {
502        ScalarPrimitive::from(&scalar)
503    }
504}
505
506impl From<&Scalar> for ScalarPrimitive<NistP256> {
507    fn from(scalar: &Scalar) -> ScalarPrimitive<NistP256> {
508        ScalarPrimitive::new(scalar.0).unwrap()
509    }
510}
511
512impl From<&SecretKey> for Scalar {
513    fn from(secret_key: &SecretKey) -> Scalar {
514        *secret_key.to_nonzero_scalar()
515    }
516}
517
518impl From<Scalar> for U256 {
519    fn from(scalar: Scalar) -> U256 {
520        scalar.0
521    }
522}
523
524impl From<&Scalar> for U256 {
525    fn from(scalar: &Scalar) -> U256 {
526        scalar.0
527    }
528}
529
530#[cfg(feature = "bits")]
531impl From<&Scalar> for ScalarBits {
532    fn from(scalar: &Scalar) -> ScalarBits {
533        scalar.0.to_words().into()
534    }
535}
536
537impl Add<Scalar> for Scalar {
538    type Output = Scalar;
539
540    fn add(self, other: Scalar) -> Scalar {
541        Scalar::add(&self, &other)
542    }
543}
544
545impl Add<&Scalar> for &Scalar {
546    type Output = Scalar;
547
548    fn add(self, other: &Scalar) -> Scalar {
549        Scalar::add(self, other)
550    }
551}
552
553impl Add<&Scalar> for Scalar {
554    type Output = Scalar;
555
556    fn add(self, other: &Scalar) -> Scalar {
557        Scalar::add(&self, other)
558    }
559}
560
561impl AddAssign<Scalar> for Scalar {
562    fn add_assign(&mut self, rhs: Scalar) {
563        *self = Scalar::add(self, &rhs);
564    }
565}
566
567impl AddAssign<&Scalar> for Scalar {
568    fn add_assign(&mut self, rhs: &Scalar) {
569        *self = Scalar::add(self, rhs);
570    }
571}
572
573impl Sub<Scalar> for Scalar {
574    type Output = Scalar;
575
576    fn sub(self, other: Scalar) -> Scalar {
577        Scalar::sub(&self, &other)
578    }
579}
580
581impl Sub<&Scalar> for &Scalar {
582    type Output = Scalar;
583
584    fn sub(self, other: &Scalar) -> Scalar {
585        Scalar::sub(self, other)
586    }
587}
588
589impl Sub<&Scalar> for Scalar {
590    type Output = Scalar;
591
592    fn sub(self, other: &Scalar) -> Scalar {
593        Scalar::sub(&self, other)
594    }
595}
596
597impl SubAssign<Scalar> for Scalar {
598    fn sub_assign(&mut self, rhs: Scalar) {
599        *self = Scalar::sub(self, &rhs);
600    }
601}
602
603impl SubAssign<&Scalar> for Scalar {
604    fn sub_assign(&mut self, rhs: &Scalar) {
605        *self = Scalar::sub(self, rhs);
606    }
607}
608
609impl Mul<Scalar> for Scalar {
610    type Output = Scalar;
611
612    fn mul(self, other: Scalar) -> Scalar {
613        Scalar::multiply(&self, &other)
614    }
615}
616
617impl Mul<&Scalar> for &Scalar {
618    type Output = Scalar;
619
620    fn mul(self, other: &Scalar) -> Scalar {
621        Scalar::multiply(self, other)
622    }
623}
624
625impl Mul<&Scalar> for Scalar {
626    type Output = Scalar;
627
628    fn mul(self, other: &Scalar) -> Scalar {
629        Scalar::multiply(&self, other)
630    }
631}
632
633impl MulAssign<Scalar> for Scalar {
634    fn mul_assign(&mut self, rhs: Scalar) {
635        *self = Scalar::multiply(self, &rhs);
636    }
637}
638
639impl MulAssign<&Scalar> for Scalar {
640    fn mul_assign(&mut self, rhs: &Scalar) {
641        *self = Scalar::multiply(self, rhs);
642    }
643}
644
645impl Neg for Scalar {
646    type Output = Scalar;
647
648    fn neg(self) -> Scalar {
649        Scalar::ZERO - self
650    }
651}
652
653impl<'a> Neg for &'a Scalar {
654    type Output = Scalar;
655
656    fn neg(self) -> Scalar {
657        Scalar::ZERO - self
658    }
659}
660
661impl Reduce<U256> for Scalar {
662    type Bytes = FieldBytes;
663
664    fn reduce(w: U256) -> Self {
665        let (r, underflow) = w.sbb(&NistP256::ORDER, Limb::ZERO);
666        let underflow = Choice::from((underflow.0 >> (Limb::BITS - 1)) as u8);
667        Self(U256::conditional_select(&w, &r, !underflow))
668    }
669
670    fn reduce_bytes(bytes: &FieldBytes) -> Self {
671        Self::reduce(U256::from_be_byte_array(*bytes))
672    }
673}
674
675impl ReduceNonZero<U256> for Scalar {
676    fn reduce_nonzero(w: U256) -> Self {
677        const ORDER_MINUS_ONE: U256 = NistP256::ORDER.wrapping_sub(&U256::ONE);
678        let (r, underflow) = w.sbb(&ORDER_MINUS_ONE, Limb::ZERO);
679        let underflow = Choice::from((underflow.0 >> (Limb::BITS - 1)) as u8);
680        Self(U256::conditional_select(&w, &r, !underflow).wrapping_add(&U256::ONE))
681    }
682
683    fn reduce_nonzero_bytes(bytes: &FieldBytes) -> Self {
684        Self::reduce_nonzero(U256::from_be_byte_array(*bytes))
685    }
686}
687
688impl Sum for Scalar {
689    fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
690        iter.reduce(core::ops::Add::add).unwrap_or(Self::ZERO)
691    }
692}
693
694impl<'a> Sum<&'a Scalar> for Scalar {
695    fn sum<I: Iterator<Item = &'a Scalar>>(iter: I) -> Self {
696        iter.copied().sum()
697    }
698}
699
700impl Product for Scalar {
701    fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
702        iter.reduce(core::ops::Mul::mul).unwrap_or(Self::ONE)
703    }
704}
705
706impl<'a> Product<&'a Scalar> for Scalar {
707    fn product<I: Iterator<Item = &'a Scalar>>(iter: I) -> Self {
708        iter.copied().product()
709    }
710}
711
712impl ConditionallySelectable for Scalar {
713    fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
714        Self(U256::conditional_select(&a.0, &b.0, choice))
715    }
716}
717
718impl ConstantTimeEq for Scalar {
719    fn ct_eq(&self, other: &Self) -> Choice {
720        self.0.ct_eq(&other.0)
721    }
722}
723
724impl Debug for Scalar {
725    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
726        write!(f, "Scalar(0x{:X})", &self.0)
727    }
728}
729
730#[cfg(feature = "serde")]
731impl Serialize for Scalar {
732    fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
733    where
734        S: ser::Serializer,
735    {
736        ScalarPrimitive::from(self).serialize(serializer)
737    }
738}
739
740#[cfg(feature = "serde")]
741impl<'de> Deserialize<'de> for Scalar {
742    fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
743    where
744        D: de::Deserializer<'de>,
745    {
746        Ok(ScalarPrimitive::deserialize(deserializer)?.into())
747    }
748}
749
750#[cfg(test)]
751mod tests {
752    use super::Scalar;
753    use crate::{FieldBytes, SecretKey};
754    use elliptic_curve::group::ff::{Field, PrimeField};
755    use primeorder::{
756        impl_field_identity_tests, impl_field_invert_tests, impl_field_sqrt_tests,
757        impl_primefield_tests,
758    };
759
760    /// t = (modulus - 1) >> S
761    const T: [u64; 4] = [
762        0x4f3b9cac2fc63255,
763        0xfbce6faada7179e8,
764        0x0fffffffffffffff,
765        0x0ffffffff0000000,
766    ];
767
768    impl_field_identity_tests!(Scalar);
769    impl_field_invert_tests!(Scalar);
770    impl_field_sqrt_tests!(Scalar);
771    impl_primefield_tests!(Scalar, T);
772
773    #[test]
774    fn from_to_bytes_roundtrip() {
775        let k: u64 = 42;
776        let mut bytes = FieldBytes::default();
777        bytes[24..].copy_from_slice(k.to_be_bytes().as_ref());
778
779        let scalar = Scalar::from_repr(bytes).unwrap();
780        assert_eq!(bytes, scalar.to_bytes());
781    }
782
783    /// Basic tests that multiplication works.
784    #[test]
785    fn multiply() {
786        let one = Scalar::ONE;
787        let two = one + &one;
788        let three = two + &one;
789        let six = three + &three;
790        assert_eq!(six, two * &three);
791
792        let minus_two = -two;
793        let minus_three = -three;
794        assert_eq!(two, -minus_two);
795
796        assert_eq!(minus_three * &minus_two, minus_two * &minus_three);
797        assert_eq!(six, minus_two * &minus_three);
798    }
799
800    /// Tests that a Scalar can be safely converted to a SecretKey and back
801    #[test]
802    fn from_ec_secret() {
803        let scalar = Scalar::ONE;
804        let secret = SecretKey::from_bytes(&scalar.to_bytes()).unwrap();
805        let rederived_scalar = Scalar::from(&secret);
806        assert_eq!(scalar.0, rederived_scalar.0);
807    }
808
809    #[test]
810    #[cfg(all(feature = "bits", target_pointer_width = "32"))]
811    fn scalar_into_scalarbits() {
812        use crate::ScalarBits;
813
814        let minus_one = ScalarBits::from([
815            0xfc63_2550,
816            0xf3b9_cac2,
817            0xa717_9e84,
818            0xbce6_faad,
819            0xffff_ffff,
820            0xffff_ffff,
821            0x0000_0000,
822            0xffff_ffff,
823        ]);
824
825        let scalar_bits = ScalarBits::from(&-Scalar::from(1u32));
826        assert_eq!(minus_one, scalar_bits);
827    }
828}
p256/arithmetic/scalar.rs

p256/arithmetic/
scalar.rs
