ark-r1cs-std/src/groups/curves/short_weierstrass/mod.rs
src/groups/curves/short_weierstrass/mod.rsRaw
1use ark_ec::{
2 short_weierstrass::{Affine as SWAffine, Projective as SWProjective, SWCurveConfig},
3 AffineRepr, CurveConfig, CurveGroup,
4};
5use ark_ff::{AdditiveGroup, BitIteratorBE, Field, One, PrimeField, Zero};
6use ark_relations::r1cs::{ConstraintSystemRef, Namespace, SynthesisError};
7use ark_std::{borrow::Borrow, marker::PhantomData, ops::Mul};
8use non_zero_affine::NonZeroAffineVar;
9
10use crate::{
11 convert::{ToBitsGadget, ToBytesGadget, ToConstraintFieldGadget},
12 fields::{emulated_fp::EmulatedFpVar, fp::FpVar},
13 prelude::*,
14 Vec,
15};
16
17/// This module provides a generic implementation of G1 and G2 for
18/// the [\[BLS12]\](<https://eprint.iacr.org/2002/088.pdf>) family of bilinear groups.
19pub mod bls12;
20
21/// This module provides a generic implementation of G1 and G2 for
22/// the [\[MNT4]\](<https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.20.8113&rep=rep1&type=pdf>)
23/// family of bilinear groups.
24pub mod mnt4;
25/// This module provides a generic implementation of G1 and G2 for
26/// the [\[MNT6]\](<https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.20.8113&rep=rep1&type=pdf>)
27/// family of bilinear groups.
28pub mod mnt6;
29
30/// This module provides a generic implementation of elliptic curve operations
31/// for points on short-weierstrass curves in affine coordinates that **are
32/// not** equal to zero.
33///
34/// Note: this module is **unsafe** in general: it can synthesize unsatisfiable
35/// or underconstrained constraint systems when a represented point _is_ equal
36/// to zero. The [ProjectiveVar] gadget is the recommended way of working with
37/// elliptic curve points.
38pub mod non_zero_affine;
39
40type BasePrimeField<P> = <<P as CurveConfig>::BaseField as Field>::BasePrimeField;
41
42/// An implementation of arithmetic for Short Weierstrass curves that relies on
43/// the complete formulae derived in the paper of
44/// [[Renes, Costello, Batina 2015]](<https://eprint.iacr.org/2015/1060>).
45#[derive(Derivative)]
46#[derivative(Debug, Clone)]
47#[must_use]
48pub struct ProjectiveVar<P: SWCurveConfig, F: FieldVar<P::BaseField, BasePrimeField<P>>>
49where
50 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
51{
52 /// The x-coordinate.
53 pub x: F,
54 /// The y-coordinate.
55 pub y: F,
56 /// The z-coordinate.
57 pub z: F,
58 #[derivative(Debug = "ignore")]
59 _params: PhantomData<P>,
60}
61
62/// An affine representation of a curve point.
63#[derive(Derivative)]
64#[derivative(Debug(bound = "F: ark_std::fmt::Debug"), Clone(bound = "F: Clone"))]
65#[must_use]
66pub struct AffineVar<P: SWCurveConfig, F: FieldVar<P::BaseField, BasePrimeField<P>>>
67where
68 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
69{
70 /// The x-coordinate.
71 pub x: F,
72 /// The y-coordinate.
73 pub y: F,
74 /// Is `self` the point at infinity.
75 pub infinity: Boolean<BasePrimeField<P>>,
76 #[derivative(Debug = "ignore")]
77 _params: PhantomData<P>,
78}
79
80impl<P, F> AffineVar<P, F>
81where
82 P: SWCurveConfig,
83 F: FieldVar<P::BaseField, BasePrimeField<P>>,
84 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
85{
86 fn new(x: F, y: F, infinity: Boolean<BasePrimeField<P>>) -> Self {
87 Self {
88 x,
89 y,
90 infinity,
91 _params: PhantomData,
92 }
93 }
94
95 /// Returns the value assigned to `self` in the underlying
96 /// constraint system.
97 pub fn value(&self) -> Result<SWAffine<P>, SynthesisError> {
98 Ok(match self.infinity.value()? {
99 true => SWAffine::identity(),
100 false => SWAffine::new(self.x.value()?, self.y.value()?),
101 })
102 }
103}
104
105impl<P, F> ToConstraintFieldGadget<BasePrimeField<P>> for AffineVar<P, F>
106where
107 P: SWCurveConfig,
108 F: FieldVar<P::BaseField, BasePrimeField<P>>,
109 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
110 F: ToConstraintFieldGadget<BasePrimeField<P>>,
111{
112 fn to_constraint_field(&self) -> Result<Vec<FpVar<BasePrimeField<P>>>, SynthesisError> {
113 let mut res = Vec::<FpVar<BasePrimeField<P>>>::new();
114
115 res.extend_from_slice(&self.x.to_constraint_field()?);
116 res.extend_from_slice(&self.y.to_constraint_field()?);
117 res.extend_from_slice(&self.infinity.to_constraint_field()?);
118
119 Ok(res)
120 }
121}
122
123impl<P, F> R1CSVar<BasePrimeField<P>> for ProjectiveVar<P, F>
124where
125 P: SWCurveConfig,
126 F: FieldVar<P::BaseField, BasePrimeField<P>>,
127 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
128{
129 type Value = SWProjective<P>;
130
131 fn cs(&self) -> ConstraintSystemRef<BasePrimeField<P>> {
132 self.x.cs().or(self.y.cs()).or(self.z.cs())
133 }
134
135 fn value(&self) -> Result<Self::Value, SynthesisError> {
136 let (x, y, z) = (self.x.value()?, self.y.value()?, self.z.value()?);
137 let result = if let Some(z_inv) = z.inverse() {
138 SWAffine::new(x * &z_inv, y * &z_inv)
139 } else {
140 SWAffine::identity()
141 };
142 Ok(result.into())
143 }
144}
145
146impl<P: SWCurveConfig, F: FieldVar<P::BaseField, BasePrimeField<P>>> ProjectiveVar<P, F>
147where
148 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
149{
150 /// Constructs `Self` from an `(x, y, z)` coordinate triple.
151 pub fn new(x: F, y: F, z: F) -> Self {
152 Self {
153 x,
154 y,
155 z,
156 _params: PhantomData,
157 }
158 }
159
160 /// Convert this point into affine form.
161 #[tracing::instrument(target = "r1cs")]
162 pub fn to_affine(&self) -> Result<AffineVar<P, F>, SynthesisError> {
163 if self.is_constant() {
164 let point = self.value()?.into_affine();
165 let x = F::new_constant(ConstraintSystemRef::None, point.x)?;
166 let y = F::new_constant(ConstraintSystemRef::None, point.y)?;
167 let infinity = Boolean::constant(point.infinity);
168 Ok(AffineVar::new(x, y, infinity))
169 } else {
170 let cs = self.cs();
171 let infinity = self.is_zero()?;
172 let zero_affine = SWAffine::<P>::zero();
173 let zero_x = F::new_constant(cs.clone(), &zero_affine.x)?;
174 let zero_y = F::new_constant(cs.clone(), &zero_affine.y)?;
175 // Allocate a variable whose value is either `self.z.inverse()` if the inverse
176 // exists, and is zero otherwise.
177 let z_inv = F::new_witness(ark_relations::ns!(cs, "z_inverse"), || {
178 Ok(self.z.value()?.inverse().unwrap_or_else(P::BaseField::zero))
179 })?;
180 // The inverse exists if `!self.is_zero()`.
181 // This means that `z_inv * self.z = 1` if `self.is_not_zero()`, and
182 // `z_inv * self.z = 0` if `self.is_zero()`.
183 //
184 // Thus, `z_inv * self.z = !self.is_zero()`.
185 z_inv.mul_equals(&self.z, &F::from(!&infinity))?;
186
187 let non_zero_x = &self.x * &z_inv;
188 let non_zero_y = &self.y * &z_inv;
189
190 let x = infinity.select(&zero_x, &non_zero_x)?;
191 let y = infinity.select(&zero_y, &non_zero_y)?;
192
193 Ok(AffineVar::new(x, y, infinity))
194 }
195 }
196
197 /// Allocates a new variable without performing an on-curve check, which is
198 /// useful if the variable is known to be on the curve (eg., if the point
199 /// is a constant or is a public input).
200 #[tracing::instrument(target = "r1cs", skip(cs, f))]
201 pub fn new_variable_omit_on_curve_check(
202 cs: impl Into<Namespace<BasePrimeField<P>>>,
203 f: impl FnOnce() -> Result<SWProjective<P>, SynthesisError>,
204 mode: AllocationMode,
205 ) -> Result<Self, SynthesisError> {
206 let ns = cs.into();
207 let cs = ns.cs();
208
209 let (x, y, z) = match f() {
210 Ok(ge) => {
211 let ge = ge.into_affine();
212 if ge.is_zero() {
213 // These values are convenient since the point satisfies
214 // curve equation.
215 (
216 Ok(P::BaseField::zero()),
217 Ok(P::BaseField::one()),
218 Ok(P::BaseField::zero()),
219 )
220 } else {
221 (Ok(ge.x), Ok(ge.y), Ok(P::BaseField::one()))
222 }
223 },
224 _ => (
225 Err(SynthesisError::AssignmentMissing),
226 Err(SynthesisError::AssignmentMissing),
227 Err(SynthesisError::AssignmentMissing),
228 ),
229 };
230
231 let x = F::new_variable(ark_relations::ns!(cs, "x"), || x, mode)?;
232 let y = F::new_variable(ark_relations::ns!(cs, "y"), || y, mode)?;
233 let z = F::new_variable(ark_relations::ns!(cs, "z"), || z, mode)?;
234
235 Ok(Self::new(x, y, z))
236 }
237
238 /// Mixed addition, which is useful when `other = (x2, y2)` is known to have
239 /// z = 1.
240 #[tracing::instrument(target = "r1cs", skip(self, other))]
241 pub(crate) fn add_mixed(&self, other: &NonZeroAffineVar<P, F>) -> Result<Self, SynthesisError> {
242 // Complete mixed addition formula from Renes-Costello-Batina 2015
243 // Algorithm 2
244 // (https://eprint.iacr.org/2015/1060).
245 // Below, comments at the end of a line denote the corresponding
246 // step(s) of the algorithm
247 //
248 // Adapted from code in
249 // https://github.com/RustCrypto/elliptic-curves/blob/master/p256/src/arithmetic/projective.rs
250 let three_b = P::COEFF_B.double() + &P::COEFF_B;
251 let (x1, y1, z1) = (&self.x, &self.y, &self.z);
252 let (x2, y2) = (&other.x, &other.y);
253
254 let xx = x1 * x2; // 1
255 let yy = y1 * y2; // 2
256 let xy_pairs = ((x1 + y1) * &(x2 + y2)) - (&xx + &yy); // 4, 5, 6, 7, 8
257 let xz_pairs = (x2 * z1) + x1; // 8, 9
258 let yz_pairs = (y2 * z1) + y1; // 10, 11
259
260 let axz = mul_by_coeff_a::<P, F>(&xz_pairs); // 12
261
262 let bz3_part = &axz + z1 * three_b; // 13, 14
263
264 let yy_m_bz3 = &yy - &bz3_part; // 15
265 let yy_p_bz3 = &yy + &bz3_part; // 16
266
267 let azz = mul_by_coeff_a::<P, F>(z1); // 20
268 let xx3_p_azz = xx.double().unwrap() + &xx + &azz; // 18, 19, 22
269
270 let bxz3 = &xz_pairs * three_b; // 21
271 let b3_xz_pairs = mul_by_coeff_a::<P, F>(&(&xx - &azz)) + &bxz3; // 23, 24, 25
272
273 let x = (&yy_m_bz3 * &xy_pairs) - &yz_pairs * &b3_xz_pairs; // 28,29, 30
274 let y = (&yy_p_bz3 * &yy_m_bz3) + &xx3_p_azz * b3_xz_pairs; // 17, 26, 27
275 let z = (&yy_p_bz3 * &yz_pairs) + xy_pairs * xx3_p_azz; // 31, 32, 33
276
277 Ok(ProjectiveVar::new(x, y, z))
278 }
279
280 /// Computes a scalar multiplication with a little-endian scalar of size
281 /// `P::ScalarField::MODULUS_BITS`.
282 #[tracing::instrument(
283 target = "r1cs",
284 skip(self, mul_result, multiple_of_power_of_two, bits)
285 )]
286 fn fixed_scalar_mul_le(
287 &self,
288 mul_result: &mut Self,
289 multiple_of_power_of_two: &mut NonZeroAffineVar<P, F>,
290 bits: &[&Boolean<BasePrimeField<P>>],
291 ) -> Result<(), SynthesisError> {
292 let scalar_modulus_bits = <P::ScalarField as PrimeField>::MODULUS_BIT_SIZE as usize;
293
294 assert!(scalar_modulus_bits >= bits.len());
295 let split_len = ark_std::cmp::min(scalar_modulus_bits - 2, bits.len());
296 let (affine_bits, proj_bits) = bits.split_at(split_len);
297 // Computes the standard little-endian double-and-add algorithm
298 // (Algorithm 3.26, Guide to Elliptic Curve Cryptography)
299 //
300 // We rely on *incomplete* affine formulae for partially computing this.
301 // However, we avoid exceptional edge cases because we partition the scalar
302 // into two chunks: one guaranteed to be less than p - 2, and the rest.
303 // We only use incomplete formulae for the first chunk, which means we avoid
304 // exceptions:
305 //
306 // `add_unchecked(a, b)` is incomplete when either `b.is_zero()`, or when
307 // `b = ±a`. During scalar multiplication, we don't hit either case:
308 // * `b = ±a`: `b = accumulator = k * a`, where `2 <= k < p - 1`. This implies
309 // that `k != p ± 1`, and so `b != (p ± 1) * a`. Because the group is finite,
310 // this in turn means that `b != ±a`, as required.
311 // * `a` or `b` is zero: for `a`, we handle the zero case after the loop; for
312 // `b`, notice that it is monotonically increasing, and furthermore, equals `k
313 // * a`, where `k != p = 0 mod p`.
314
315 // Unlike normal double-and-add, here we start off with a non-zero
316 // `accumulator`, because `NonZeroAffineVar::add_unchecked` doesn't
317 // support addition with `zero`. In more detail, we initialize
318 // `accumulator` to be the initial value of `multiple_of_power_of_two`.
319 // This ensures that all unchecked additions of `accumulator` with later
320 // values of `multiple_of_power_of_two` are safe. However, to do this
321 // correctly, we need to perform two steps:
322 // * We must skip the LSB, and instead proceed assuming that it was 1. Later, we
323 // will conditionally subtract the initial value of `accumulator`: if LSB ==
324 // 0: subtract initial_acc_value; else, subtract 0.
325 // * Because we are assuming the first bit, we must double
326 // `multiple_of_power_of_two`.
327
328 let mut accumulator = multiple_of_power_of_two.clone();
329 let initial_acc_value = accumulator.into_projective();
330
331 // The powers start at 2 (instead of 1) because we're skipping the first bit.
332 multiple_of_power_of_two.double_in_place()?;
333
334 // As mentioned, we will skip the LSB, and will later handle it via a
335 // conditional subtraction.
336 for bit in affine_bits.iter().skip(1) {
337 if bit.is_constant() {
338 if *bit == &Boolean::TRUE {
339 accumulator = accumulator.add_unchecked(multiple_of_power_of_two)?;
340 }
341 } else {
342 let temp = accumulator.add_unchecked(multiple_of_power_of_two)?;
343 accumulator = bit.select(&temp, &accumulator)?;
344 }
345 multiple_of_power_of_two.double_in_place()?;
346 }
347 // Perform conditional subtraction:
348
349 // We can convert to projective safely because the result is guaranteed to be
350 // non-zero by the condition on `affine_bits.len()`, and by the fact
351 // that `accumulator` is non-zero
352 let result = accumulator.into_projective();
353 // If bits[0] is 0, then we have to subtract `self`; else, we subtract zero.
354 let subtrahend = bits[0].select(&Self::zero(), &initial_acc_value)?;
355 *mul_result += result - subtrahend;
356
357 // Now, let's finish off the rest of the bits using our complete formulae
358 for bit in proj_bits {
359 if bit.is_constant() {
360 if *bit == &Boolean::TRUE {
361 *mul_result += &multiple_of_power_of_two.into_projective();
362 }
363 } else {
364 let temp = &*mul_result + &multiple_of_power_of_two.into_projective();
365 *mul_result = bit.select(&temp, &mul_result)?;
366 }
367 multiple_of_power_of_two.double_in_place()?;
368 }
369 Ok(())
370 }
371}
372
373impl<P, F> CurveVar<SWProjective<P>, BasePrimeField<P>> for ProjectiveVar<P, F>
374where
375 P: SWCurveConfig,
376 F: FieldVar<P::BaseField, BasePrimeField<P>>,
377 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
378{
379 fn constant(g: SWProjective<P>) -> Self {
380 let cs = ConstraintSystemRef::None;
381 Self::new_variable_omit_on_curve_check(cs, || Ok(g), AllocationMode::Constant).unwrap()
382 }
383
384 fn zero() -> Self {
385 Self::new(F::zero(), F::one(), F::zero())
386 }
387
388 fn is_zero(&self) -> Result<Boolean<BasePrimeField<P>>, SynthesisError> {
389 self.z.is_zero()
390 }
391
392 #[tracing::instrument(target = "r1cs", skip(cs, f))]
393 fn new_variable_omit_prime_order_check(
394 cs: impl Into<Namespace<BasePrimeField<P>>>,
395 f: impl FnOnce() -> Result<SWProjective<P>, SynthesisError>,
396 mode: AllocationMode,
397 ) -> Result<Self, SynthesisError> {
398 let ns = cs.into();
399 let cs = ns.cs();
400 // Curve equation in projective form:
401 // E: Y² * Z = X³ + aX * Z² + bZ³
402 //
403 // This can be re-written as
404 // E: Y² * Z - bZ³ = X³ + aX * Z²
405 // E: Z * (Y² - bZ²) = X * (X² + aZ²)
406 // so, compute X², Y², Z²,
407 // compute temp = X * (X² + aZ²)
408 // check Z.mul_equals((Y² - bZ²), temp)
409 //
410 // A total of 5 multiplications
411
412 let g = Self::new_variable_omit_on_curve_check(cs, f, mode)?;
413
414 if mode != AllocationMode::Constant {
415 // Perform on-curve check.
416 let b = P::COEFF_B;
417 let a = P::COEFF_A;
418
419 let x2 = g.x.square()?;
420 let y2 = g.y.square()?;
421 let z2 = g.z.square()?;
422 let t = &g.x * (x2 + &z2 * a);
423
424 g.z.mul_equals(&(y2 - z2 * b), &t)?;
425 }
426 Ok(g)
427 }
428
429 /// Enforce that `self` is in the prime-order subgroup.
430 ///
431 /// Does so by multiplying by the prime order, and checking that the result
432 /// is unchanged.
433 // TODO: at the moment this doesn't work, because the addition and doubling
434 // formulae are incomplete for even-order points.
435 #[tracing::instrument(target = "r1cs")]
436 fn enforce_prime_order(&self) -> Result<(), SynthesisError> {
437 unimplemented!("cannot enforce prime order");
438 // let r_minus_1 = (-P::ScalarField::one()).into_bigint();
439
440 // let mut result = Self::zero();
441 // for b in BitIteratorBE::without_leading_zeros(r_minus_1) {
442 // result.double_in_place()?;
443
444 // if b {
445 // result += self;
446 // }
447 // }
448 // self.negate()?.enforce_equal(&result)?;
449 // Ok(())
450 }
451
452 #[inline]
453 #[tracing::instrument(target = "r1cs")]
454 fn double_in_place(&mut self) -> Result<(), SynthesisError> {
455 // Complete doubling formula from Renes-Costello-Batina 2015
456 // Algorithm 3
457 // (https://eprint.iacr.org/2015/1060).
458 // Below, comments at the end of a line denote the corresponding
459 // step(s) of the algorithm
460 //
461 // Adapted from code in
462 // https://github.com/RustCrypto/elliptic-curves/blob/master/p256/src/arithmetic/projective.rs
463 let three_b = P::COEFF_B.double() + &P::COEFF_B;
464
465 let xx = self.x.square()?; // 1
466 let yy = self.y.square()?; // 2
467 let zz = self.z.square()?; // 3
468 let xy2 = (&self.x * &self.y).double()?; // 4, 5
469 let xz2 = (&self.x * &self.z).double()?; // 6, 7
470
471 let axz2 = mul_by_coeff_a::<P, F>(&xz2); // 8
472
473 let bzz3_part = &axz2 + &zz * three_b; // 9, 10
474 let yy_m_bzz3 = &yy - &bzz3_part; // 11
475 let yy_p_bzz3 = &yy + &bzz3_part; // 12
476 let y_frag = yy_p_bzz3 * &yy_m_bzz3; // 13
477 let x_frag = yy_m_bzz3 * &xy2; // 14
478
479 let bxz3 = xz2 * three_b; // 15
480 let azz = mul_by_coeff_a::<P, F>(&zz); // 16
481 let b3_xz_pairs = mul_by_coeff_a::<P, F>(&(&xx - &azz)) + &bxz3; // 15, 16, 17, 18, 19
482 let xx3_p_azz = (xx.double()? + &xx + &azz) * &b3_xz_pairs; // 23, 24, 25
483
484 let y = y_frag + &xx3_p_azz; // 26, 27
485 let yz2 = (&self.y * &self.z).double()?; // 28, 29
486 let x = x_frag - &(b3_xz_pairs * &yz2); // 30, 31
487 let z = (yz2 * &yy).double()?.double()?; // 32, 33, 34
488 self.x = x;
489 self.y = y;
490 self.z = z;
491 Ok(())
492 }
493
494 #[tracing::instrument(target = "r1cs")]
495 fn negate(&self) -> Result<Self, SynthesisError> {
496 Ok(Self::new(self.x.clone(), self.y.negate()?, self.z.clone()))
497 }
498
499 /// Computes `bits * self`, where `bits` is a little-endian
500 /// `Boolean` representation of a scalar.
501 #[tracing::instrument(target = "r1cs", skip(bits))]
502 fn scalar_mul_le<'a>(
503 &self,
504 bits: impl Iterator<Item = &'a Boolean<BasePrimeField<P>>>,
505 ) -> Result<Self, SynthesisError> {
506 if self.is_constant() {
507 if self.value().unwrap().is_zero() {
508 return Ok(self.clone());
509 }
510 }
511 let self_affine = self.to_affine()?;
512 let (x, y, infinity) = (self_affine.x, self_affine.y, self_affine.infinity);
513 // We first handle the non-zero case, and then later will conditionally select
514 // zero if `self` was zero. However, we also want to make sure that generated
515 // constraints are satisfiable in both cases.
516 //
517 // In particular, using non-sensible values for `x` and `y` in zero-case may
518 // cause `unchecked` operations to generate constraints that can never
519 // be satisfied, depending on the curve equation coefficients.
520 //
521 // The safest approach is to use coordinates of some point from the curve, thus
522 // not violating assumptions of `NonZeroAffine`. For instance, generator
523 // point.
524 let x = infinity.select(&F::constant(P::GENERATOR.x), &x)?;
525 let y = infinity.select(&F::constant(P::GENERATOR.y), &y)?;
526 let non_zero_self = NonZeroAffineVar::new(x, y);
527
528 let mut bits = bits.collect::<Vec<_>>();
529 if bits.len() == 0 {
530 return Ok(Self::zero());
531 }
532 // Remove unnecessary constant zeros in the most-significant positions.
533 bits = bits
534 .into_iter()
535 // We iterate from the MSB down.
536 .rev()
537 // Skip leading zeros, if they are constants.
538 .skip_while(|b| b.is_constant() && (b.value().unwrap() == false))
539 .collect();
540 // After collecting we are in big-endian form; we have to reverse to get back to
541 // little-endian.
542 bits.reverse();
543
544 let scalar_modulus_bits = <P::ScalarField as PrimeField>::MODULUS_BIT_SIZE;
545 let mut mul_result = Self::zero();
546 let mut power_of_two_times_self = non_zero_self;
547 // We chunk up `bits` into `p`-sized chunks.
548 for bits in bits.chunks(scalar_modulus_bits as usize) {
549 self.fixed_scalar_mul_le(&mut mul_result, &mut power_of_two_times_self, bits)?;
550 }
551
552 // The foregoing algorithm relies on incomplete addition, and so does not
553 // work when the input (`self`) is zero. We hence have to perform
554 // a check to ensure that if the input is zero, then so is the output.
555 // The cost of this check should be less than the benefit of using
556 // mixed addition in almost all cases.
557 infinity.select(&Self::zero(), &mul_result)
558 }
559
560 #[tracing::instrument(target = "r1cs", skip(scalar_bits_with_bases))]
561 fn precomputed_base_scalar_mul_le<'a, I, B>(
562 &mut self,
563 scalar_bits_with_bases: I,
564 ) -> Result<(), SynthesisError>
565 where
566 I: Iterator<Item = (B, &'a SWProjective<P>)>,
567 B: Borrow<Boolean<BasePrimeField<P>>>,
568 {
569 // We just ignore the provided bases and use the faster scalar multiplication.
570 let (bits, bases): (Vec<_>, Vec<_>) = scalar_bits_with_bases
571 .map(|(b, c)| (b.borrow().clone(), *c))
572 .unzip();
573 let base = bases[0];
574 *self += Self::constant(base).scalar_mul_le(bits.iter())?;
575 Ok(())
576 }
577}
578
579impl<P, F> ToConstraintFieldGadget<BasePrimeField<P>> for ProjectiveVar<P, F>
580where
581 P: SWCurveConfig,
582 F: FieldVar<P::BaseField, BasePrimeField<P>>,
583 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
584 F: ToConstraintFieldGadget<BasePrimeField<P>>,
585{
586 fn to_constraint_field(&self) -> Result<Vec<FpVar<BasePrimeField<P>>>, SynthesisError> {
587 self.to_affine()?.to_constraint_field()
588 }
589}
590
591fn mul_by_coeff_a<P: SWCurveConfig, F: FieldVar<P::BaseField, BasePrimeField<P>>>(f: &F) -> F
592where
593 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
594{
595 if !P::COEFF_A.is_zero() {
596 f * P::COEFF_A
597 } else {
598 F::zero()
599 }
600}
601
602impl_bounded_ops!(
603 ProjectiveVar<P, F>,
604 SWProjective<P>,
605 Add,
606 add,
607 AddAssign,
608 add_assign,
609 |mut this: &'a ProjectiveVar<P, F>, mut other: &'a ProjectiveVar<P, F>| {
610 // Implement complete addition for Short Weierstrass curves, following
611 // the complete addition formula from Renes-Costello-Batina 2015
612 // (https://eprint.iacr.org/2015/1060).
613 //
614 // We special case handling of constants to get better constraint weight.
615 if this.is_constant() {
616 // we'll just act like `other` is constant.
617 core::mem::swap(&mut this, &mut other);
618 }
619
620 if other.is_constant() {
621 // The value should exist because `other` is a constant.
622 let other = other.value().unwrap();
623 if other.is_zero() {
624 // this + 0 = this
625 this.clone()
626 } else {
627 // We'll use mixed addition to add non-zero constants.
628 let x = F::constant(other.x);
629 let y = F::constant(other.y);
630 this.add_mixed(&NonZeroAffineVar::new(x, y)).unwrap()
631 }
632 } else {
633 // Complete addition formula from Renes-Costello-Batina 2015
634 // Algorithm 1
635 // (https://eprint.iacr.org/2015/1060).
636 // Below, comments at the end of a line denote the corresponding
637 // step(s) of the algorithm
638 //
639 // Adapted from code in
640 // https://github.com/RustCrypto/elliptic-curves/blob/master/p256/src/arithmetic/projective.rs
641 let three_b = P::COEFF_B.double() + &P::COEFF_B;
642 let (x1, y1, z1) = (&this.x, &this.y, &this.z);
643 let (x2, y2, z2) = (&other.x, &other.y, &other.z);
644
645 let xx = x1 * x2; // 1
646 let yy = y1 * y2; // 2
647 let zz = z1 * z2; // 3
648 let xy_pairs = ((x1 + y1) * &(x2 + y2)) - (&xx + &yy); // 4, 5, 6, 7, 8
649 let xz_pairs = ((x1 + z1) * &(x2 + z2)) - (&xx + &zz); // 9, 10, 11, 12, 13
650 let yz_pairs = ((y1 + z1) * &(y2 + z2)) - (&yy + &zz); // 14, 15, 16, 17, 18
651
652 let axz = mul_by_coeff_a::<P, F>(&xz_pairs); // 19
653
654 let bzz3_part = &axz + &zz * three_b; // 20, 21
655
656 let yy_m_bzz3 = &yy - &bzz3_part; // 22
657 let yy_p_bzz3 = &yy + &bzz3_part; // 23
658
659 let azz = mul_by_coeff_a::<P, F>(&zz);
660 let xx3_p_azz = xx.double().unwrap() + &xx + &azz; // 25, 26, 27, 29
661
662 let bxz3 = &xz_pairs * three_b; // 28
663 let b3_xz_pairs = mul_by_coeff_a::<P, F>(&(&xx - &azz)) + &bxz3; // 30, 31, 32
664
665 let x = (&yy_m_bzz3 * &xy_pairs) - &yz_pairs * &b3_xz_pairs; // 35, 39, 40
666 let y = (&yy_p_bzz3 * &yy_m_bzz3) + &xx3_p_azz * b3_xz_pairs; // 24, 36, 37, 38
667 let z = (&yy_p_bzz3 * &yz_pairs) + xy_pairs * xx3_p_azz; // 41, 42, 43
668
669 ProjectiveVar::new(x, y, z)
670 }
671
672 },
673 |this: &'a ProjectiveVar<P, F>, other: SWProjective<P>| {
674 this + ProjectiveVar::constant(other)
675 },
676 (F: FieldVar<P::BaseField, BasePrimeField<P>>, P: SWCurveConfig),
677 for <'b> &'b F: FieldOpsBounds<'b, P::BaseField, F>,
678);
679
680impl_bounded_ops!(
681 ProjectiveVar<P, F>,
682 SWProjective<P>,
683 Sub,
684 sub,
685 SubAssign,
686 sub_assign,
687 |this: &'a ProjectiveVar<P, F>, other: &'a ProjectiveVar<P, F>| this + other.negate().unwrap(),
688 |this: &'a ProjectiveVar<P, F>, other: SWProjective<P>| this - ProjectiveVar::constant(other),
689 (F: FieldVar<P::BaseField, BasePrimeField<P>>, P: SWCurveConfig),
690 for <'b> &'b F: FieldOpsBounds<'b, P::BaseField, F>
691);
692
693impl_bounded_ops_diff!(
694 ProjectiveVar<P, F>,
695 SWProjective<P>,
696 EmulatedFpVar<P::ScalarField, BasePrimeField<P>>,
697 P::ScalarField,
698 Mul,
699 mul,
700 MulAssign,
701 mul_assign,
702 |this: &'a ProjectiveVar<P, F>, other: &'a EmulatedFpVar<P::ScalarField, BasePrimeField<P>>| {
703 if this.is_constant() && other.is_constant() {
704 assert!(this.is_constant() && other.is_constant());
705 ProjectiveVar::constant(this.value().unwrap() * &other.value().unwrap())
706 } else {
707 let bits = other.to_bits_le().unwrap();
708 this.scalar_mul_le(bits.iter()).unwrap()
709 }
710 },
711 |this: &'a ProjectiveVar<P, F>, other: P::ScalarField| this * EmulatedFpVar::constant(other),
712 (F: FieldVar<P::BaseField, BasePrimeField<P>>, P: SWCurveConfig),
713 for <'b> &'b F: FieldOpsBounds<'b, P::BaseField, F>,
714);
715
716impl<'a, P, F> GroupOpsBounds<'a, SWProjective<P>, ProjectiveVar<P, F>> for ProjectiveVar<P, F>
717where
718 P: SWCurveConfig,
719 F: FieldVar<P::BaseField, BasePrimeField<P>>,
720 for<'b> &'b F: FieldOpsBounds<'b, P::BaseField, F>,
721{
722}
723
724impl<'a, P, F> GroupOpsBounds<'a, SWProjective<P>, ProjectiveVar<P, F>> for &'a ProjectiveVar<P, F>
725where
726 P: SWCurveConfig,
727 F: FieldVar<P::BaseField, BasePrimeField<P>>,
728 for<'b> &'b F: FieldOpsBounds<'b, P::BaseField, F>,
729{
730}
731
732impl<P, F> CondSelectGadget<BasePrimeField<P>> for ProjectiveVar<P, F>
733where
734 P: SWCurveConfig,
735 F: FieldVar<P::BaseField, BasePrimeField<P>>,
736 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
737{
738 #[inline]
739 #[tracing::instrument(target = "r1cs")]
740 fn conditionally_select(
741 cond: &Boolean<BasePrimeField<P>>,
742 true_value: &Self,
743 false_value: &Self,
744 ) -> Result<Self, SynthesisError> {
745 let x = cond.select(&true_value.x, &false_value.x)?;
746 let y = cond.select(&true_value.y, &false_value.y)?;
747 let z = cond.select(&true_value.z, &false_value.z)?;
748
749 Ok(Self::new(x, y, z))
750 }
751}
752
753impl<P, F> EqGadget<BasePrimeField<P>> for ProjectiveVar<P, F>
754where
755 P: SWCurveConfig,
756 F: FieldVar<P::BaseField, BasePrimeField<P>>,
757 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
758{
759 #[tracing::instrument(target = "r1cs")]
760 fn is_eq(&self, other: &Self) -> Result<Boolean<BasePrimeField<P>>, SynthesisError> {
761 let x_equal = (&self.x * &other.z).is_eq(&(&other.x * &self.z))?;
762 let y_equal = (&self.y * &other.z).is_eq(&(&other.y * &self.z))?;
763 let coordinates_equal = x_equal & y_equal;
764 let both_are_zero = self.is_zero()? & other.is_zero()?;
765 Ok(both_are_zero | coordinates_equal)
766 }
767
768 #[inline]
769 #[tracing::instrument(target = "r1cs")]
770 fn conditional_enforce_equal(
771 &self,
772 other: &Self,
773 condition: &Boolean<BasePrimeField<P>>,
774 ) -> Result<(), SynthesisError> {
775 let x_equal = (&self.x * &other.z).is_eq(&(&other.x * &self.z))?;
776 let y_equal = (&self.y * &other.z).is_eq(&(&other.y * &self.z))?;
777 let coordinates_equal = x_equal & y_equal;
778 let both_are_zero = self.is_zero()? & other.is_zero()?;
779 (both_are_zero | coordinates_equal).conditional_enforce_equal(&Boolean::TRUE, condition)
780 }
781
782 #[inline]
783 #[tracing::instrument(target = "r1cs")]
784 fn conditional_enforce_not_equal(
785 &self,
786 other: &Self,
787 condition: &Boolean<BasePrimeField<P>>,
788 ) -> Result<(), SynthesisError> {
789 let is_equal = self.is_eq(other)?;
790 (is_equal & condition).enforce_equal(&Boolean::FALSE)
791 }
792}
793
794impl<P, F> AllocVar<SWAffine<P>, BasePrimeField<P>> for ProjectiveVar<P, F>
795where
796 P: SWCurveConfig,
797 F: FieldVar<P::BaseField, BasePrimeField<P>>,
798 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
799{
800 fn new_variable<T: Borrow<SWAffine<P>>>(
801 cs: impl Into<Namespace<BasePrimeField<P>>>,
802 f: impl FnOnce() -> Result<T, SynthesisError>,
803 mode: AllocationMode,
804 ) -> Result<Self, SynthesisError> {
805 Self::new_variable(
806 cs,
807 || f().map(|b| SWProjective::from((*b.borrow()).clone())),
808 mode,
809 )
810 }
811}
812
813impl<P, F> AllocVar<SWProjective<P>, BasePrimeField<P>> for ProjectiveVar<P, F>
814where
815 P: SWCurveConfig,
816 F: FieldVar<P::BaseField, BasePrimeField<P>>,
817 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
818{
819 fn new_variable<T: Borrow<SWProjective<P>>>(
820 cs: impl Into<Namespace<BasePrimeField<P>>>,
821 f: impl FnOnce() -> Result<T, SynthesisError>,
822 mode: AllocationMode,
823 ) -> Result<Self, SynthesisError> {
824 let ns = cs.into();
825 let cs = ns.cs();
826 let f = || Ok(*f()?.borrow());
827 match mode {
828 AllocationMode::Constant => Self::new_variable_omit_prime_order_check(cs, f, mode),
829 AllocationMode::Input => Self::new_variable_omit_prime_order_check(cs, f, mode),
830 AllocationMode::Witness => {
831 // if cofactor.is_even():
832 // divide until you've removed all even factors
833 // else:
834 // just directly use double and add.
835 let mut power_of_2: u32 = 0;
836 let mut cofactor = P::COFACTOR.to_vec();
837 while cofactor[0] % 2 == 0 {
838 div2(&mut cofactor);
839 power_of_2 += 1;
840 }
841
842 let cofactor_weight = BitIteratorBE::new(cofactor.as_slice())
843 .filter(|b| *b)
844 .count();
845 let modulus_minus_1 = (-P::ScalarField::one()).into_bigint(); // r - 1
846 let modulus_minus_1_weight =
847 BitIteratorBE::new(modulus_minus_1).filter(|b| *b).count();
848
849 // We pick the most efficient method of performing the prime order check:
850 // If the cofactor has lower hamming weight than the scalar field's modulus,
851 // we first multiply by the inverse of the cofactor, and then, after allocating,
852 // multiply by the cofactor. This ensures the resulting point has no cofactors
853 //
854 // Else, we multiply by the scalar field's modulus and ensure that the result
855 // equals the identity.
856
857 let (mut ge, iter) = if cofactor_weight < modulus_minus_1_weight {
858 let ge = Self::new_variable_omit_prime_order_check(
859 ark_relations::ns!(cs, "Witness without subgroup check with cofactor mul"),
860 || f().map(|g| g.into_affine().mul_by_cofactor_inv().into()),
861 mode,
862 )?;
863 (
864 ge,
865 BitIteratorBE::without_leading_zeros(cofactor.as_slice()),
866 )
867 } else {
868 let ge = Self::new_variable_omit_prime_order_check(
869 ark_relations::ns!(cs, "Witness without subgroup check with `r` check"),
870 || {
871 f().map(|g| {
872 let g = g.into_affine();
873 let power_of_two = P::ScalarField::ONE.into_bigint() << power_of_2;
874 let power_of_two_inv = P::ScalarField::from_bigint(power_of_two)
875 .and_then(|n| n.inverse())
876 .unwrap();
877 g.mul(power_of_two_inv)
878 })
879 },
880 mode,
881 )?;
882
883 (
884 ge,
885 BitIteratorBE::without_leading_zeros(modulus_minus_1.as_ref()),
886 )
887 };
888 // Remove the even part of the cofactor
889 for _ in 0..power_of_2 {
890 ge.double_in_place()?;
891 }
892
893 let mut result = Self::zero();
894 for b in iter {
895 result.double_in_place()?;
896
897 if b {
898 result += &ge
899 }
900 }
901 if cofactor_weight < modulus_minus_1_weight {
902 Ok(result)
903 } else {
904 ge.enforce_equal(&ge)?;
905 Ok(ge)
906 }
907 },
908 _ => unimplemented!(),
909 }
910 }
911}
912
913#[inline]
914fn div2(limbs: &mut [u64]) {
915 let mut t = 0;
916 for i in limbs.iter_mut().rev() {
917 let t2 = *i << 63;
918 *i >>= 1;
919 *i |= t;
920 t = t2;
921 }
922}
923
924impl<P, F> ToBitsGadget<BasePrimeField<P>> for ProjectiveVar<P, F>
925where
926 P: SWCurveConfig,
927 F: FieldVar<P::BaseField, BasePrimeField<P>>,
928 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
929{
930 #[tracing::instrument(target = "r1cs")]
931 fn to_bits_le(&self) -> Result<Vec<Boolean<BasePrimeField<P>>>, SynthesisError> {
932 let g = self.to_affine()?;
933 let mut bits = g.x.to_bits_le()?;
934 let y_bits = g.y.to_bits_le()?;
935 bits.extend_from_slice(&y_bits);
936 bits.push(g.infinity);
937 Ok(bits)
938 }
939
940 #[tracing::instrument(target = "r1cs")]
941 fn to_non_unique_bits_le(&self) -> Result<Vec<Boolean<BasePrimeField<P>>>, SynthesisError> {
942 let g = self.to_affine()?;
943 let mut bits = g.x.to_non_unique_bits_le()?;
944 let y_bits = g.y.to_non_unique_bits_le()?;
945 bits.extend_from_slice(&y_bits);
946 bits.push(g.infinity);
947 Ok(bits)
948 }
949}
950
951impl<P, F> ToBytesGadget<BasePrimeField<P>> for ProjectiveVar<P, F>
952where
953 P: SWCurveConfig,
954 F: FieldVar<P::BaseField, BasePrimeField<P>>,
955 for<'a> &'a F: FieldOpsBounds<'a, P::BaseField, F>,
956{
957 #[tracing::instrument(target = "r1cs")]
958 fn to_bytes_le(&self) -> Result<Vec<UInt8<BasePrimeField<P>>>, SynthesisError> {
959 let g = self.to_affine()?;
960 let mut bytes = g.x.to_bytes_le()?;
961 let y_bytes = g.y.to_bytes_le()?;
962 let inf_bytes = g.infinity.to_bytes_le()?;
963 bytes.extend_from_slice(&y_bytes);
964 bytes.extend_from_slice(&inf_bytes);
965 Ok(bytes)
966 }
967
968 #[tracing::instrument(target = "r1cs")]
969 fn to_non_unique_bytes_le(&self) -> Result<Vec<UInt8<BasePrimeField<P>>>, SynthesisError> {
970 let g = self.to_affine()?;
971 let mut bytes = g.x.to_non_unique_bytes_le()?;
972 let y_bytes = g.y.to_non_unique_bytes_le()?;
973 let inf_bytes = g.infinity.to_non_unique_bytes_le()?;
974 bytes.extend_from_slice(&y_bytes);
975 bytes.extend_from_slice(&inf_bytes);
976 Ok(bytes)
977 }
978}
979
980#[cfg(test)]
981mod test_sw_curve {
982 use crate::{
983 alloc::AllocVar,
984 convert::ToBitsGadget,
985 eq::EqGadget,
986 fields::{emulated_fp::EmulatedFpVar, fp::FpVar},
987 groups::{curves::short_weierstrass::ProjectiveVar, CurveVar},
988 };
989 use ark_ec::{
990 short_weierstrass::{Projective, SWCurveConfig},
991 CurveGroup,
992 };
993 use ark_ff::PrimeField;
994 use ark_relations::r1cs::{ConstraintSystem, Result};
995 use ark_std::UniformRand;
996 use num_traits::Zero;
997
998 fn zero_point_scalar_mul_satisfied<G>() -> Result<bool>
999 where
1000 G: CurveGroup,
1001 G::BaseField: PrimeField,
1002 G::Config: SWCurveConfig,
1003 {
1004 let mut rng = ark_std::test_rng();
1005
1006 let cs = ConstraintSystem::new_ref();
1007 let point_in = Projective::<G::Config>::zero();
1008 let point_out = Projective::<G::Config>::zero();
1009 let scalar = G::ScalarField::rand(&mut rng);
1010
1011 let point_in =
1012 ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
1013 Ok(point_in)
1014 })?;
1015 let point_out =
1016 ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_input(cs.clone(), || {
1017 Ok(point_out)
1018 })?;
1019 let scalar = EmulatedFpVar::new_input(cs.clone(), || Ok(scalar))?;
1020
1021 let mul = point_in.scalar_mul_le(scalar.to_bits_le().unwrap().iter())?;
1022
1023 point_out.enforce_equal(&mul)?;
1024
1025 cs.is_satisfied()
1026 }
1027
1028 #[test]
1029 fn test_zero_point_scalar_mul() {
1030 assert!(zero_point_scalar_mul_satisfied::<ark_bls12_381::G1Projective>().unwrap());
1031 assert!(zero_point_scalar_mul_satisfied::<ark_pallas::Projective>().unwrap());
1032 assert!(zero_point_scalar_mul_satisfied::<ark_mnt4_298::G1Projective>().unwrap());
1033 assert!(zero_point_scalar_mul_satisfied::<ark_mnt6_298::G1Projective>().unwrap());
1034 assert!(zero_point_scalar_mul_satisfied::<ark_bn254::G1Projective>().unwrap());
1035 }
1036}