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