Implement Fp/Fq for the Pallas and Vesta curves.
Co-authored-by: Kris Nuttycombe <kris@electriccoin.co>
This commit is contained in:
Sean Bowe
parent
ae20f75f7a
commit
7536af8b69
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@ -9,9 +9,9 @@ use crate::arithmetic::{adc, mac, sbb, FieldExt, Group};
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/// This represents an element of $\mathbb{F}_p$ where
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///
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/// `p = 0x40000000000000000000000000000000038aa1276c3f59b9a14064e200000001`
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/// `p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001`
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///
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/// is the base field of the Tweedledum curve.
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/// is the base field of the Pallas curve.
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// The internal representation of this type is four 64-bit unsigned
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// integers in little-endian order. `Fp` values are always in
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// Montgomery form; i.e., Fp(a) = aR mod p, with R = 2^256.
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@ -94,11 +94,11 @@ impl ConditionallySelectable for Fp {
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}
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/// Constant representing the modulus
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/// p = 0x40000000000000000000000000000000038aa1276c3f59b9a14064e200000001
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/// p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
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const MODULUS: Fp = Fp([
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0xa14064e200000001,
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0x38aa1276c3f59b9,
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0x0,
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0x992d30ed00000001,
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0x224698fc094cf91b,
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0x0000000000000000,
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0x4000000000000000,
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]);
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@ -106,9 +106,9 @@ const MODULUS: Fp = Fp([
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#[cfg(not(target_pointer_width = "64"))]
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const MODULUS_LIMBS_32: [u32; 8] = [
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0x0000_0001,
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0xa140_64e2,
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0x6c3f_59b9,
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0x038a_a127,
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0x992d_30ed,
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0x094c_f91b,
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0x2246_98fc,
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0x0000_0000,
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0x0000_0000,
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0x0000_0000,
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@ -164,34 +164,34 @@ impl_binops_additive!(Fp, Fp);
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impl_binops_multiplicative!(Fp, Fp);
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/// INV = -(p^{-1} mod 2^64) mod 2^64
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const INV: u64 = 0xa14064e1ffffffff;
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const INV: u64 = 0x992d30ecffffffff;
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/// R = 2^256 mod p
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const R: Fp = Fp([
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0x1c3ed159fffffffd,
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0xf5601c89bb41f2d3,
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0x34786d38fffffffd,
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0x992c350be41914ad,
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0xffffffffffffffff,
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0x3fffffffffffffff,
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]);
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/// R^2 = 2^512 mod p
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const R2: Fp = Fp([
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0x280c9c4000000010,
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0x91a4409b5400af74,
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0xdd7b28e19094c659,
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0xc8ad9107ccca0e,
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0x8c78ecb30000000f,
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0xd7d30dbd8b0de0e7,
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0x7797a99bc3c95d18,
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0x96d41af7b9cb714,
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]);
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/// R^3 = 2^768 mod p
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const R3: Fp = Fp([
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0x98fb3d144380a737,
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0xf9fdbeb55b7eb87c,
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0x63f75cb999eafa89,
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0x217cb214ebb8fc72,
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0xf185a5993a9e10f9,
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0xf6a68f3b6ac5b1d1,
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0xdf8d1014353fd42c,
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0x2ae309222d2d9910,
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]);
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/// `GENERATOR = 5 mod p` is a generator of the `p - 1` order multiplicative subgroup, and
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/// is also a quadratic non-residue.
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/// `GENERATOR = 5 mod p` is a generator of the `p - 1` order multiplicative
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/// subgroup, or in other words a primitive root of the field.
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const GENERATOR: Fp = Fp::from_raw([
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0x0000_0000_0000_0005,
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0x0000_0000_0000_0000,
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@ -199,34 +199,26 @@ const GENERATOR: Fp = Fp::from_raw([
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0x0000_0000_0000_0000,
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]);
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const S: u32 = 33;
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const S: u32 = 32;
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/// GENERATOR^t where t * 2^s + 1 = p
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/// with t odd. In other words, this
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/// is a 2^s root of unity.
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///
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/// `GENERATOR = 5 mod p` is a generator
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/// of the p - 1 order multiplicative
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/// subgroup.
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const ROOT_OF_UNITY: Fp = Fp::from_raw([
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0x53de9f31b88837ce,
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0xff46e8f3f3ea99d6,
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0xf624f2eaaf8c2d57,
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0x2ae45117890ee2fc,
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0xbdad6fabd87ea32f,
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0xea322bf2b7bb7584,
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0x362120830561f81a,
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0x2bce74deac30ebda,
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]);
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/// GENERATOR^{2^s} where t * 2^s + 1 = p
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/// with t odd. In other words, this
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/// is a t root of unity.
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///
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/// `GENERATOR = 5 mod p` is a generator
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/// of the p - 1 order multiplicative
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/// subgroup.
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const DELTA: Fp = Fp::from_raw([
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0x48796f6fde98a425,
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0xa99d8b67e918805e,
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0x671383de08b5fe3c,
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0x1e9372724e80300d,
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0x6a6ccd20dd7b9ba2,
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0xf5e4f3f13eee5636,
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0xbd455b7112a5049d,
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0xa757d0f0006ab6c,
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]);
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impl Default for Fp {
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@ -503,7 +495,7 @@ impl ff::Field for Fp {
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// https://eprint.iacr.org/2012/685.pdf (page 12, algorithm 5)
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// w = self^((t - 1) // 2)
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let w = self.pow_vartime(&[0xdb0fd66e68501938, 0xe2a849, 0x0, 0x10000000]);
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let w = self.pow_vartime(&[0x04a67c8dcc969876, 0x11234c7e, 0x0, 0x20000000]);
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let mut v = S;
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let mut x = self * w;
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@ -544,8 +536,8 @@ impl ff::Field for Fp {
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/// failing if the element is zero.
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fn invert(&self) -> CtOption<Self> {
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let tmp = self.pow_vartime(&[
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0xa14064e1ffffffff,
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0x38aa1276c3f59b9,
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0x992d30ecffffffff,
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0x224698fc094cf91b,
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0x0,
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0x4000000000000000,
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]);
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@ -646,43 +638,43 @@ impl ff::PrimeField for Fp {
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impl FieldExt for Fp {
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const ROOT_OF_UNITY: Self = ROOT_OF_UNITY;
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const ROOT_OF_UNITY_INV: Self = Fp::from_raw([
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0x9246674078fa45bb,
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0xd822ebd60888c5ea,
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0x56d579133a11731f,
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0x1c88fa9e942120bb,
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0xf0b87c7db2ce91f6,
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0x84a0a1d8859f066f,
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0xb4ed8e647196dad1,
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0x2cd5282c53116b5c,
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]);
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const UNROLL_T_EXPONENT: [u64; 4] = [
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0x3b3a6633d1897d83,
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0x0000000000c93d5b,
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0xf000000000000000,
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0xe34ab16,
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0x955a0a417453113c,
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0x0000000022016b89,
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0xc000000000000000,
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0x3f7ed4c6,
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];
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const T_EXPONENT: [u64; 4] = [
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0xb61facdcd0a03271,
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0x0000000001c55093,
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0x094cf91b992d30ed,
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0x00000000224698fc,
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0x0000000000000000,
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0x20000000,
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0x40000000,
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];
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const DELTA: Self = DELTA;
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const UNROLL_S_EXPONENT: u64 = 0x11cb54e91;
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const UNROLL_S_EXPONENT: u64 = 0x204ace5;
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const TWO_INV: Self = Fp::from_raw([
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0xd0a0327100000001,
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0x01c55093b61facdc,
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0xcc96987680000001,
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0x11234c7e04a67c8d,
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0x0000000000000000,
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0x2000000000000000,
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]);
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const RESCUE_ALPHA: u64 = 5;
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const RESCUE_INVALPHA: [u64; 4] = [
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0x810050b4cccccccd,
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0x360880ec56991494,
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0xe0f0f3f0cccccccd,
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0x4e9ee0c9a10a60e2,
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0x3333333333333333,
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0x3333333333333333,
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];
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const ZETA: Self = Fp::from_raw([
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0x8598abb3a410c9c8,
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0x7881fb239ba41a26,
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0x9bebc9146ef83d9a,
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0x1508415ab5e97c94,
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0x7b7fd22f0201b547,
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0x05270d29d19fc7d2,
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0xd3552a23a8554e50,
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0x2d33357cb532458e,
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]);
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fn ct_is_zero(&self) -> Choice {
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@ -781,11 +773,36 @@ fn test_inv() {
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assert_eq!(inv, INV);
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}
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#[test]
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fn test_rescue() {
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// NB: TWO_INV is standing in as a "random" field element
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assert_eq!(
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Fp::TWO_INV
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.pow_vartime(&[Fp::RESCUE_ALPHA, 0, 0, 0])
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.pow_vartime(&Fp::RESCUE_INVALPHA),
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Fp::TWO_INV
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);
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}
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#[test]
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fn test_sqrt() {
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// NB: TWO_INV is standing in as a "random" field element
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let v = (Fp::TWO_INV).square().sqrt().unwrap();
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assert!(v == Fp::TWO_INV || (-v) == Fp::TWO_INV);
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}
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#[test]
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fn test_deterministic_sqrt() {
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// NB: TWO_INV is standing in as a "random" field element
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let v = (Fp::TWO_INV).square().deterministic_sqrt().unwrap();
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assert!(v == Fp::TWO_INV || (-v) == Fp::TWO_INV);
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}
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#[test]
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fn test_zeta() {
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assert_eq!(
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format!("{:?}", Fp::ZETA),
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"0x1508415ab5e97c949bebc9146ef83d9a7881fb239ba41a268598abb3a410c9c8"
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"0x2d33357cb532458ed3552a23a8554e5005270d29d19fc7d27b7fd22f0201b547"
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);
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let a = Fp::ZETA;
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@ -796,6 +813,14 @@ fn test_zeta() {
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assert!(c == Fp::one());
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}
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#[test]
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fn test_root_of_unity() {
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assert_eq!(
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Fp::ROOT_OF_UNITY.pow_vartime(&[1 << 32, 0, 0, 0]),
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Fp::one()
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);
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}
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#[test]
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fn test_inv_root_of_unity() {
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assert_eq!(Fp::ROOT_OF_UNITY_INV, Fp::ROOT_OF_UNITY.invert().unwrap());
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@ -810,3 +835,42 @@ fn test_inv_2() {
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fn test_delta() {
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assert_eq!(Fp::DELTA, Fp::from(5).pow(&[1u64 << Fp::S, 0, 0, 0]));
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}
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#[cfg(not(target_pointer_width = "64"))]
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#[test]
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fn consistent_modulus_limbs() {
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for (a, &b) in MODULUS
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.0
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.iter()
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.flat_map(|&limb| {
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Some(limb as u32)
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.into_iter()
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.chain(Some((limb >> 32) as u32))
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})
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.zip(MODULUS_LIMBS_32.iter())
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{
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assert_eq!(a, b);
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}
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}
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#[test]
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fn test_from_u512() {
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assert_eq!(
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Fp::from_raw([
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0x3daec14d565241d9,
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0x0b7af45b6073944b,
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0xea5b8bd611a5bd4c,
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0x150160330625db3d
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]),
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Fp::from_u512([
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0xee155641297678a1,
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0xd83e156bdbfdbe65,
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0xd9ccd834c68ba0b5,
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0xf508ede312272758,
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0x038df7cbf8228e89,
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0x3505a1e4a3c74b41,
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0xbfa46f775eb82db3,
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0x26ebe27e262f471d
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])
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);
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}
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@ -9,9 +9,9 @@ use crate::arithmetic::{adc, mac, sbb, FieldExt, Group};
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/// This represents an element of $\mathbb{F}_q$ where
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///
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/// `q = 0x40000000000000000000000000000000038aa127696286c9842cafd400000001`
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/// `q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001`
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///
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/// is the base field of the Tweedledee curve.
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/// is the base field of the Vesta curve.
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// The internal representation of this type is four 64-bit unsigned
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// integers in little-endian order. `Fq` values are always in
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// Montgomery form; i.e., Fq(a) = aR mod q, with R = 2^256.
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@ -94,10 +94,10 @@ impl ConditionallySelectable for Fq {
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}
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/// Constant representing the modulus
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/// q = 0x40000000000000000000000000000000038aa127696286c9842cafd400000001
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/// q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
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const MODULUS: Fq = Fq([
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0x842cafd400000001,
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0x38aa127696286c9,
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0x8c46eb2100000001,
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0x224698fc0994a8dd,
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0x0,
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0x4000000000000000,
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]);
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@ -106,9 +106,9 @@ const MODULUS: Fq = Fq([
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#[cfg(not(target_pointer_width = "64"))]
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const MODULUS_LIMBS_32: [u32; 8] = [
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0x0000_0001,
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0x842c_afd4,
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0x6962_86c9,
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0x038a_a127,
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0x8c46_eb21,
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0x0994_a8dd,
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0x2246_98fc,
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0x0000_0000,
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0x0000_0000,
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0x0000_0000,
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@ -164,30 +164,30 @@ impl_binops_additive!(Fq, Fq);
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impl_binops_multiplicative!(Fq, Fq);
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/// INV = -(q^{-1} mod 2^64) mod 2^64
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const INV: u64 = 0x842cafd3ffffffff;
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const INV: u64 = 0x8c46eb20ffffffff;
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/// R = 2^256 mod q
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const R: Fq = Fq([
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0x7379f083fffffffd,
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0xf5601c89c3d86ba3,
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0x5b2b3e9cfffffffd,
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0x992c350be3420567,
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0xffffffffffffffff,
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0x3fffffffffffffff,
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]);
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/// R^2 = 2^512 mod q
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const R2: Fq = Fq([
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0x8595fa8000000010,
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0x7e16a565c6895230,
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0xf4c0e6fcb03aa0a2,
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0xc8ad9106886013,
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0xfc9678ff0000000f,
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0x67bb433d891a16e3,
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0x7fae231004ccf590,
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0x96d41af7ccfdaa9,
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]);
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/// R^3 = 2^768 mod q
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const R3: Fq = Fq([
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0xa624f338075cdb5e,
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0x57964eacb8fe21f2,
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0xcb266d18c0413bc2,
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0xa42cdf95f959577,
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0x008b421c249dae4c,
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0xe13bda50dba41326,
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0x88fececb8e15cb63,
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0x7dd97a06e6792c8,
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]);
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/// `GENERATOR = 5 mod q` is a generator of the `q - 1` order multiplicative subgroup, and
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@ -199,34 +199,26 @@ const GENERATOR: Fq = Fq::from_raw([
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0x0000_0000_0000_0000,
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]);
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const S: u32 = 34;
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const S: u32 = 32;
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/// GENERATOR^t where t * 2^s + 1 = q
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/// with t odd. In other words, this
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/// is a 2^s root of unity.
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///
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/// `GENERATOR = 5 mod q` is a generator
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/// of the q - 1 order multiplicative
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/// subgroup.
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const ROOT_OF_UNITY: Fq = Fq::from_raw([
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0x1cbd3234869d57ec,
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0xa287dd1b8084fbf,
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0xf1dbcb645a987293,
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0x113efc510dc03c0b,
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0xa70e2c1102b6d05f,
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0x9bb97ea3c106f049,
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0x9e5c4dfd492ae26e,
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0x2de6a9b8746d3f58,
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]);
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/// GENERATOR^{2^s} where t * 2^s + 1 = q
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/// with t odd. In other words, this
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/// is a t root of unity.
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///
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/// `GENERATOR = 5 mod q` is a generator
|
||||
/// of the q - 1 order multiplicative
|
||||
/// subgroup.
|
||||
const DELTA: Fq = Fq::from_raw([
|
||||
0x83d2833d15f2bbf9,
|
||||
0x5127e2ce24a8e69c,
|
||||
0x4243423589e0a9b5,
|
||||
0x20daec44973be920,
|
||||
0x8494392472d1683c,
|
||||
0xe3ac3376541d1140,
|
||||
0x06f0a88e7f7949f8,
|
||||
0x2237d54423724166,
|
||||
]);
|
||||
|
||||
impl Default for Fq {
|
||||
|
|
@ -518,7 +510,7 @@ impl ff::Field for Fq {
|
|||
// https://eprint.iacr.org/2012/685.pdf (page 12, algorithm 5)
|
||||
|
||||
// w = self^((t - 1) // 2)
|
||||
let w = self.pow_vartime(&[0xed2c50d9308595fa, 0x715424, 0x0, 0x8000000]);
|
||||
let w = self.pow_vartime(&[0x04ca546ec6237590, 0x11234c7e, 0x0, 0x20000000]);
|
||||
|
||||
let mut v = S;
|
||||
let mut x = self * w;
|
||||
|
|
@ -559,8 +551,8 @@ impl ff::Field for Fq {
|
|||
/// failing if the element is zero.
|
||||
fn invert(&self) -> CtOption<Self> {
|
||||
let tmp = self.pow_vartime(&[
|
||||
0x842cafd3ffffffff,
|
||||
0x38aa127696286c9,
|
||||
0x8c46eb20ffffffff,
|
||||
0x224698fc0994a8dd,
|
||||
0x0,
|
||||
0x4000000000000000,
|
||||
]);
|
||||
|
|
@ -661,43 +653,43 @@ impl ff::PrimeField for Fq {
|
|||
impl FieldExt for Fq {
|
||||
const ROOT_OF_UNITY: Self = ROOT_OF_UNITY;
|
||||
const ROOT_OF_UNITY_INV: Self = Fq::from_raw([
|
||||
0x824860d3eb30de02,
|
||||
0xad9f0afd0ea63acc,
|
||||
0xd250318c11a16fe1,
|
||||
0x12a9f5e1dd62dabc,
|
||||
0x57eecda0a84b6836,
|
||||
0x4ad38b9084b8a80c,
|
||||
0xf4c8f353124086c1,
|
||||
0x2235e1a7415bf936,
|
||||
]);
|
||||
const UNROLL_T_EXPONENT: [u64; 4] = [
|
||||
0x9b71de17e6d2d5a0,
|
||||
0x0000000000296ee0,
|
||||
0x8c00000000000000,
|
||||
0x2ecc05e,
|
||||
0xcc771cc2ac1e1664,
|
||||
0x00000000062dfe9e,
|
||||
0xc000000000000000,
|
||||
0xb89e9c7,
|
||||
];
|
||||
const T_EXPONENT: [u64; 4] = [
|
||||
0xda58a1b2610b2bf5,
|
||||
0x0000000000e2a849,
|
||||
0x0994a8dd8c46eb21,
|
||||
0x00000000224698fc,
|
||||
0x0000000000000000,
|
||||
0x10000000,
|
||||
0x40000000,
|
||||
];
|
||||
const DELTA: Self = DELTA;
|
||||
const UNROLL_S_EXPONENT: u64 = 0x344cfe85d;
|
||||
const UNROLL_S_EXPONENT: u64 = 0xd1d858e1;
|
||||
const TWO_INV: Self = Fq::from_raw([
|
||||
0xc21657ea00000001,
|
||||
0x01c55093b4b14364,
|
||||
0xc623759080000001,
|
||||
0x11234c7e04ca546e,
|
||||
0x0000000000000000,
|
||||
0x2000000000000000,
|
||||
]);
|
||||
const RESCUE_ALPHA: u64 = 5;
|
||||
const RESCUE_INVALPHA: [u64; 4] = [
|
||||
0xd023bfdccccccccd,
|
||||
0x360880ec544ed23a,
|
||||
0xd69f2280cccccccd,
|
||||
0x4e9ee0c9a143ba4a,
|
||||
0x3333333333333333,
|
||||
0x3333333333333333,
|
||||
];
|
||||
const ZETA: Self = Fq::from_raw([
|
||||
0x4394c2bd148fa4fd,
|
||||
0x69cf8de720e52ec1,
|
||||
0x87ad8b5ff9731ffe,
|
||||
0x36c66d3a1e049a58,
|
||||
0x2aa9d2e050aa0e4f,
|
||||
0x0fed467d47c033af,
|
||||
0x511db4d81cf70f5a,
|
||||
0x6819a58283e528e,
|
||||
]);
|
||||
|
||||
fn ct_is_zero(&self) -> Choice {
|
||||
|
|
@ -796,11 +788,36 @@ fn test_inv() {
|
|||
assert_eq!(inv, INV);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_rescue() {
|
||||
// NB: TWO_INV is standing in as a "random" field element
|
||||
assert_eq!(
|
||||
Fq::TWO_INV
|
||||
.pow_vartime(&[Fq::RESCUE_ALPHA, 0, 0, 0])
|
||||
.pow_vartime(&Fq::RESCUE_INVALPHA),
|
||||
Fq::TWO_INV
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_sqrt() {
|
||||
// NB: TWO_INV is standing in as a "random" field element
|
||||
let v = (Fq::TWO_INV).square().sqrt().unwrap();
|
||||
assert!(v == Fq::TWO_INV || (-v) == Fq::TWO_INV);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_deterministic_sqrt() {
|
||||
// NB: TWO_INV is standing in as a "random" field element
|
||||
let v = (Fq::TWO_INV).square().deterministic_sqrt().unwrap();
|
||||
assert!(v == Fq::TWO_INV || (-v) == Fq::TWO_INV);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_zeta() {
|
||||
assert_eq!(
|
||||
format!("{:?}", Fq::ZETA),
|
||||
"0x36c66d3a1e049a5887ad8b5ff9731ffe69cf8de720e52ec14394c2bd148fa4fd"
|
||||
"0x06819a58283e528e511db4d81cf70f5a0fed467d47c033af2aa9d2e050aa0e4f"
|
||||
);
|
||||
let a = Fq::ZETA;
|
||||
assert!(a != Fq::one());
|
||||
|
|
@ -810,6 +827,14 @@ fn test_zeta() {
|
|||
assert!(c == Fq::one());
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_root_of_unity() {
|
||||
assert_eq!(
|
||||
Fq::ROOT_OF_UNITY.pow_vartime(&[1 << 32, 0, 0, 0]),
|
||||
Fq::one()
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_inv_root_of_unity() {
|
||||
assert_eq!(Fq::ROOT_OF_UNITY_INV, Fq::ROOT_OF_UNITY.invert().unwrap());
|
||||
|
|
@ -824,3 +849,42 @@ fn test_inv_2() {
|
|||
fn test_delta() {
|
||||
assert_eq!(Fq::DELTA, Fq::from(5).pow(&[1u64 << Fq::S, 0, 0, 0]));
|
||||
}
|
||||
|
||||
#[cfg(not(target_pointer_width = "64"))]
|
||||
#[test]
|
||||
fn consistent_modulus_limbs() {
|
||||
for (a, &b) in MODULUS
|
||||
.0
|
||||
.iter()
|
||||
.flat_map(|&limb| {
|
||||
Some(limb as u32)
|
||||
.into_iter()
|
||||
.chain(Some((limb >> 32) as u32))
|
||||
})
|
||||
.zip(MODULUS_LIMBS_32.iter())
|
||||
{
|
||||
assert_eq!(a, b);
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_from_u512() {
|
||||
assert_eq!(
|
||||
Fq::from_raw([
|
||||
0xe22bd0d1b22cc43e,
|
||||
0x6b84e5b52490a7c8,
|
||||
0x264262941ac9e229,
|
||||
0x27dcfdf361ce4254
|
||||
]),
|
||||
Fq::from_u512([
|
||||
0x64a80cce0b5a2369,
|
||||
0x84f2ef0501bc783c,
|
||||
0x696e5e63c86bbbde,
|
||||
0x924072f52dc6cc62,
|
||||
0x8288a507c8d61128,
|
||||
0x3b2efb1ef697e3fe,
|
||||
0x75a4998d06855f27,
|
||||
0x52ea589e69712cc0
|
||||
])
|
||||
);
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in New Issue