Hash to curve as defined in the standard

The current implementation is not compatible with the current definition of the standard. This PR provides a hash-to-curve implementation as defined in draft-irtf-cfrg-hash-to-curve-12.

* Implementation of `hash_to_field` as defined in the standard
* Implementation of `hash_to_curve` as defined in the standard, by changing the mechanism over which we chose the sign.
* For the point above, had to change the `elligator_encode` to return whether `eps` is a square or not (required for `hash_to_curve`).
* Included test vectors of the draft.
* Included `FieldElement::from_bytes_wide(bytes: &u8; 64])` to reduce integers encoded in 64 bytes.

Author: iquerejeta

src/backend/serial/u32/constants.rs

@@ -29,6 +29,11 @@ pub(crate) const EDWARDS_D: FieldElement2625 = FieldElement2625([
     56195235, 13857412, 51736253, 6949390, 114729, 24766616, 60832955, 30306712, 48412415, 21499315,
 ]);
 
+/// sqrt(-486664)
+pub(crate) const ED25519_SQRTAM2: FieldElement2625 = FieldElement2625([
+    54885894, 25242303, 55597453, 9067496, 51808079, 33312638, 25456129, 14121551, 54921728, 3972023,
+]);
+
 /// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.
 pub(crate) const EDWARDS_D2: FieldElement2625 = FieldElement2625([
     45281625, 27714825, 36363642, 13898781, 229458, 15978800, 54557047, 27058993, 29715967, 9444199,

src/backend/serial/u32/field.rs

@@ -414,6 +414,26 @@ impl FieldElement2625 {
         FieldElement2625::reduce(h)
     }
 
+    /// Load a `FieldElement51` from 64 bytes, by reducing modulo q.
+    pub fn from_bytes_wide(hash: &[u8; 64]) -> FieldElement2625 {
+        let mut fl = [0u8; 32];
+        let mut gl = [0u8; 32];
+        fl.copy_from_slice(&hash[..32]);
+        gl.copy_from_slice(&hash[32..]);
+        fl[31] &= 0x7f;
+        gl[31] &= 0x7f;
+
+        let fe_f = Self::from_bytes(&fl);
+        let fe_g = Self::from_bytes(&gl);
+        let mut fe_f64 = fe_f.0.map(|i| i as u64);
+        let fe_g64 = fe_g.0.map(|i| i as u64);
+        fe_f64[0] = fe_f64[0] + (hash[31] >> 7) as u64 * 19 + (hash[63] >> 7) as u64 * 722;
+        for i in 0..10 {
+            fe_f64[i] += 38 * fe_g64[i];
+        }
+        Self::reduce(fe_f64)
+    }
+
     /// Serialize this `FieldElement51` to a 32-byte array.  The
     /// encoding is canonical.
     pub fn to_bytes(&self) -> [u8; 32] {

src/backend/serial/u64/constants.rs

@@ -35,6 +35,15 @@ pub(crate) const EDWARDS_D: FieldElement51 = FieldElement51([
     1442794654840575,
 ]);
 
+/// sqrt(-486664)
+pub(crate) const ED25519_SQRTAM2: FieldElement51 = FieldElement51([
+    1693982333959686,
+    608509411481997,
+    2235573344831311,
+    947681270984193,
+    266558006233600
+]);
+
 /// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.
 pub(crate) const EDWARDS_D2: FieldElement51 = FieldElement51([
     1859910466990425,

src/backend/serial/u64/field.rs

@@ -355,6 +355,24 @@ impl FieldElement51 {
         ])
     }
 
+    /// Load a `FieldElement51` from 64 bytes, by reducing modulo q.
+    pub fn from_bytes_wide(hash: &[u8; 64]) -> FieldElement51 {
+        let mut fl = [0u8; 32];
+        let mut gl = [0u8; 32];
+        fl.copy_from_slice(&hash[..32]);
+        gl.copy_from_slice(&hash[32..]);
+        fl[31] &= 0x7f;
+        gl[31] &= 0x7f;
+
+        let mut fe_f = Self::from_bytes(&fl);
+        let fe_g = Self::from_bytes(&gl);
+        fe_f.0[0] = fe_f.0[0] + (hash[31] >> 7) as u64 * 19 + (hash[63] >> 7) as u64 * 722;
+        for i in 0..5 {
+            fe_f.0[i] += 38 * fe_g.0[i];
+        }
+        Self::reduce(fe_f.0)
+    }
+
     /// Serialize this `FieldElement51` to a 32-byte array.  The
     /// encoding is canonical.
     pub fn to_bytes(&self) -> [u8; 32] {

src/edwards.rs

@@ -221,6 +221,7 @@ impl CompressedEdwardsY {
 use serde::{self, Serialize, Deserialize, Serializer, Deserializer};
 #[cfg(feature = "serde")]
 use serde::de::Visitor;
+use constants::ED25519_SQRTAM2;
 
 #[cfg(feature = "serde")]
 impl Serialize for EdwardsPoint {
@@ -526,9 +527,26 @@ impl EdwardsPoint {
         CompressedEdwardsY(s)
     }
 
-    /// Perform hashing to the group using the Elligator2 map
+    /// Perform hashing to curve, with explicit hash function and DST using the suite
+    /// edwards25519_XMD:SHA-512_ELL2_NU_
     ///
-    /// See https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-10#section-6.7.1
+    /// See https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-12#section-6.8.2
+    pub fn hash_to_curve_dst<D>(bytes: &[u8], dst: &[u8]) -> EdwardsPoint
+    where
+        D: Digest<OutputSize = U64> + Default,
+    {
+        let fe = FieldElement::hash_to_field::<D>(bytes, dst);
+        let (M1, is_sq) = crate::montgomery::elligator_encode(&fe);
+        let mut E1_opt = M1.to_edwards(0).expect("Montgomery conversion to Edwards point in Elligator failed");
+
+        // Now we recover v, to ensure that we got the sign right.
+        let mont_v = &(&ED25519_SQRTAM2*&FieldElement::from_bytes(&M1.to_bytes()))*&E1_opt.X.invert();
+        E1_opt.X.conditional_negate(is_sq ^ mont_v.is_negative());
+        E1_opt
+            .mul_by_cofactor()
+    }
+
+    /// Hash from bytes, following the elligator2 version implemented in signal
     pub fn hash_from_bytes<D>(bytes: &[u8]) -> EdwardsPoint
     where
         D: Digest<OutputSize = U64> + Default,
@@ -543,7 +561,7 @@ impl EdwardsPoint {
 
         let fe = FieldElement::from_bytes(&res);
 
-        let M1 = crate::montgomery::elligator_encode(&fe);
+        let (M1, _) = crate::montgomery::elligator_encode(&fe);
         let E1_opt = M1.to_edwards(sign_bit);
 
         E1_opt
@@ -1208,6 +1226,7 @@ mod test {
     use subtle::ConditionallySelectable;
     use constants;
     use super::*;
+    use sha2::Sha512;
 
     /// X coordinate of the basepoint.
     /// = 15112221349535400772501151409588531511454012693041857206046113283949847762202
@@ -1807,4 +1826,44 @@ mod test {
             assert_eq!(point.compress().to_bytes(), output[..]);
         }
     }
+
+    // Hash-to-curve test vectors from
+    //     https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/master/draft-irtf-cfrg-hash-to-curve.md
+    fn test_vectors_h2c() -> Vec<Vec<&'static str>> {
+        vec![
+            vec![
+                "",
+                "222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab2d687f0f9b",
+            ],
+            vec![
+                "abc",
+                "67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1f5c827fa42",
+            ],
+            vec![
+                "abcdef0123456789",
+                "af8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a0a8e1a86fb",
+            ],
+            vec![
+                "q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq",
+                "aaf6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91f57a603450",
+            ],
+            vec![
+                "a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
+                "ac90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a4245891a37",
+            ],
+        ]
+    }
+
+    #[test]
+    fn elligator_hash_to_curve_test_vectors(){
+        let dst = b"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_";
+        for (index, vector) in test_vectors_h2c().iter().enumerate() {
+            let input = vector[0].as_bytes();
+            let mut output = hex::decode(vector[1]).unwrap();
+            output.reverse();
+
+            let point = EdwardsPoint::hash_to_curve_dst::<Sha512>(&input, dst).compress().to_bytes();
+            assert!(!(point != output[..]), "Failed in test {}", index);
+        }
+    }
 }

src/field.rs

@@ -60,6 +60,8 @@ pub type FieldElement = backend::serial::u64::field::FieldElement51;
 
 #[cfg(feature = "u32_backend")]
 pub use backend::serial::u32::field::*;
+use digest::{generic_array::typenum::U64, Digest};
+
 /// A `FieldElement` represents an element of the field
 /// \\( \mathbb Z / (2\^{255} - 19)\\).
 ///
@@ -289,12 +291,44 @@ impl FieldElement {
     pub fn invsqrt(&self) -> (Choice, FieldElement) {
         FieldElement::sqrt_ratio_i(&FieldElement::one(), self)
     }
+
+    /// Hash_to_field as described in hash_to_curve standard
+    pub fn hash_to_field<D>(bytes: &[u8], dst: &[u8]) -> Self
+    where
+        D: Digest<OutputSize = U64> + Default,
+    {
+        let len_in_bytes = 48;
+        let l_i_b_str = [0u8, len_in_bytes];
+        let z_pad = [0u8; 128];
+        let dst_prime = [dst, &[dst.len() as u8]].concat();
+
+        let b_0 = D::new()
+            .chain(z_pad)
+            .chain(bytes)
+            .chain(l_i_b_str)
+            .chain([0u8])
+            .chain(&dst_prime)
+            .finalize();
+
+        let b_1 = D::new()
+            .chain(b_0.as_slice())
+            .chain([1u8])
+            .chain(&dst_prime)
+            .finalize();
+
+        let mut bytes_wide = [0u8; 64];
+        bytes_wide[..48].copy_from_slice(&b_1.as_slice()[..48]);
+        bytes_wide[..48].reverse();
+
+        FieldElement::from_bytes_wide(&bytes_wide)
+    }
 }
 
 #[cfg(test)]
 mod test {
     use field::*;
     use subtle::ConditionallyNegatable;
+    use sha2::Sha512;
 
     /// Random element a of GF(2^255-19), from Sage
     /// a = 1070314506888354081329385823235218444233221\
@@ -473,4 +507,51 @@ mod test {
     fn batch_invert_empty() {
         FieldElement::batch_invert(&mut []);
     }
+
+    #[test]
+    fn from_hash_wide() {
+        let mut test_vec= [0x6d, 0x2f, 0x2f, 0xfa, 0x94, 0x12, 0xb4, 0x15, 0x2f, 0x6a, 0xb6, 0x28, 0x41, 0xb8, 0x25, 0x92, 0x4a, 0x44, 0x90, 0x65, 0x15, 0x2b, 0x95, 0x47, 0x6f, 0x12, 0x1d, 0xe8, 0x99, 0xbb, 0x77, 0xbd, 0x48, 0x24, 0x6a, 0x37, 0x8e, 0x31, 0x33, 0xfb, 0x30, 0x23, 0x2a, 0xad, 0xa9, 0x20, 0xae, 0x04];
+        let mut hash_wide = [0u8; 64];
+        hash_wide[..48].copy_from_slice(&test_vec);
+
+        let mut reduce_fe = FieldElement::from_bytes_wide(&hash_wide);
+        let mut expected_reduced = [0x30, 0x92, 0xf0, 0x33, 0xb1, 0x6d, 0x4d, 0x5f, 0x74, 0xa3, 0xf7, 0xdc, 0x70, 0x91, 0xfe, 0x43, 0x4b, 0x44, 0x90, 0x65, 0x15, 0x2b, 0x95, 0x47, 0x6f, 0x12, 0x1d, 0xe8, 0x99, 0xbb, 0x77, 0x3d];
+
+        assert_eq!(reduce_fe.to_bytes(), expected_reduced);
+
+        test_vec= [0xae, 0x69, 0x22, 0xd7, 0x28, 0xc1, 0x21, 0xf6, 0x90, 0x48, 0x61, 0xbd, 0x67, 0x49, 0x67, 0xb3, 0x19, 0xd4, 0x6d, 0xee, 0x9d, 0x04, 0x7f, 0x86, 0xc4, 0x27, 0xc5, 0x3f, 0x8b, 0x29, 0xa5, 0x5c, 0xdb, 0xe1, 0x5e, 0xae, 0x23, 0x4e, 0xb7, 0x84, 0xe5, 0x9d, 0x6d, 0x20, 0x2a, 0x78, 0x20, 0xd6];
+        hash_wide = [0u8; 64];
+        hash_wide[..48].copy_from_slice(&test_vec);
+
+        reduce_fe = FieldElement::from_bytes_wide(&hash_wide);
+        expected_reduced = [0x30, 0xf0, 0x37, 0xb9, 0x74, 0x5a, 0x57, 0xa9, 0xa2, 0xb8, 0xa6, 0x8d, 0xa8, 0x1f, 0x39, 0x7c, 0x39, 0xd4, 0x6d, 0xee, 0x9d, 0x04, 0x7f, 0x86, 0xc4, 0x27, 0xc5, 0x3f, 0x8b, 0x29, 0xa5, 0x5c];
+
+        assert_eq!(reduce_fe.to_bytes(), expected_reduced);
+
+        test_vec= [0x3a, 0x7a, 0x2b, 0x29, 0x83, 0xe8, 0x88, 0x61, 0x25, 0x20, 0xcf, 0x6a, 0xfe, 0xbb, 0xea, 0x6b, 0x21, 0x8b, 0x58, 0x15, 0xf2, 0x38, 0x80, 0x09, 0x2a, 0x92, 0x5a, 0xf9, 0x4c, 0xd6, 0xfa, 0x24, 0x6a, 0x35, 0xb7, 0xdb, 0xed, 0x1e, 0x8f, 0xdf, 0xfd, 0xcd, 0x36, 0x6e, 0x55, 0xed, 0x0a, 0xbe];
+        hash_wide = [0u8; 64];
+        hash_wide[..48].copy_from_slice(&test_vec);
+
+        reduce_fe = FieldElement::from_bytes_wide(&hash_wide);
+        expected_reduced = [0xf6, 0x67, 0x5d, 0xc6, 0xd1, 0x7f, 0xc7, 0x90, 0xd4, 0xb3, 0xf1, 0xc6, 0xac, 0xf6, 0x89, 0xa1, 0x3d, 0x8b, 0x58, 0x15, 0xf2, 0x38, 0x80, 0x09, 0x2a, 0x92, 0x5a, 0xf9, 0x4c, 0xd6, 0xfa, 0x24];
+
+        assert_eq!(reduce_fe.to_bytes(), expected_reduced);
+    }
+
+    #[test]
+    fn hash_to_field() {
+        let message = [0xfc, 0x51, 0xcd, 0x8e, 0x62, 0x18, 0xa1, 0xa3, 0x8d, 0xa4, 0x7e, 0xd0, 0x02, 0x30, 0xf0, 0x58, 0x08, 0x16, 0xed, 0x13, 0xba, 0x33, 0x03, 0xac, 0x5d, 0xeb, 0x91, 0x15, 0x48, 0x90, 0x80, 0x25, 0xaf, 0x82];
+        let dst = b"ECVRF_edwards25519_XMD:SHA-512_ELL2_NU_\x04";
+        let fe = FieldElement::hash_to_field::<Sha512>(&message, dst);
+        let expected_fe = FieldElement::from_bytes(&[0xf6, 0x67, 0x5d, 0xc6, 0xd1, 0x7f, 0xc7, 0x90, 0xd4, 0xb3, 0xf1, 0xc6, 0xac, 0xf6, 0x89, 0xa1, 0x3d, 0x8b, 0x58, 0x15, 0xf2, 0x38, 0x80, 0x09, 0x2a, 0x92, 0x5a, 0xf9, 0x4c, 0xd6, 0xfa, 0x24]);
+        assert_eq!(fe, expected_fe);
+
+        let message = "";
+        let dst = "QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_";
+        let fe = FieldElement::hash_to_field::<Sha512>(message.as_bytes(), dst.as_bytes());
+        let mut expected_fe_bytes = [0x7f, 0x3e, 0x7f, 0xb9, 0x42, 0x81, 0x03, 0xad, 0x7f, 0x52, 0xdb, 0x32, 0xf9, 0xdf, 0x32, 0x50, 0x5d, 0x7b, 0x42, 0x7d, 0x89, 0x4c, 0x50, 0x93, 0xf7, 0xa0, 0xf0, 0x37, 0x4a, 0x30, 0x64, 0x1d];
+        expected_fe_bytes.reverse();
+        let expected_fe = FieldElement::from_bytes(&expected_fe_bytes);
+        assert_eq!(fe, expected_fe);
+    }
 }

src/lib.rs

@@ -272,6 +272,8 @@ extern crate fiat_crypto;
 // Used for traits related to constant-time code.
 extern crate subtle;
 
+#[cfg(test)]
+extern crate sha2;
 #[cfg(all(test, feature = "serde"))]
 extern crate bincode;
 #[cfg(feature = "serde")]

src/montgomery.rs

@@ -164,14 +164,16 @@ impl MontgomeryPoint {
     }
 }
 
-/// Perform the Elligator2 mapping to a Montgomery point.
+/// Perform the Elligator2 mapping to a Montgomery point. Returns a Montgomery point and a `Choice`
+/// determining whether eps is a square. This is required by the standard to determine the
+/// sign of the v coordinate.
 ///
 /// See <https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-10#section-6.7.1>
 //
 // TODO Determine how much of the hash-to-group API should be exposed after the CFRG
 //      draft gets into a more polished/accepted state.
 #[allow(unused)]
-pub(crate) fn elligator_encode(r_0: &FieldElement) -> MontgomeryPoint {
+pub(crate) fn elligator_encode(r_0: &FieldElement) -> (MontgomeryPoint, Choice) {
     let one = FieldElement::one();
     let d_1 = &one + &r_0.square2(); /* 2r^2 */
 
@@ -190,7 +192,7 @@ pub(crate) fn elligator_encode(r_0: &FieldElement) -> MontgomeryPoint {
     let mut u = &d + &Atemp; /* d, or d+A if nonsquare */
     u.conditional_negate(!eps_is_sq); /* d, or -d-A if nonsquare */
 
-    MontgomeryPoint(u.to_bytes())
+    (MontgomeryPoint(u.to_bytes()), eps_is_sq)
 }
 
 /// A `ProjectivePoint` holds a point on the projective line
@@ -465,15 +467,15 @@ mod test {
         let bits_in: [u8; 32] = (&bytes[..]).try_into().expect("Range invariant broken");
 
         let fe = FieldElement::from_bytes(&bits_in);
-        let eg = elligator_encode(&fe);
+        let (eg, _) = elligator_encode(&fe);
         assert_eq!(eg.to_bytes(), ELLIGATOR_CORRECT_OUTPUT);
     }
 
     #[test]
     fn montgomery_elligator_zero_zero() {
         let zero = [0u8; 32];
         let fe = FieldElement::from_bytes(&zero);
-        let eg = elligator_encode(&fe);
+        let (eg, _) = elligator_encode(&fe);
         assert_eq!(eg.to_bytes(), zero);
     }
 }