pasta_curves-source/src/arithmetic/hashtocurve.rs

//! This module implements "simplified SWU" hashing to short Weierstrass curves
//! with a = 0.

use byteorder::{BigEndian, WriteBytesExt};
use core::fmt::Debug;
use core::marker::PhantomData;
use subtle::ConstantTimeEq;

use super::{Curve, CurveAffine, Field, FieldExt};

/// Implementation of the "simplified SWU" hashing to short Weierstrass curves
/// with a = 0. Internally uses SHAKE128.
#[derive(Debug)]
pub struct SimplifiedSWUWithDegree3Isogeny<
    F: FieldExt,
    C: CurveAffine<Base = F>,
    I: CurveAffine<Base = F>,
> {
    /// `Z` parameter (ξ in [WB2019](https://eprint.iacr.org/2019/403)).
    pub z: F,

    /// Precomputed -b/a for the isogenous curve.
    pub minus_b_over_a: F,

    /// Precomputed b/Za for the isogenous curve.
    pub b_over_za: F,

    /// Precomputed sqrt(Z / ROOT_OF_UNITY).
    pub theta: F,

    /// Constants for the isogeny.
    pub isogeny_constants: [F; 13],

    _marker_c: PhantomData<C>,
    _marker_i: PhantomData<I>,
}

impl<F: FieldExt, C: CurveAffine<Base = F>, I: CurveAffine<Base = F>>
    SimplifiedSWUWithDegree3Isogeny<F, C, I>
{
    /// Create a SimplifiedSWUWithDegree3Isogeny method for the given parameters.
    ///
    /// # Panics
    /// Panics if z is square.
    pub fn new(z: &F, isogeny_constants: [F; 13]) -> Self {
        let a = I::a();
        let b = I::b();

        SimplifiedSWUWithDegree3Isogeny {
            z: *z,
            minus_b_over_a: (-b) * &(a.invert().unwrap()),
            b_over_za: b * &((*z * a).invert().unwrap()),
            theta: (F::ROOT_OF_UNITY.invert().unwrap() * z).sqrt().unwrap(),
            isogeny_constants: isogeny_constants,
            _marker_c: PhantomData,
            _marker_i: PhantomData,
        }
    }

    /// The full hash from an input message to a curve point.
    ///
    /// `domain_prefix` should identify the application protocol, usage
    /// within that protocol, and version, e.g. "z.cash:Orchard-V1".
    /// Other fields required to conform to [IRTF-CFRG-Hash-to-Curve]
    /// will be added automatically. There may be a length limitation on
    /// `domain_prefix`.
    ///
    /// For example, the resulting full domain separation tag for the
    /// Pallas curve using `Shake128` and the simplified SWU map might be
    /// b"z.cash:Orchard-V1-pallas_XOF:SHAKE128_SSWU_RO_".
    pub fn hash_to_curve(&self, domain_prefix: &str) -> Box<dyn Fn(&[u8]) -> C::Projective + '_> {
        let domain_separation_tag: String = format!(
            "{}-{}_{}_{}_RO_",
            domain_prefix,
            C::CURVE_ID,
            "XOF:SHAKE128",
            "SSWU"
        );

        Box::new(move |message| {
            let mut us = [Field::zero(); 2];
            Self::hash_to_field(message, domain_separation_tag.as_bytes(), &mut us);
            self.field_elements_to_curve(&us[0], &us[1])
        })
    }

    /// A non-uniform hash from an input message to a curve point.
    /// This is *not* suitable for applications requiring a random oracle.
    /// Use `hash_to_curve` instead unless you are really sure that a
    /// non-uniform map is sufficient.
    ///
    /// `domain_prefix` is as described for `hash_to_curve`.
    ///
    /// For example, the resulting full domain separation tag for the
    /// Pallas curve using `Shake128` and the simplified SWU map might be
    /// b"z.cash:Orchard-V1-pallas_XOF:SHAKE128_SSWU_NU_".
    pub fn encode_to_curve(&self, domain_prefix: &str) -> Box<dyn Fn(&[u8]) -> C::Projective + '_> {
        let domain_separation_tag: String = format!(
            "{}-{}_{}_{}_NU_",
            domain_prefix,
            C::CURVE_ID,
            "XOF:SHAKE128",
            "SSWU"
        );

        Box::new(move |message| {
            let mut us = [Field::zero(); 1];
            Self::hash_to_field(message, domain_separation_tag.as_bytes(), &mut us);
            let r = self.map_to_curve(&us[0]);
            self.iso_map(&r)
        })
    }

    /// Hashes over a message and writes the output to all of `buf`.
    pub fn hash_to_field(message: &[u8], domain_separation_tag: &[u8], buf: &mut [F]) {
        use sha3::digest::{ExtendableOutput, Update};
        assert!(domain_separation_tag.len() < 256);

        // Assume that the field size is 32 bytes and k is 256, where k is defined in
        // <https://www.ietf.org/archive/id/draft-irtf-cfrg-hash-to-curve-10.html#name-security-considerations-3>.
        const CHUNKLEN: usize = 64;

        let outlen = buf.len() * CHUNKLEN;
        let mut outlen_enc = vec![];
        outlen_enc.write_u32::<BigEndian>(outlen as u32).unwrap();

        let mut xof = sha3::Shake128::default();
        xof.update(message);
        xof.update(outlen_enc);
        xof.update([domain_separation_tag.len() as u8]);
        xof.update(domain_separation_tag);

        for (big, buf) in xof
            .finalize_boxed(outlen)
            .chunks(CHUNKLEN)
            .zip(buf.iter_mut())
        {
            let mut little = [0u8; CHUNKLEN];
            little.copy_from_slice(big);
            little.reverse();
            *buf = F::from_bytes_wide(&little);
        }
    }

    /// Maps a field element to the isogenous curve.
    pub fn map_to_curve(&self, u: &F) -> I::Projective {
        // 1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
        // 2. x1 = (-B / A) * (1 + tv1)
        // 3. If tv1 == 0, set x1 = B / (Z * A)
        // 4. gx1 = x1^3 + A * x1 + B
        //
        // We use the "Avoiding inversions" optimization in [WB2019, section 4.2]
        // (not to be confused with section 4.3):
        //
        //   here       [WB2019]
        //   -------    ---------------------------------
        //   Z          ξ
        //   u          t
        //   Z * u^2    ξ * t^2 (called u, confusingly)
        //   x1         X_0(t)
        //   x2         X_1(t)
        //   gx1        g(X_0(t))
        //   gx2        g(X_1(t))
        //
        // Using the "here" names:
        //    x1 = num_x1/div      = [B*(Z^2 * u^4 + Z * u^2 + 1)] / [-A*(Z^2 * u^4 + Z * u^2]
        //   gx1 = num_gx1/div_gx1 = [num_x1^3 + A * num_x1 * div^2 + B * div^3] / div^3

        let a = I::a();
        let b = I::b();
        let z_u2 = self.z * u.square();
        let ta = z_u2.square() + z_u2;
        let num_x1 = b * (ta + F::one());
        let div = -a * ta;
        let num2_x1 = num_x1.square();
        let div2 = div.square();
        let div3 = div2 * div;
        let ta_is_zero = ta.ct_is_zero();
        let num_gx1 = F::conditional_select(
            &((num2_x1 + a * div2) * num_x1 + b * div3),
            &self.b_over_za,
            ta_is_zero,
        );
        let div_gx1 = F::conditional_select(&div3, &F::one(), ta_is_zero);

        // 5. x2 = Z * u^2 * x1
        let num_x2 = z_u2 * num_x1; // same div

        // 6. gx2 = x2^3 + A * x2 + B  [optimized out; see below]
        // 7. If is_square(gx1), set x = x1 and y = sqrt(gx1)
        // 8. Else set x = x2 and y = sqrt(gx2)
        let (gx1_square, y1) = F::sqrt_ratio(&num_gx1, &div_gx1);

        // This magic also comes from a generalization of [WB2019, section 4.2].
        //
        // The Sarkar square root algorithm with input s gives us a square root of
        // h * s for free when s is not square, where h is a fixed nonsquare.
        // In our implementation, h = ROOT_OF_UNITY.
        // We know that Z / h is a square since both Z and h are
        // nonsquares. Precompute theta as a square root of Z / ROOT_OF_UNITY.
        //
        // We have gx2 = g(Z * u^2 * x1) = Z^3 * u^6 * gx1
        //                               = (Z * u^3)^2 * (Z/h * h * gx1)
        //                               = (Z * theta * u^3)^2 * (h * gx1)
        //
        // When gx1 is not square, y1 is a square root of h * gx1, and so Z * theta * u^3 * y1
        // is a square root of gx2. Note that we don't actually need to compute gx2.

        let y2 = self.theta * z_u2 * u * y1;
        let num_x = F::conditional_select(&num_x2, &num_x1, gx1_square);
        let y = F::conditional_select(&y2, &y1, gx1_square);

        // 9. If sgn0(u) != sgn0(y), set y = -y
        let y = F::conditional_select(
            &(-y),
            &y,
            (u.get_lower_32() % 2).ct_eq(&(y.get_lower_32() % 2)),
        );

        I::Projective::new_jacobian(num_x * div, y * div3, div).unwrap()
    }

    /// Implements a degree 3 isogeny map.
    pub fn iso_map(&self, p: &I::Projective) -> C::Projective {
        // The input and output are in Jacobian coordinates, using the method
        // in "Avoiding inversions" [WB2019, section 4.3].

        let iso = self.isogeny_constants;
        let (x, y, z) = p.jacobian_coordinates();

        let z2 = z.square();
        let z3 = z2 * z;
        let z4 = z2.square();
        let z6 = z3.square();

        let num_x = ((iso[0] * x + iso[1] * z2) * x + iso[2] * z4) * x + iso[3] * z6;
        let div_x = (z2 * x + iso[4] * z4) * x + iso[5] * z6;

        let num_y = (((iso[6] * x + iso[7] * z2) * x + iso[8] * z4) * x + iso[9] * z6) * y;
        let div_y = (((x + iso[10] * z2) * x + iso[11] * z4) * x + iso[12] * z6) * z3;

        let zo = div_x * div_y;
        let xo = num_x * div_y * zo;
        let yo = num_y * div_x * zo.square();

        C::Projective::new_jacobian(xo, yo, zo).unwrap()
    }

    /// Map two field elements to a curve point.
    pub fn field_elements_to_curve(&self, u0: &C::Base, u1: &C::Base) -> C::Projective {
        let q0 = self.map_to_curve(u0);
        let q1 = self.map_to_curve(u1);
        let r: I::Projective = q0 + &q1;
        assert!(bool::from(r.is_on_curve()));
        // here is where we would scale by the cofactor if we supported nonprime-order curves
        self.iso_map(&r)
    }
}