Endo

.gitignore

@@ -7,3 +7,4 @@ Cargo.lock
 *.orig
 coeffs_*
 msm_bases_*
+*.py

algebra/src/curves/bls12_377/g1.rs

@@ -3,7 +3,7 @@ use crate::{
     biginteger::{BigInteger256, BigInteger384},
     curves::models::{ModelParameters, SWModelParameters},
     fields::{
-        bls12_377::{Fq, Fr},
+        bls12_377::*,
         Field,
     },
 };

algebra/src/curves/bls12_377/g2.rs

@@ -4,7 +4,7 @@ use crate::{
     biginteger::{BigInteger256, BigInteger384},
     curves::models::{ModelParameters, SWModelParameters},
     fields::{
-        bls12_377::{Fq, Fq2, Fr},
+        bls12_377::*,
         Field,
     },
 };

algebra/src/curves/bls12_381/g1.rs

@@ -7,7 +7,7 @@ use crate::{
         models::{ModelParameters, SWModelParameters},
     },
     fields::{
-        bls12_381::{Fq, Fr},
+        bls12_381::*,
         Field,
     },
 };

algebra/src/curves/bls12_381/g2.rs

@@ -7,7 +7,7 @@ use crate::{
         models::{ModelParameters, SWModelParameters},
     },
     fields::{
-        bls12_381::{Fq, Fq2, Fr},
+        bls12_381::*,
         Field,
     },
 };

algebra/src/curves/check_curve_parameters.sage

@@ -1,9 +1,13 @@
 # The following Sage script check the consistency of the following curves parameters:
 #
-#   1) P=(GENERATOR_X,GENERATOR_Y)  must belongs to the curve of equation E: y^2 = x^3 + Ax + B   
+#   1) P=(GENERATOR_X,GENERATOR_Y)      must belongs to the curve of equation E: y^2 = x^3 + Ax + B   
 #   2) P                                must have order equal to the MODULUS of the scalar field
 #   3) COFACTOR                         must be equal to Order(E)/Order(P)
 #   4) COFACTOR_INV                     must be the inverse of COFACTOR in the scalar Field
+#   5) ENDO_COEFF                       must be a cube root in the base field.
+#   6) ENDO_SCALAR                      must be a cube root in the scalar field and satisfy ENDO_SCALAR * (X, Y) == (ENDO_COEFF * X, Y)        
+#   7) The intersection of the plane lattice spanned by {(1, ENDO_SCALAR), (0, SCALAR_FIELD_MODULUS)} with the square [-A,A]^2 must be empty,
+#       where A = 2^(LAMBDA/2 + 1) + 2^(LAMBDA/2) + 1.                         
 # Open Sage Shell in the corresponding folder and run the command 
 #       "sage check_curve_paramaters sage [file_path_curve] [file_path_basefield] [file_path_scalarfield]".
 
@@ -65,10 +69,10 @@ scalar_field_name = re.findall(pattern, readfile)[0]
 fn = "(?:" + base_field_name + "|" + scalar_field_name + ")" #fn = field name = "(:?Fr|Fq)". Useful declaration for the pattern
 
 #### Reading the big integers list and extracting names and values
-pattern = "const\s+(\w+):\s*" + fn + "\s*=\s*field_new!\(\s*" + fn + "\s*,\s*BigInteger\d*\s*\(\s*\[" + "([0-9a-fA-Fxu\s,]+)\s*" + "\]\s*\)"
+pattern = "const\s+(\w+)[:\w\s]*=\s*field_new!\([\s\w,]*\(\s*\[" + "([0-9a-fA-Fxu\s,]+)\s*" + "\]\s*\)"
 big_int_ls = re.findall(pattern,readfile)    #####list of couples of the form ('[VARIABLE_NAME]',"[u64],..,[u64]")
 
-big_int_names = [b[0] for b in big_int_ls] 
+big_int_names = [b[0] for b in big_int_ls]
 big_int_values = [BigInteger_to_number(b[1]) for b in big_int_ls]
 
 BigIntegerLen = BigInteger_len(big_int_ls[0][1])
@@ -87,6 +91,10 @@ for s in big_int_names:
 pattern =  "const\s+COFACTOR:\s*&'static\s*\[u64\]\s*=\s*&\[([0-9a-fA-Fxu\s,]+)\]\s*;"
 COFACTOR = BigInteger_to_number(re.findall(pattern,readfile)[0])
 
+####Reading the value of LAMBDA
+pattern =  "const\s+LAMBDA[:\w\s]*=\s*([\d]+)\s*;"
+LAMBDA = int(re.findall(pattern,readfile)[0])
+
 #######################################Reading the values from the file containing the Base Field parameters########################
 filename = sys.argv[2]
 
@@ -163,4 +171,60 @@ else:
 if Fr(COFACTOR) * Fr(COFACTOR_INV) == Fr(SCALAR_FIELD_R):
     print("Correct. COFACTOR_INV is the inverse of COFACTOR in the the scalar field.")
 else:
-    print("WARNING! COFACTOR_INV IS NOT THE INVERSE OF COFACTOR IN THE SCALAR FIELD!")
+    print("WARNING! COFACTOR_INV IS NOT THE INVERSE OF COFACTOR IN THE SCALAR FIELD!")
+####### Checking the correctness of ENDO_COEFF and ENDO_FACTOR ############
+endo_mul_is_used = False
+if 'ENDO_COEFF' in locals() and 'ENDO_SCALAR' in locals():
+    zeta_q = Fq(ENDO_COEFF) * Fq(BASE_FIELD_R)**(-1)
+    if zeta_q**2 + zeta_q == Fq(-1):
+        endo_mul_is_used = True
+        print("Correct. ENDO_COEFF is a primitive cube root of unity.")
+    else:
+        print("WARNING! ENDO_COEFF IS NOT A PRIMITIVE CUBE ROOT OF UNITY.")
+    zeta_r = Fr(ENDO_SCALAR) * Fr(SCALAR_FIELD_R)**(-1)
+    if zeta_r**2 + zeta_r == Fr(-1):
+        print("Correct. ENDO_SCALAR is a primitive cube root of unity.")
+    else:
+        print("WARNING! ENDO_SCALAR IS NOT A PRIMITIVE CUBE ROOT OF UNITY.")
+
+
+####### Checking the consistency of ENDO_COEFF and ENDO_SCALAR #############
+if endo_mul_is_used:
+    Q = int(zeta_r) * P
+    if Q == E([zeta_q * X, Y]):
+        print("Correct. ENDO_COEFF and ENDO_SCALAR are consistent.")
+    else:
+        print("WARNING! ENDO_COEFF AND ENDO_SCALAR ARE NOT CONSISTENT!")
+
+
+########## Checking that shortest vector in the lattice ([1,zeta_r),[0,r]) is long enough #########
+## The Halo paper (https://eprint.iacr.org/2019/1021.pdf) proves the injectivity of the endo_mul map.
+## The injectivity of the map (a,b) |-> a\zeta + b for a,b in [0,A] (essential for using add_unsafe)
+## is equivalent the lattice condition below.
+## a*zeta + b = a'*zeta_r + b' mod r   for a,a',b,b' in [0,A] 
+## is equivalent to the fact that there are non-zero solutions to 
+##      a * zeta_r = b mod r      for a,b in [-A,A].
+## Then it would exists c such that
+##      b = a * zeta_r + c * r.
+## Any such solution correspond to a point of the lattice spanned by (1, zeta_r) and (0, r).
+##      (a, b) = (a, c) * (1  zeta_r)
+##                        (0    r )
+## The injectivity is equivalent to the fact that the intersection between this lattice and [-A, A]^2
+## is trivial. To verify this we first compute a LLL reduced basis {v,w} and
+## then check if at least one of v, w, v + w, v - w is belongs to such a square.
+## If not, there can't be other lattice points in the square.
+if endo_mul_is_used:
+    A = 2**(LAMBDA//2 + 1) + 2**(LAMBDA//2) + 1 
+    L = Matrix([[1,Integer(zeta_r)],[0,SCALAR_FIELD_MODULUS]])
+    Lred = L.LLL()
+    set = [Lred.row(0), Lred.row(1), Lred.row(0) - Lred.row(1), Lred.row(0) + Lred.row(1)]
+    add_unsafe = True
+    for v in set:
+        if abs(v[0]) <= A and abs(v[1]) <= A:
+            add_unsafe = False
+    if add_unsafe:
+        print("We can use add_unsafe for endo_mul.")
+    else:
+        print("WARNING! WE CAN'T USE add_unsafe FOR endo_mul!")
+else:
+    print("endo_mul is not used for this curve.")

algebra/src/curves/mnt6/g1.rs

@@ -7,7 +7,7 @@ use crate::{
         short_weierstrass_projective::{GroupAffine, GroupProjective},
         AffineCurve,
     },
-    fields::mnt6::{Fq, Fq3, Fr},
+    fields::mnt6::*,
 };
 use crate::{field_new, FromBytes};
 use serde::{Deserialize, Serialize};

algebra/src/curves/mnt6/g2.rs

@@ -7,7 +7,7 @@ use crate::{
         short_weierstrass_projective::{GroupAffine, GroupProjective},
         AffineCurve,
     },
-    fields::mnt6::{Fq, Fq3, Fr},
+    fields::mnt6::*,
 };
 use crate::{field_new, FromBytes};
 use byteorder::{BigEndian, ReadBytesExt, WriteBytesExt};

algebra/src/curves/mod.rs

@@ -321,6 +321,21 @@ pub trait AffineCurve:
     fn mul_by_cofactor_inv(&self) -> Self;
 }
 
+/// The `EndoMulCurve` trait for curves that have a non-trivial endomorphism 
+/// `Phi` of the form `Phi(x,y) = (zeta*x,y)`. 
+pub trait EndoMulCurve: AffineCurve {
+    /// Apply `Phi`
+    fn apply_endomorphism(&self) -> Self;
+
+    /// Conversion of a bit sequence used in `endo_mul()` into its equivalent
+    /// scalar
+    fn endo_rep_to_scalar(bits: Vec<bool>) -> Result<Self::ScalarField, Error>;
+
+    /// Endomorphism-based multiplication of `&self` with `bits`, a little-endian 
+    /// endomorphism representation. 
+    fn endo_mul(&self, bits: Vec<bool>) -> Result<Self::Projective, Error>;
+}
+
 impl<C: ProjectiveCurve> Group for C {
     type ScalarField = C::ScalarField;
     #[must_use]

algebra/src/curves/models/mod.rs

@@ -122,3 +122,22 @@ pub trait MontgomeryModelParameters: ModelParameters {
 
     type TEModelParameters: TEModelParameters<BaseField = Self::BaseField>;
 }
+
+pub trait EndoMulParameters: SWModelParameters {
+    /// Parameters for endomorphism-based scalar multiplication [Halo](https://eprint.iacr.org/2019/1021).
+    /// A non-trivial cubic root of unity `ENDO_COEFF` for a curve endomorphism of the form
+    ///     (x, y) -> (ENDO_COEFF * x, y).
+    const ENDO_COEFF: Self::BaseField;
+
+    /// The scalar representation `zeta_r` of `ENDO_COEFF`.
+    /// NOTE : If one wants to use the endo mul circuit with `lambda` many bits,
+    /// then `zeta_r` MUST satisfy the minimal distance property
+    ///     D = min { d(n*zeta_r, m*zeta_r) : n,m in [0, T] } >= R + 1,
+    /// where `T = 2^{lambda/2 + 1} + 2^{lambda/2} - 1` is the output
+    /// bound for the coefficients a, b of the equivalent scalar
+    /// representation `a*zeta_r + b`.
+    const ENDO_SCALAR: Self::ScalarField;
+
+    /// Maximum number of bits for which security of endo mul is proven. MUST be an even number.
+    const LAMBDA: usize;
+}

algebra/src/curves/models/short_weierstrass_jacobian.rs

@@ -1,6 +1,9 @@
 use crate::{
     bytes::{FromBytes, ToBytes},
-    curves::{models::SWModelParameters as Parameters, AffineCurve, ProjectiveCurve},
+    curves::{models::{
+        SWModelParameters as Parameters,
+        EndoMulParameters as EndoParameters,
+    }, AffineCurve, ProjectiveCurve, EndoMulCurve},
     fields::{BitIterator, Field, PrimeField, SquareRootField},
     BitSerializationError, CanonicalDeserialize, CanonicalDeserializeWithFlags, CanonicalSerialize,
     CanonicalSerializeWithFlags, Error, FromBytesChecked, FromCompressedBits, SWFlags,
@@ -295,6 +298,99 @@ impl<P: Parameters> AffineCurve for GroupAffine<P> {
     }
 }
 
+impl<P: EndoParameters> EndoMulCurve for GroupAffine<P> {
+
+    fn apply_endomorphism(&self) -> Self {
+        let mut self_e = self.clone();
+        self_e.x.mul_assign(P::ENDO_COEFF);
+        self_e
+    }
+
+    fn endo_rep_to_scalar(bits: Vec<bool>) -> Result<Self::ScalarField, Error> {
+
+        let mut a : P::ScalarField = 2u64.into();
+        let mut b : P::ScalarField = 2u64.into();
+
+        let one = P::ScalarField::one();
+        let one_neg = one.neg();
+
+        let mut bits = bits;
+        if bits.len() % 2 == 1 {
+            bits.push(false);
+        }
+
+        if bits.len() > P::LAMBDA {
+            Err("Endo mul bits length exceeds LAMBDA")?
+        }
+
+        for i in (0..(bits.len() / 2)).rev() {
+            a.double_in_place();
+            b.double_in_place();
+
+            let s =
+                if bits[i * 2] {
+                    &one
+                } else {
+                    &one_neg
+                };
+
+            if bits[i * 2 + 1] {
+                a.add_assign(s);
+            } else {
+                b.add_assign(s);
+            }
+        }
+
+        Ok(a.mul(P::ENDO_SCALAR) + &b)
+    }
+
+    /// Endomorphism-based multiplication of a curve point 
+    /// with a scalar in little-endian endomorphism representation.
+    fn endo_mul(&self, bits: Vec<bool>) -> Result<Self::Projective, Error> {
+
+        let self_neg = self.neg();
+
+        let self_e = self.apply_endomorphism();
+        let self_e_neg = self_e.neg();
+
+        let mut acc = self_e.into_projective();
+        acc.add_assign_mixed(&self);
+        acc.double_in_place();
+
+        let mut bits = bits;
+        if bits.len() % 2 == 1 {
+            bits.push(false);
+        }
+
+        if bits.len() > P::LAMBDA {
+            Err("Endo mul bits length exceeds LAMBDA")?
+        }
+
+        for i in (0..(bits.len() / 2)).rev() {
+
+            let s =
+                if bits[i * 2 + 1] {
+                    if bits[i * 2] {
+                        &self_e
+                    } else {
+                        &self_e_neg
+                    }
+                } else {
+                    if bits[i * 2] {
+                        &self
+                    } else {
+                        &self_neg
+                    }
+                };
+
+            acc.double_in_place();
+            acc.add_assign_mixed(s);
+        }
+
+        Ok(acc)
+    }
+}
+
 impl<P: Parameters> SemanticallyValid for GroupAffine<P> {
     fn is_valid(&self) -> bool {
         self.x.is_valid() && self.y.is_valid() && self.group_membership_test()