Add the relaxed NAF (#302)

* add the relaxed NAF

* fmt; rename

* simplify the algorithm

* CHANGELOG

* make find_relaxed_naf public

* Update CHANGELOG.md

* cleaned up

* show the usefulness

CHANGELOG.md

@@ -5,13 +5,14 @@
 ### Breaking changes
 
 - [\#300](https://github.com/arkworks-rs/algebra/pull/300) (`ark-ec`) Change the implementation of `Hash` trait of `GroupProjective` to use the affine co-ordinates.
+- [\#302](https://github.com/arkworks-rs/algebra/pull/302) (`ark-ff`) Rename `find_wnaf` to `find_naf`.
 - [\#310](https://github.com/arkworks-rs/algebra/pull/310) (`ark-ec`, `ark-ff`) Remove unnecessary internal `PhantomData`.
 - [\#333](https://github.com/arkworks-rs/algebra/pull/333) (`ark-poly`) Expose more properties of `EvaluationDomain`s.
 - [\#338](https://github.com/arkworks-rs/algebra/pull/338) (`ark-ec`) Add missing `UniformRand` trait bound to `GroupAffine`.
 - [\#338](https://github.com/arkworks-rs/algebra/pull/338) (workspace) Change to Rust 2021 edition.
 - [\#345](https://github.com/arkworks-rs/algebra/pull/345) (`ark-ec`, `ark-serialize`) Change the serialization format for Twisted Edwards Curves. We now encode the Y co-ordinate and take the sign bit of the X co-ordinate, the default flag is also now the Positive X value. The old methods for backwards compatibility are located [here](https://github.com/arkworks-rs/algebra/pull/345/files#diff-3621a48bb33f469144044d8d5fc663f767e103132a764812cda6be6c25877494R860)
 - [\#348](https://github.com/arkworks-rs/algebra/pull/348) (`ark-ec`) Rename `msm:{Fixed,Variable}BaseMSM:multi_scalar_mul` to `msm:{Fixed,Variable}:msm` to avoid redundancy.
-- [\#359](https://github.com/arkworks-rs/algebra/pull/359) (ark-test-templates) Simplify the field and curve test macros.
+- [\#359](https://github.com/arkworks-rs/algebra/pull/359) (`ark-test-templates`) Simplify the field and curve test macros.
 - [\#365](https://github.com/arkworks-rs/algebra/pull/365) (`ark-ec`)
     - Move `COFACTOR`, `COFACTOR_INV`, and `is_in_correct_subgroup_assuming_on_curve()` from `{SW,TE}ModelParameters` to `ModelParameters`.
     - Add `mul_bits()` to `AffineCurve` and provide a default implementation of `mul()` using this.
@@ -42,6 +43,7 @@
 
 ### Improvements
 
+- [\#302](https://github.com/arkworks-rs/algebra/pull/302) (`ark-ff`) Add the relaxed NAF computation.
 - [\#306](https://github.com/arkworks-rs/algebra/pull/306) (`ark-ff`, `ark-ff-asm`) Make the assembly backend available on `stable`.
 - [\#339](https://github.com/arkworks-rs/algebra/pull/339) (`ark-ff`) Remove duplicated code from `test_field` module and replace its usage with `ark-test-curves` crate.
 - [\#352](https://github.com/arkworks-rs/algebra/pull/352) (`ark-ff`) Update `QuadExtField::sqrt` for better performance.

ff/src/biginteger/arithmetic.rs

@@ -1,4 +1,4 @@
-use ark_std::vec::Vec;
+use ark_std::{vec, vec::Vec};
 
 macro_rules! adc {
     ($a:expr, $b:expr, &mut $carry:expr$(,)?) => {{
@@ -66,8 +66,8 @@ pub(crate) fn mac_discard(a: u64, b: u64, c: u64, carry: &mut u64) {
     *carry = (tmp >> 64) as u64;
 }
 
-/// Compute the Window NAF (non-adjacent form) of num
-pub fn find_wnaf(num: &[u64]) -> Vec<i64> {
+/// Compute the NAF (non-adjacent form) of num
+pub fn find_naf(num: &[u64]) -> Vec<i64> {
     let is_zero = |num: &[u64]| num.iter().all(|x| *x == 0u64);
     let is_odd = |num: &[u64]| num[0] & 1 == 1;
     let sub_noborrow = |num: &mut [u64], z: u64| {
@@ -119,3 +119,78 @@ pub fn find_wnaf(num: &[u64]) -> Vec<i64> {
 
     res
 }
+
+/// We define relaxed NAF as a variant of NAF with a very small tweak.
+///
+/// Note that the cost of scalar multiplication grows with the length of the sequence (for doubling)
+/// plus the Hamming weight of the sequence (for addition, or subtraction).
+///
+/// NAF is optimizing for the Hamming weight only and therefore can be suboptimal.
+/// For example, NAF may generate a sequence (in little-endian) of the form ...0 -1 0 1.
+///
+/// This can be rewritten as ...0 1 1 to avoid one doubling, at the cost that we are making an
+/// exception of non-adjacence for the most significant bit.
+///
+/// Since this representation is no longer a strict NAF, we call it ``relaxed NAF''.
+///
+pub fn find_relaxed_naf(num: &[u64]) -> Vec<i64> {
+    let mut res = find_naf(num);
+
+    let len = res.len();
+    if res[len - 2] == 0 && res[len - 3] == -1 {
+        res[len - 3] = 1;
+        res[len - 2] = 1;
+        res.resize(len - 1, 0);
+    }
+
+    res
+}
+
+#[test]
+fn test_find_relaxed_naf_usefulness() {
+    let vec = find_naf(&[12u64]);
+    assert_eq!(vec.len(), 5);
+
+    let vec = find_relaxed_naf(&[12u64]);
+    assert_eq!(vec.len(), 4);
+}
+
+#[test]
+fn test_find_relaxed_naf_correctness() {
+    use ark_std::{One, UniformRand, Zero};
+    use num_bigint::BigInt;
+
+    let mut rng = ark_std::test_rng();
+
+    for _ in 0..10 {
+        let num = [
+            u64::rand(&mut rng),
+            u64::rand(&mut rng),
+            u64::rand(&mut rng),
+            u64::rand(&mut rng),
+        ];
+        let relaxed_naf = find_relaxed_naf(&num);
+
+        let test = {
+            let mut sum = BigInt::zero();
+            let mut cur = BigInt::one();
+            for v in relaxed_naf {
+                sum += cur.clone() * v;
+                cur = cur * 2;
+            }
+            sum
+        };
+
+        let test_expected = {
+            let mut sum = BigInt::zero();
+            let mut cur = BigInt::one();
+            for v in num.iter() {
+                sum += cur.clone() * v;
+                cur = cur << 64;
+            }
+            sum
+        };
+
+        assert_eq!(test, test_expected);
+    }
+}

ff/src/fields/models/fp12_2over3over2.rs

@@ -59,7 +59,7 @@ impl<P: Fp12Parameters> QuadExtConfig for Fp12ParamsWrapper<P> {
         fe_inverse.conjugate();
 
         let mut found_nonzero = false;
-        let naf = crate::biginteger::arithmetic::find_wnaf(exponent.as_ref());
+        let naf = crate::biginteger::arithmetic::find_naf(exponent.as_ref());
 
         for &value in naf.iter().rev() {
             if found_nonzero {

ff/src/fields/models/quadratic_extension.rs

@@ -103,7 +103,7 @@ pub trait QuadExtConfig: 'static + Send + Sync + Sized {
         self_inverse.conjugate();
 
         let mut found_nonzero = false;
-        let naf = crate::biginteger::arithmetic::find_wnaf(exponent.as_ref());
+        let naf = crate::biginteger::arithmetic::find_naf(exponent.as_ref());
 
         for &value in naf.iter().rev() {
             if found_nonzero {