Implement u256 prime field (#137)

* fix mul_two_256_bit_integers_works_4() test

Test mul_two_256_bit_integers_works_4() fixed after the bug  in the * operator patched

* Update element.rs

* Update element.rs

* get rid of space

* Update element.rs

* Generalize IsMontgomeryConfiguration and MontgomeryBackendPrimeField

Solves #126

* Extended U256 Montgomery backed prime fields test

* Implemented U256PrimeField

solves #95

* Conflicts fixed

* fmt

* Get rid of unused commented lines

* Update montgomery_backed_prime_fields.rs

Author: GianfrancoBazzani

crypto/src/hash/poseidon/mod.rs

@@ -147,20 +147,22 @@ where
 #[cfg(test)]
 mod tests {
     use lambdaworks_math::{
-        field::fields::u384_prime_field::{IsMontgomeryConfiguration, MontgomeryBackendPrimeField},
+        field::fields::montgomery_backed_prime_fields::{
+            IsMontgomeryConfiguration, U384PrimeField,
+        },
         unsigned_integer::element::U384,
     };
 
     use super::*;
 
     #[derive(Clone, Debug)]
     pub struct TestFieldConfig;
-    impl IsMontgomeryConfiguration for TestFieldConfig {
+    impl IsMontgomeryConfiguration<6> for TestFieldConfig {
         const MODULUS: U384 =
             U384::from("2000000000000080000000000000000000000000000000000000000000000001");
     }
 
-    pub type PoseidonTestField = MontgomeryBackendPrimeField<TestFieldConfig>;
+    pub type PoseidonTestField = U384PrimeField<TestFieldConfig>;
     type TestFieldElement = FieldElement<PoseidonTestField>;
 
     pub fn load_test_parameters() -> Result<Parameters<PoseidonTestField>, String> {

math/src/elliptic_curve/short_weierstrass/curves/bls12_377/field_extension.rs

@@ -1,6 +1,8 @@
 use crate::field::{
     element::FieldElement,
-    fields::u384_prime_field::{IsMontgomeryConfiguration, MontgomeryBackendPrimeField},
+    fields::montgomery_backed_prime_fields::{
+        IsMontgomeryConfiguration, MontgomeryBackendPrimeField,
+    },
 };
 use crate::unsigned_integer::element::U384;
 
@@ -9,11 +11,11 @@ pub const BLS12377_PRIME_FIELD_ORDER: U384 = U384::from("1ae3a4617c510eac63b05c0
 // FPBLS12377
 #[derive(Clone, Debug)]
 pub struct BLS12377FieldConfig;
-impl IsMontgomeryConfiguration for BLS12377FieldConfig {
+impl IsMontgomeryConfiguration<6> for BLS12377FieldConfig {
     const MODULUS: U384 = BLS12377_PRIME_FIELD_ORDER;
 }
 
-pub type BLS12377PrimeField = MontgomeryBackendPrimeField<BLS12377FieldConfig>;
+pub type BLS12377PrimeField = MontgomeryBackendPrimeField<BLS12377FieldConfig, 6>;
 
 impl FieldElement<BLS12377PrimeField> {
     pub fn new_base(a_hex: &str) -> Self {

math/src/elliptic_curve/short_weierstrass/curves/bls12_381/field_extension.rs

@@ -4,7 +4,9 @@ use crate::field::{
         cubic::{CubicExtensionField, HasCubicNonResidue},
         quadratic::{HasQuadraticNonResidue, QuadraticExtensionField},
     },
-    fields::u384_prime_field::{IsMontgomeryConfiguration, MontgomeryBackendPrimeField},
+    fields::montgomery_backed_prime_fields::{
+        IsMontgomeryConfiguration, MontgomeryBackendPrimeField,
+    },
 };
 use crate::unsigned_integer::element::U384;
 
@@ -13,11 +15,11 @@ pub const BLS12381_PRIME_FIELD_ORDER: U384 = U384::from("1a0111ea397fe69a4b1ba7b
 // FPBLS12381
 #[derive(Clone, Debug)]
 pub struct BLS12381FieldConfig;
-impl IsMontgomeryConfiguration for BLS12381FieldConfig {
+impl IsMontgomeryConfiguration<6> for BLS12381FieldConfig {
     const MODULUS: U384 = BLS12381_PRIME_FIELD_ORDER;
 }
 
-pub type BLS12381PrimeField = MontgomeryBackendPrimeField<BLS12381FieldConfig>;
+pub type BLS12381PrimeField = MontgomeryBackendPrimeField<BLS12381FieldConfig, 6>;
 
 #[derive(Debug, Clone)]
 pub struct LevelOneResidue;

math/src/elliptic_curve/short_weierstrass/curves/test_curve_2.rs

@@ -1,6 +1,6 @@
 use crate::elliptic_curve::short_weierstrass::point::ShortWeierstrassProjectivePoint;
 use crate::elliptic_curve::traits::IsEllipticCurve;
-use crate::field::fields::u384_prime_field::{
+use crate::field::fields::montgomery_backed_prime_fields::{
     IsMontgomeryConfiguration, MontgomeryBackendPrimeField,
 };
 use crate::unsigned_integer::element::U384;
@@ -21,11 +21,11 @@ pub const TEST_CURVE_2_MAIN_SUBGROUP_ORDER: U384 = U384::from("40a065fb5a76390de
 // FPBLS12381
 #[derive(Clone, Debug)]
 pub struct TestCurve2MontgomeryConfig;
-impl IsMontgomeryConfiguration for TestCurve2MontgomeryConfig {
+impl IsMontgomeryConfiguration<6> for TestCurve2MontgomeryConfig {
     const MODULUS: U384 = TEST_CURVE_2_PRIME_FIELD_ORDER;
 }
 
-type TestCurve2PrimeField = MontgomeryBackendPrimeField<TestCurve2MontgomeryConfig>;
+type TestCurve2PrimeField = MontgomeryBackendPrimeField<TestCurve2MontgomeryConfig, 6>;
 
 /// In F59 the element -1 is not a square. We use this property
 /// to construct a Quadratic Field Extension out of it by adding

math/src/field/fields/mod.rs

@@ -1,3 +1,3 @@
-pub mod u384_prime_field;
+pub mod montgomery_backed_prime_fields;
 /// Implementation of prime fields over 64 bit unsigned integers.
 pub mod u64_prime_field;