[RFC] Convert identity functions in Field, Group, and {Projective,Affine}Curve traits with One/Zero traits from num_traits.

- contributes to #50,
- depends on #53 and builds on it,
- due to coherence & requirements of `num_traits::{Zero, One}` to implement `std::ops::Add<Self, ..>` and (resp.) `std::ops::Mul<Self, ..>`, I've had to replace the afferent `impl<'a, P: ..> (Add|Mul)<&'a Self> for Group(Affine|Projective)<P>` by direct implementations on `Self`,
- I did not have to fight the borrow checker for this conversion => I think this hints arithmetic operations are called in contexts where the operand is owned,
- hence should this end up on a merge track, we may want to open an issue to convert the `impl<'a, P:..> (Neg|Sub|..)<&'a Self> for ..<P>` trait usage to direct `impl<P:..> (Neg|Sub|..)<Self> for ..<P>`
- the `impl AddAssign for GroupAffine<P>` in curves/models/short_weierstrass_jacobian.rs is provided to fit trait bounds, and without any guarantee of suitability for any particular purpose
- and that, even though I don't think it's used.

crypto-primitives/Cargo.toml

@@ -32,6 +32,7 @@ bench-utils = { path = "../bench-utils" }
 digest = "0.7"
 blake2 = "0.7"
 
+num-traits = { version = "0.2.11" }
 rand = { version = "0.7" }
 derivative = "1"
 rayon = "1"

crypto-primitives/src/merkle_tree/mod.rs

@@ -372,6 +372,7 @@ mod test {
         merkle_tree::*,
     };
     use algebra::curves::jubjub::JubJubAffine as JubJub;
+    use num_traits::Zero;
     use rand::SeedableRng;
     use rand_xorshift::XorShiftRng;
 
@@ -419,7 +420,6 @@ mod test {
     }
 
     fn bad_merkle_tree_verify<L: ToBytes + Clone + Eq>(leaves: &[L]) -> () {
-        use algebra::groups::Group;
         let mut rng = XorShiftRng::seed_from_u64(13423423u64);
 
         let crh_parameters = Rc::new(H::setup(&mut rng).unwrap());

crypto-primitives/src/nizk/mod.rs

@@ -59,7 +59,8 @@ mod test {
     #[test]
     fn test_gm17() {
         use crate::nizk::{gm17::Gm17, NIZK};
-        use algebra::{curves::bls12_381::Bls12_381, fields::bls12_381::Fr, Field};
+        use algebra::{curves::bls12_381::Bls12_381, fields::bls12_381::Fr};
+        use num_traits::One;
         use r1cs_core::{ConstraintSynthesizer, ConstraintSystem, SynthesisError};
 
         #[derive(Copy, Clone)]

crypto-primitives/src/signature/schnorr/mod.rs

@@ -6,6 +6,7 @@ use algebra::{
     to_bytes, ToConstraintField, UniformRand,
 };
 use digest::Digest;
+use num_traits::{One, Zero};
 use rand::Rng;
 use std::{
     hash::Hash,

r1cs-std/Cargo.toml

@@ -25,6 +25,7 @@ edition = "2018"
 algebra = { path = "../algebra" }
 r1cs-core = { path = "../r1cs-core" }
 derivative = "1"
+num-traits = { version = "0.2.11" }
 radix_trie = "0.1"
 
 [dev-dependencies]

r1cs-std/src/bits/boolean.rs

@@ -832,13 +832,12 @@ impl CondSelectGadget for Boolean {
     }
 }
 
-    
-
 #[cfg(test)]
 mod test {
     use super::{AllocatedBit, Boolean};
     use crate::{prelude::*, test_constraint_system::TestConstraintSystem};
     use algebra::{fields::bls12_381::Fr, BitIterator, Field, PrimeField, UniformRand};
+    use num_traits::{One, Zero};
     use r1cs_core::ConstraintSystem;
     use rand::SeedableRng;
     use rand_xorshift::XorShiftRng;
@@ -903,8 +902,8 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val | *b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
             }
         }
     }
@@ -920,14 +919,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val & *b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("and result")
                         == if *a_val & *b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -935,9 +934,9 @@ mod test {
                 cs.set(
                     "and result",
                     if *a_val & *b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -956,14 +955,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val & !*b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("and not result")
                         == if *a_val & !*b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -971,9 +970,9 @@ mod test {
                 cs.set(
                     "and not result",
                     if *a_val & !*b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -992,14 +991,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), !*a_val & !*b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("nor result")
                         == if !*a_val & !*b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -1007,9 +1006,9 @@ mod test {
                 cs.set(
                     "nor result",
                     if !*a_val & !*b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -1235,7 +1234,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1243,7 +1242,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1251,7 +1250,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1259,7 +1258,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1270,7 +1269,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1278,7 +1277,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1286,7 +1285,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1294,7 +1293,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1305,7 +1304,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1313,7 +1312,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1321,7 +1320,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1329,7 +1328,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1340,7 +1339,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1348,7 +1347,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1356,7 +1355,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1364,7 +1363,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1732,7 +1731,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::one());
+                        assert!(cs.get("and result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1740,7 +1739,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1748,7 +1747,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1756,7 +1755,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::one());
+                        assert!(cs.get("and not result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1768,7 +1767,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1776,7 +1775,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1784,7 +1783,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1792,7 +1791,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1807,7 +1806,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1815,7 +1814,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1823,7 +1822,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1831,7 +1830,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1846,7 +1845,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::one());
+                        assert!(cs.get("and not result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1854,7 +1853,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1862,7 +1861,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1870,7 +1869,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::one());
+                        assert!(cs.get("nor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 

r1cs-std/src/bits/uint32.rs

@@ -344,7 +344,8 @@ impl ConditionalEqGadget for UInt32 {
 mod test {
     use super::UInt32;
     use crate::{bits::boolean::Boolean, test_constraint_system::TestConstraintSystem};
-    use algebra::fields::{bls12_381::Fr, Field};
+    use algebra::fields::bls12_381::Fr;
+    use num_traits::{One, Zero};
     use r1cs_core::ConstraintSystem;
     use rand::{Rng, SeedableRng};
     use rand_xorshift::XorShiftRng;
@@ -500,9 +501,9 @@ mod test {
 
             // Flip a bit_gadget and see if the addition constraint still works
             if cs.get("addition/result bit_gadget 0/boolean").is_zero() {
-                cs.set("addition/result bit_gadget 0/boolean", Field::one());
+                cs.set("addition/result bit_gadget 0/boolean", Fr::one());
             } else {
-                cs.set("addition/result bit_gadget 0/boolean", Field::zero());
+                cs.set("addition/result bit_gadget 0/boolean", Fr::zero());
             }
 
             assert!(!cs.is_satisfied());

r1cs-std/src/fields/fp12.rs

@@ -8,6 +8,7 @@ use algebra::{
     },
     BitIterator, Field, PrimeField,
 };
+use num_traits::One;
 use std::{borrow::Borrow, marker::PhantomData};
 
 use crate::{prelude::*, Assignment};

r1cs-std/src/groups/curves/short_weierstrass/bls12/mod.rs

@@ -3,6 +3,7 @@ use algebra::{
     fields::Field,
     BitIterator, ProjectiveCurve,
 };
+use num_traits::One;
 use r1cs_core::{ConstraintSystem, SynthesisError};
 
 use crate::{

r1cs-std/src/groups/curves/short_weierstrass/mod.rs

@@ -5,6 +5,7 @@ use algebra::{
     },
     AffineCurve, BitIterator, Field, PrimeField, ProjectiveCurve,
 };
+use num_traits::{One, Zero};
 use r1cs_core::{ConstraintSystem, SynthesisError};
 use std::{borrow::Borrow, marker::PhantomData, ops::Neg};
 

r1cs-std/src/groups/curves/twisted_edwards/mod.rs

@@ -5,6 +5,7 @@ use algebra::{
     },
     BitIterator, Field,
 };
+use num_traits::{One, Zero};
 
 use r1cs_core::{ConstraintSystem, SynthesisError};
 
@@ -38,7 +39,7 @@ pub struct MontgomeryAffineGadget<
 mod montgomery_affine_impl {
     use super::*;
     use crate::Assignment;
-    use algebra::{twisted_edwards_extended::GroupAffine, AffineCurve, Field};
+    use algebra::{twisted_edwards_extended::GroupAffine, Field};
     use std::ops::{AddAssign, MulAssign, SubAssign};
 
     impl<P: TEModelParameters, ConstraintF: Field, F: FieldGadget<P::BaseField, ConstraintF>>

r1cs-std/src/pairing/mod.rs

@@ -60,6 +60,7 @@ mod test {
     // use rand;
     use crate::test_constraint_system::TestConstraintSystem;
     use algebra::{BitIterator, Field};
+    use num_traits::One;
     use r1cs_core::ConstraintSystem;
 
     #[test]