Implement out-of-circuit repeated_frobenius using hardcoded constants

src/field/crandall_field.rs

@@ -6,7 +6,7 @@ use num::Integer;
 
 use crate::field::extension_field::quadratic::QuadraticCrandallField;
 use crate::field::extension_field::quartic::QuarticCrandallField;
-use crate::field::extension_field::Extendable;
+use crate::field::extension_field::{Extendable, Frobeniable};
 use crate::field::field::Field;
 use std::hash::{Hash, Hasher};
 use std::iter::{Product, Sum};
@@ -444,6 +444,17 @@ fn split(x: u128) -> (u64, u64) {
     (x as u64, (x >> 64) as u64)
 }
 
+impl Frobeniable for CrandallField {
+    const FROBENIUS_CONSTANTS_2: [u64; 1] = [9223372035646816256];
+    /// Placeholder since there's no OEF of degree 3 for `CrandallField`.
+    const FROBENIUS_CONSTANTS_3: [u64; 2] = [0, 0];
+    const FROBENIUS_CONSTANTS_4: [u64; 3] = [
+        4611686017823408128,
+        9223372035646816256,
+        13835058053470224384,
+    ];
+}
+
 #[cfg(test)]
 mod tests {
     use crate::test_arithmetic;

src/field/extension_field/mod.rs

@@ -13,12 +13,32 @@ pub mod target;
 pub trait OEF<const D: usize>: FieldExtension<D> {
     // Element W of BaseField, such that `X^d - W` is irreducible over BaseField.
     const W: Self::BaseField;
+}
 
-    /// Frobenius automorphisms: x -> x^p, where p is the order of BaseField.
+impl<F: Field> OEF<1> for F {
+    const W: Self::BaseField = F::ZERO;
+}
+
+pub trait Frobenius<BF: Frobeniable, const D: usize>: OEF<D, BaseField = BF> {
+    /// FrobeniusField automorphisms: x -> x^p, where p is the order of BaseField.
     fn frobenius(&self) -> Self {
+        self.repeated_frobenius(1)
+    }
+
+    /// Repeated Frobenius automorphisms: x -> x^(p^k).
+    fn repeated_frobenius(&self, k: usize) -> Self {
+        if k == 0 {
+            return *self;
+        } else if k >= D {
+            return self.repeated_frobenius(k % D);
+        }
         let arr = self.to_basefield_array();
-        let k = (Self::BaseField::ORDER - 1) / (D as u64);
-        let z0 = Self::W.exp(k);
+        let z0 = match D {
+            2 => Self::W.exp(BF::FROBENIUS_CONSTANTS_2[k - 1]),
+            3 => Self::W.exp(BF::FROBENIUS_CONSTANTS_3[k - 1]),
+            4 => Self::W.exp(BF::FROBENIUS_CONSTANTS_4[k - 1]),
+            _ => unimplemented!("Only extensions of degree 2, 3, or 4 are allowed for now."),
+        };
         let mut z = Self::BaseField::ONE;
         let mut res = [Self::BaseField::ZERO; D];
         for i in 0..D {
@@ -28,25 +48,30 @@ pub trait OEF<const D: usize>: FieldExtension<D> {
 
         Self::from_basefield_array(res)
     }
+}
 
-    /// Repeated Frobenius automorphisms: x -> x^(p^k).
-    // TODO: Implement this. Is basically the same as `frobenius` above, but using
-    // `z = W^floor(j*p^k/D)`. I'm not sure there is a closed form for these so
-    // might require to hardcode them.
-    fn repeated_frobenius(&self, k: usize) -> Self {
-        todo!()
+impl<F: Frobeniable> Frobenius<F, 1> for F {
+    fn frobenius(&self) -> Self {
+        *self
+    }
+    fn repeated_frobenius(&self, _k: usize) -> Self {
+        *self
     }
 }
 
-impl<F: Field> OEF<1> for F {
-    const W: Self::BaseField = F::ZERO;
+/// Trait to hardcode constants used in the Frobenius automorphism.
+pub trait Frobeniable: Field {
+    //! `FROBENIUS_CONSTANTS_D[i-1] = floor( p^i / D) mod p-1`
+    const FROBENIUS_CONSTANTS_2: [u64; 1];
+    const FROBENIUS_CONSTANTS_3: [u64; 2];
+    const FROBENIUS_CONSTANTS_4: [u64; 3];
 }
 
-pub trait Extendable<const D: usize>: Field + Sized {
-    type Extension: Field + OEF<D, BaseField = Self> + From<Self>;
+pub trait Extendable<const D: usize>: Frobeniable + Sized {
+    type Extension: Field + OEF<D, BaseField = Self> + Frobenius<Self, D> + From<Self>;
 }
 
-impl<F: Field> Extendable<1> for F {
+impl<F: Frobeniable + Frobenius<F, 1>> Extendable<1> for F {
     type Extension = F;
 }
 

src/field/extension_field/quadratic.rs

@@ -1,5 +1,5 @@
 use crate::field::crandall_field::CrandallField;
-use crate::field::extension_field::{FieldExtension, OEF};
+use crate::field::extension_field::{FieldExtension, Frobenius, OEF};
 use crate::field::field::Field;
 use rand::Rng;
 use std::fmt::{Debug, Display, Formatter};
@@ -16,6 +16,8 @@ impl OEF<2> for QuadraticCrandallField {
     const W: CrandallField = CrandallField(3);
 }
 
+impl Frobenius<CrandallField, 2> for QuadraticCrandallField {}
+
 impl FieldExtension<2> for QuadraticCrandallField {
     type BaseField = CrandallField;
 
@@ -63,7 +65,7 @@ impl Field for QuadraticCrandallField {
             return None;
         }
 
-        let a_pow_r_minus_1 = OEF::<2>::frobenius(self);
+        let a_pow_r_minus_1 = self.frobenius();
         let a_pow_r = a_pow_r_minus_1 * *self;
         debug_assert!(FieldExtension::<2>::is_in_basefield(&a_pow_r));
 
@@ -190,7 +192,7 @@ impl DivAssign for QuadraticCrandallField {
 #[cfg(test)]
 mod tests {
     use crate::field::extension_field::quadratic::QuadraticCrandallField;
-    use crate::field::extension_field::{FieldExtension, OEF};
+    use crate::field::extension_field::{FieldExtension, Frobenius, OEF};
     use crate::field::field::Field;
 
     #[test]
@@ -231,7 +233,7 @@ mod tests {
         let x = F::rand();
         assert_eq!(
             x.exp(<F as FieldExtension<2>>::BaseField::ORDER),
-            OEF::<2>::frobenius(&x)
+            x.frobenius()
         );
     }
 

src/field/extension_field/quartic.rs

@@ -6,7 +6,7 @@ use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssi
 use rand::Rng;
 
 use crate::field::crandall_field::CrandallField;
-use crate::field::extension_field::{FieldExtension, OEF};
+use crate::field::extension_field::{FieldExtension, Frobenius, OEF};
 use crate::field::field::Field;
 
 /// A quartic extension of `CrandallField`.
@@ -20,6 +20,8 @@ impl OEF<4> for QuarticCrandallField {
     const W: CrandallField = CrandallField(3);
 }
 
+impl Frobenius<CrandallField, 4> for QuarticCrandallField {}
+
 impl FieldExtension<4> for QuarticCrandallField {
     type BaseField = CrandallField;
 
@@ -93,9 +95,9 @@ impl Field for QuarticCrandallField {
             return None;
         }
 
-        let a_pow_p = OEF::<4>::frobenius(self);
+        let a_pow_p = self.frobenius();
         let a_pow_p_plus_1 = a_pow_p * *self;
-        let a_pow_p3_plus_p2 = OEF::<4>::frobenius(&OEF::<4>::frobenius(&a_pow_p_plus_1));
+        let a_pow_p3_plus_p2 = a_pow_p_plus_1.repeated_frobenius(2);
         let a_pow_r_minus_1 = a_pow_p3_plus_p2 * a_pow_p;
         let a_pow_r = a_pow_r_minus_1 * *self;
         debug_assert!(FieldExtension::<4>::is_in_basefield(&a_pow_r));
@@ -241,7 +243,7 @@ impl DivAssign for QuarticCrandallField {
 #[cfg(test)]
 mod tests {
     use crate::field::extension_field::quartic::QuarticCrandallField;
-    use crate::field::extension_field::{FieldExtension, OEF};
+    use crate::field::extension_field::{FieldExtension, Frobenius, OEF};
     use crate::field::field::Field;
 
     fn exp_naive<F: Field>(x: F, power: u128) -> F {
@@ -295,7 +297,12 @@ mod tests {
         let x = F::rand();
         assert_eq!(
             exp_naive(x, <F as FieldExtension<4>>::BaseField::ORDER as u128),
-            OEF::<4>::frobenius(&x)
+            x.frobenius()
+        );
+        assert_eq!(x.repeated_frobenius(2), x.frobenius().frobenius());
+        assert_eq!(
+            x.repeated_frobenius(3),
+            x.frobenius().frobenius().frobenius()
         );
     }
 

src/field/field.rs

@@ -7,7 +7,7 @@ use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssi
 use num::Integer;
 use rand::Rng;
 
-use crate::field::extension_field::OEF;
+use crate::field::extension_field::{Extendable, FieldExtension, Frobeniable, Frobenius, OEF};
 use crate::util::bits_u64;
 
 /// A finite field with prime order less than 2^64.
@@ -287,15 +287,14 @@ impl<F: Field> Iterator for Powers<F> {
 
 impl<F: Field> Powers<F> {
     /// Apply the Frobenius automorphism `k` times.
-    // TODO: Use `OEF::repeated_frobenius` when it is implemented.
-    pub fn repeated_frobenius<const D: usize>(self, k: usize) -> Self
+    pub fn repeated_frobenius<BF: Frobeniable, const D: usize>(self, k: usize) -> Self
     where
-        F: OEF<D>,
+        F: Frobenius<BF, D>,
     {
         let Self { base, current } = self;
         Self {
-            base: (0..k).fold(base, |acc, _| acc.frobenius()),
-            current: (0..k).fold(current, |acc, _| acc.frobenius()),
+            base: base.repeated_frobenius(k),
+            current: base.repeated_frobenius(k),
         }
     }
 }

src/fri/verifier.rs

@@ -1,7 +1,7 @@
 use anyhow::{ensure, Result};
 use itertools::izip;
 
-use crate::field::extension_field::{flatten, Extendable, FieldExtension, OEF};
+use crate::field::extension_field::{flatten, Extendable, FieldExtension, Frobenius, OEF};
 use crate::field::field::Field;
 use crate::field::lagrange::{barycentric_weights, interpolant, interpolate};
 use crate::fri::FriConfig;

src/polynomial/commitment.rs

@@ -1,8 +1,8 @@
 use anyhow::Result;
 use rayon::prelude::*;
 
-use crate::field::extension_field::FieldExtension;
 use crate::field::extension_field::{Extendable, OEF};
+use crate::field::extension_field::{FieldExtension, Frobenius};
 use crate::field::field::Field;
 use crate::field::lagrange::interpolant;
 use crate::fri::{prover::fri_proof, verifier::verify_fri_proof, FriConfig};