binius_field/

aes_field.rs

// Copyright 2024-2025 Irreducible Inc.
// Copyright 2026 The Binius Developers

use std::{
	fmt::{Debug, Display, Formatter},
	iter::{Product, Sum},
	marker::PhantomData,
	ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign},
};

use binius_utils::{
	DeserializeBytes, FixedSizeSerializeBytes, SerializationError, SerializeBytes,
	bytes::{Buf, BufMut},
};
use bytemuck::{Pod, Zeroable};

use super::{
	PackedExtension, PackedSubfield,
	arithmetic_traits::InvertOrZero,
	binary_field::{BinaryField, BinaryField1b, binary_field, impl_field_extension},
	mul_by_binary_field_1b,
};
use crate::{
	ExtensionField, Field, binary_field_arithmetic::impl_arithmetic_using_packed,
	linear_transformation::Transformation, underlier::U1,
};

// These fields represent a tower based on AES GF(2^8) field (GF(256)/x^8+x^4+x^3+x+1)
// that is isomorphically included into binary tower, i.e.:
//  - AESTowerField16b is GF(2^16) / (x^2 + x * x_2 + 1) where `x_2` is 0x10 from
// BinaryField8b isomorphically projected to AESTowerField8b.
//  - AESTowerField32b is GF(2^32) / (x^2 + x * x_3 + 1), where `x_3` is 0x1000 from
//    AESTowerField16b.
//  ...
binary_field!(pub AESTowerField8b(u8), 0xD0);

impl AESTowerField8b {
	pub const fn new(value: u8) -> Self {
		Self(value)
	}
}

// `WideMul` for the AES tower field is a direct log/exp-table multiply that already produces the
// reduced 8-bit element, so the wide product is `Self` and `reduce` is the identity. This is also
// the single source of truth for the portable packed AES multiply (see `AesLookupWideMul` and the
// elementwise strategy in the packed AES modules).
impl crate::arithmetic_traits::WideMul for AESTowerField8b {
	type Output = Self;

	#[inline]
	fn wide_mul(a: Self, b: Self) -> Self {
		Self::new(aes_mul_8b(a.val(), b.val()))
	}

	#[inline]
	fn reduce(wide: Self) -> Self {
		wide
	}
}

/// Multiplies two `AESTowerField8b` elements (as raw bytes) via the tower-field log/exp tables,
/// returning the reduced product. Shared by the scalar `WideMul` and the portable packed AES
/// widening multiply.
#[inline]
pub(crate) fn aes_mul_8b(lhs: u8, rhs: u8) -> u8 {
	if lhs != 0 && rhs != 0 {
		let log_index = AES_LOG_TABLE[lhs as usize] as usize + AES_LOG_TABLE[rhs as usize] as usize;
		// Each log table value is at most 254, so the sum is at most 508; subtracting 255 when it
		// exceeds 254 keeps the index in `0..=253`.
		let log_index = if log_index > 254 {
			log_index - 255
		} else {
			log_index
		};
		// Safety: `log_index <= 253 < 256` by the bound above.
		unsafe { *AES_EXP_TABLE.get_unchecked(log_index) }
	} else {
		0
	}
}

#[rustfmt::skip]
const AES_EXP_TABLE: [u8; 256] = [
	0x01, 0x03, 0x05, 0x0f, 0x11, 0x33, 0x55, 0xff, 0x1a, 0x2e, 0x72, 0x96, 0xa1, 0xf8, 0x13, 0x35,
	0x5f, 0xe1, 0x38, 0x48, 0xd8, 0x73, 0x95, 0xa4, 0xf7, 0x02, 0x06, 0x0a, 0x1e, 0x22, 0x66, 0xaa,
	0xe5, 0x34, 0x5c, 0xe4, 0x37, 0x59, 0xeb, 0x26, 0x6a, 0xbe, 0xd9, 0x70, 0x90, 0xab, 0xe6, 0x31,
	0x53, 0xf5, 0x04, 0x0c, 0x14, 0x3c, 0x44, 0xcc, 0x4f, 0xd1, 0x68, 0xb8, 0xd3, 0x6e, 0xb2, 0xcd,
	0x4c, 0xd4, 0x67, 0xa9, 0xe0, 0x3b, 0x4d, 0xd7, 0x62, 0xa6, 0xf1, 0x08, 0x18, 0x28, 0x78, 0x88,
	0x83, 0x9e, 0xb9, 0xd0, 0x6b, 0xbd, 0xdc, 0x7f, 0x81, 0x98, 0xb3, 0xce, 0x49, 0xdb, 0x76, 0x9a,
	0xb5, 0xc4, 0x57, 0xf9, 0x10, 0x30, 0x50, 0xf0, 0x0b, 0x1d, 0x27, 0x69, 0xbb, 0xd6, 0x61, 0xa3,
	0xfe, 0x19, 0x2b, 0x7d, 0x87, 0x92, 0xad, 0xec, 0x2f, 0x71, 0x93, 0xae, 0xe9, 0x20, 0x60, 0xa0,
	0xfb, 0x16, 0x3a, 0x4e, 0xd2, 0x6d, 0xb7, 0xc2, 0x5d, 0xe7, 0x32, 0x56, 0xfa, 0x15, 0x3f, 0x41,
	0xc3, 0x5e, 0xe2, 0x3d, 0x47, 0xc9, 0x40, 0xc0, 0x5b, 0xed, 0x2c, 0x74, 0x9c, 0xbf, 0xda, 0x75,
	0x9f, 0xba, 0xd5, 0x64, 0xac, 0xef, 0x2a, 0x7e, 0x82, 0x9d, 0xbc, 0xdf, 0x7a, 0x8e, 0x89, 0x80,
	0x9b, 0xb6, 0xc1, 0x58, 0xe8, 0x23, 0x65, 0xaf, 0xea, 0x25, 0x6f, 0xb1, 0xc8, 0x43, 0xc5, 0x54,
	0xfc, 0x1f, 0x21, 0x63, 0xa5, 0xf4, 0x07, 0x09, 0x1b, 0x2d, 0x77, 0x99, 0xb0, 0xcb, 0x46, 0xca,
	0x45, 0xcf, 0x4a, 0xde, 0x79, 0x8b, 0x86, 0x91, 0xa8, 0xe3, 0x3e, 0x42, 0xc6, 0x51, 0xf3, 0x0e,
	0x12, 0x36, 0x5a, 0xee, 0x29, 0x7b, 0x8d, 0x8c, 0x8f, 0x8a, 0x85, 0x94, 0xa7, 0xf2, 0x0d, 0x17,
	0x39, 0x4b, 0xdd, 0x7c, 0x84, 0x97, 0xa2, 0xfd, 0x1c, 0x24, 0x6c, 0xb4, 0xc7, 0x52, 0xf6, 0x01,
];

#[rustfmt::skip]
const AES_LOG_TABLE: [u8; 256] = [
	0x00, 0x00, 0x19, 0x01, 0x32, 0x02, 0x1a, 0xc6, 0x4b, 0xc7, 0x1b, 0x68, 0x33, 0xee, 0xdf, 0x03,
	0x64, 0x04, 0xe0, 0x0e, 0x34, 0x8d, 0x81, 0xef, 0x4c, 0x71, 0x08, 0xc8, 0xf8, 0x69, 0x1c, 0xc1,
	0x7d, 0xc2, 0x1d, 0xb5, 0xf9, 0xb9, 0x27, 0x6a, 0x4d, 0xe4, 0xa6, 0x72, 0x9a, 0xc9, 0x09, 0x78,
	0x65, 0x2f, 0x8a, 0x05, 0x21, 0x0f, 0xe1, 0x24, 0x12, 0xf0, 0x82, 0x45, 0x35, 0x93, 0xda, 0x8e,
	0x96, 0x8f, 0xdb, 0xbd, 0x36, 0xd0, 0xce, 0x94, 0x13, 0x5c, 0xd2, 0xf1, 0x40, 0x46, 0x83, 0x38,
	0x66, 0xdd, 0xfd, 0x30, 0xbf, 0x06, 0x8b, 0x62, 0xb3, 0x25, 0xe2, 0x98, 0x22, 0x88, 0x91, 0x10,
	0x7e, 0x6e, 0x48, 0xc3, 0xa3, 0xb6, 0x1e, 0x42, 0x3a, 0x6b, 0x28, 0x54, 0xfa, 0x85, 0x3d, 0xba,
	0x2b, 0x79, 0x0a, 0x15, 0x9b, 0x9f, 0x5e, 0xca, 0x4e, 0xd4, 0xac, 0xe5, 0xf3, 0x73, 0xa7, 0x57,
	0xaf, 0x58, 0xa8, 0x50, 0xf4, 0xea, 0xd6, 0x74, 0x4f, 0xae, 0xe9, 0xd5, 0xe7, 0xe6, 0xad, 0xe8,
	0x2c, 0xd7, 0x75, 0x7a, 0xeb, 0x16, 0x0b, 0xf5, 0x59, 0xcb, 0x5f, 0xb0, 0x9c, 0xa9, 0x51, 0xa0,
	0x7f, 0x0c, 0xf6, 0x6f, 0x17, 0xc4, 0x49, 0xec, 0xd8, 0x43, 0x1f, 0x2d, 0xa4, 0x76, 0x7b, 0xb7,
	0xcc, 0xbb, 0x3e, 0x5a, 0xfb, 0x60, 0xb1, 0x86, 0x3b, 0x52, 0xa1, 0x6c, 0xaa, 0x55, 0x29, 0x9d,
	0x97, 0xb2, 0x87, 0x90, 0x61, 0xbe, 0xdc, 0xfc, 0xbc, 0x95, 0xcf, 0xcd, 0x37, 0x3f, 0x5b, 0xd1,
	0x53, 0x39, 0x84, 0x3c, 0x41, 0xa2, 0x6d, 0x47, 0x14, 0x2a, 0x9e, 0x5d, 0x56, 0xf2, 0xd3, 0xab,
	0x44, 0x11, 0x92, 0xd9, 0x23, 0x20, 0x2e, 0x89, 0xb4, 0x7c, 0xb8, 0x26, 0x77, 0x99, 0xe3, 0xa5,
	0x67, 0x4a, 0xed, 0xde, 0xc5, 0x31, 0xfe, 0x18, 0x0d, 0x63, 0x8c, 0x80, 0xc0, 0xf7, 0x70, 0x07,
];

unsafe impl Pod for AESTowerField8b {}

impl_field_extension!(BinaryField1b(U1) < @3 => AESTowerField8b(u8));

mul_by_binary_field_1b!(AESTowerField8b);

impl_arithmetic_using_packed!(AESTowerField8b);

/// A 3- step transformation :
/// 1. Cast to base b-bit packed field
/// 2. Apply linear transformation between aes and binary b8 tower fields
/// 3. Cast back to the target field
pub struct SubfieldTransformer<IF, OF, T> {
	inner_transform: T,
	_ip_pd: PhantomData<IF>,
	_op_pd: PhantomData<OF>,
}

impl<IF, OF, IEP, OEP, T> Transformation<IEP, OEP> for SubfieldTransformer<IF, OF, T>
where
	IF: Field,
	OF: Field,
	IEP: PackedExtension<IF>,
	OEP: PackedExtension<OF>,
	T: Transformation<PackedSubfield<IEP, IF>, PackedSubfield<OEP, OF>>,
{
	fn transform(&self, input: &IEP) -> OEP {
		OEP::cast_ext(self.inner_transform.transform(IEP::cast_base_ref(input)))
	}
}

impl SerializeBytes for AESTowerField8b {
	fn serialize(&self, write_buf: impl BufMut) -> Result<(), SerializationError> {
		self.0.serialize(write_buf)
	}
}

impl DeserializeBytes for AESTowerField8b {
	fn deserialize(read_buf: impl Buf) -> Result<Self, SerializationError>
	where
		Self: Sized,
	{
		Ok(Self(DeserializeBytes::deserialize(read_buf)?))
	}
}

impl FixedSizeSerializeBytes for AESTowerField8b {
	const BYTE_SIZE: usize = 1;
}

#[cfg(test)]
mod tests {
	use binius_utils::{SerializeBytes, bytes::BytesMut};
	use proptest::{arbitrary::any, proptest};
	use rand::prelude::*;

	use super::*;
	use crate::{Random, binary_field::tests::is_binary_field_valid_generator};

	fn check_square(f: impl Field) {
		assert_eq!(f.square(), f * f);
	}

	proptest! {
		#[test]
		fn test_square_8(a in any::<u8>()) {
			check_square(AESTowerField8b::from(a))
		}
	}

	fn check_invert(f: impl Field) {
		let inversed = f.invert_or_zero();
		if f.is_zero() {
			assert!(inversed.is_zero());
		} else {
			assert_eq!(inversed * f, Field::ONE);
		}
	}

	proptest! {
		#[test]
		fn test_invert_8(a in any::<u8>()) {
			check_invert(AESTowerField8b::from(a))
		}
	}

	fn check_mul_by_one<F: Field>(f: F) {
		assert_eq!(F::ONE * f, f);
		assert_eq!(f * F::ONE, f);
	}

	fn check_commutative<F: Field>(f_1: F, f_2: F) {
		assert_eq!(f_1 * f_2, f_2 * f_1);
	}

	fn check_associativity_and_lineraity<F: Field>(f_1: F, f_2: F, f_3: F) {
		assert_eq!(f_1 * (f_2 * f_3), (f_1 * f_2) * f_3);
		assert_eq!(f_1 * (f_2 + f_3), f_1 * f_2 + f_1 * f_3);
	}

	fn check_mul<F: Field>(f_1: F, f_2: F, f_3: F) {
		check_mul_by_one(f_1);
		check_mul_by_one(f_2);
		check_mul_by_one(f_3);

		check_commutative(f_1, f_2);
		check_commutative(f_1, f_3);
		check_commutative(f_2, f_3);

		check_associativity_and_lineraity(f_1, f_2, f_3);
		check_associativity_and_lineraity(f_1, f_3, f_2);
		check_associativity_and_lineraity(f_2, f_1, f_3);
		check_associativity_and_lineraity(f_2, f_3, f_1);
		check_associativity_and_lineraity(f_3, f_1, f_2);
		check_associativity_and_lineraity(f_3, f_2, f_1);
	}

	proptest! {
		#[test]
		fn test_mul_8(a in any::<u8>(), b in any::<u8>(), c in any::<u8>()) {
			check_mul(AESTowerField8b::from(a), AESTowerField8b::from(b), AESTowerField8b::from(c))
		}
	}

	#[test]
	fn test_multiplicative_generators() {
		assert!(is_binary_field_valid_generator::<AESTowerField8b>());
	}

	#[test]
	fn test_serialization() {
		let mut buffer = BytesMut::new();
		let mut rng = StdRng::seed_from_u64(0);
		let aes8 = AESTowerField8b::random(&mut rng);

		SerializeBytes::serialize(&aes8, &mut buffer).unwrap();

		let mut read_buffer = buffer.freeze();

		assert_eq!(AESTowerField8b::deserialize(&mut read_buffer).unwrap(), aes8);
	}
}