hekate_math/
algebra.rs

1// SPDX-License-Identifier: Apache-2.0
2// This file is part of the hekate-math project.
3// Copyright (C) 2026 Andrei Kochergin <andrei@oumuamua.dev>
4// Copyright (C) 2026 Oumuamua Labs <info@oumuamua.dev>.
5//
6// Licensed under the Apache License, Version 2.0 (the "License");
7// you may not use this file except in compliance with the License.
8// You may obtain a copy of the License at
9//
10//     http://www.apache.org/licenses/LICENSE-2.0
11//
12// Unless required by applicable law or agreed to in writing, software
13// distributed under the License is distributed on an "AS IS" BASIS,
14// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
15// See the License for the specific language governing permissions and
16// limitations under the License.
17
18//! `BinaryFieldExtras` (square, Frobenius, trace, Artin–Schreier
19//! solver) and the macro that implements it for Block16 to Block128.
20
21use crate::{Bit, TowerField};
22
23/// Square, Frobenius, absolute trace and the Artin–Schreier solver of
24/// a binary field; the solver builds the Cantor basis of the additive FFT.
25pub trait BinaryFieldExtras: TowerField {
26    /// Returns `self^2`; squaring is additive in characteristic 2.
27    fn square(&self) -> Self {
28        *self * *self
29    }
30
31    /// Returns `self^(2^k)`, `k` reduced mod `BITS` since
32    /// `x^(2^BITS) = x`; costs `k % BITS` squarings.
33    fn frobenius(&self, k: u32) -> Self {
34        let reps = (k % Self::BITS as u32) as usize;
35
36        let mut acc = *self;
37        for _ in 0..reps {
38            acc = acc.square();
39        }
40
41        acc
42    }
43
44    /// Returns the absolute trace Σ_{i<BITS} self^(2^i), always
45    /// 0 or 1; the tower fields compute it as a masked parity.
46    fn trace(&self) -> Bit {
47        let mut acc = Self::ZERO;
48        let mut p = *self;
49
50        for _ in 0..Self::BITS {
51            acc += p;
52            p = p.square();
53        }
54
55        Bit::new((acc == Self::ONE) as u8)
56    }
57
58    /// Returns a root of x^2 + x = c, or `None` iff Tr(c) = 1; the other
59    /// root is `result + ONE`. The fixed choice defines the Cantor basis.
60    /// The value path is constant time; only `Some`/`None` reveals Tr(c).
61    fn solve_quadratic(c: Self) -> Option<Self>;
62}
63
64macro_rules! impl_binary_field_extras {
65    ($block:ty, $map_ct:ident, $trace_mask:ident, $solve_basis:ident) => {
66        impl BinaryFieldExtras for $block {
67            #[inline(always)]
68            fn square(&self) -> Self {
69                // char 2:
70                // (lo + hi·X)^2 = lo^2 + hi^2·X^2,
71                // no cross term.
72                let (lo, hi) = self.split();
73                let hi2 = hi.square();
74
75                Self::new(lo.square() + hi2.mul_tau(), hi2)
76            }
77
78            #[inline(always)]
79            fn trace(&self) -> Bit {
80                Bit::new(((self.0 & constants::$trace_mask).count_ones() & 1) as u8)
81            }
82
83            #[inline(always)]
84            fn solve_quadratic(c: Self) -> Option<Self> {
85                if c.trace() != Bit::ZERO {
86                    return None;
87                }
88
89                Some(Self($map_ct(c.0, &constants::$solve_basis)))
90            }
91        }
92    };
93}
94
95pub(crate) use impl_binary_field_extras;
hekate_math/algebra.rs

hekate_math/
algebra.rs
