rlnc 0.8.7

Random Linear Network Coding
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216

//! Following GF(2**8) logarithm and exponentiation tables are generated using
//! Python script @ <https://gist.github.com/itzmeanjan/0b2ec3f378de2c2e911bd4bb5505d45a>.

use rand::Rng;
use rand::distr::{Distribution, StandardUniform};
use std::ops::{Add, AddAssign, Div, Mul, Neg, Sub};

pub const GF256_ORDER: usize = u8::MAX as usize + 1;

#[cfg(any(target_arch = "x86", target_arch = "x86_64", target_arch = "aarch64"))]
pub const GF256_BIT_WIDTH: usize = u8::BITS as usize;

#[cfg(any(target_arch = "x86", target_arch = "x86_64", target_arch = "aarch64"))]
pub const GF256_HALF_ORDER: usize = 1usize << (GF256_BIT_WIDTH / 2);

const GF256_LOG_TABLE: [u8; GF256_ORDER] = [
    0, 0, 25, 1, 50, 2, 26, 198, 75, 199, 27, 104, 51, 238, 223, 3, 100, 4, 224, 14, 52, 141, 129, 239, 76, 113, 8, 200, 248, 105, 28, 193, 125, 194, 29, 181,
    249, 185, 39, 106, 77, 228, 166, 114, 154, 201, 9, 120, 101, 47, 138, 5, 33, 15, 225, 36, 18, 240, 130, 69, 53, 147, 218, 142, 150, 143, 219, 189, 54, 208,
    206, 148, 19, 92, 210, 241, 64, 70, 131, 56, 102, 221, 253, 48, 191, 6, 139, 98, 179, 37, 226, 152, 34, 136, 145, 16, 126, 110, 72, 195, 163, 182, 30, 66,
    58, 107, 40, 84, 250, 133, 61, 186, 43, 121, 10, 21, 155, 159, 94, 202, 78, 212, 172, 229, 243, 115, 167, 87, 175, 88, 168, 80, 244, 234, 214, 116, 79,
    174, 233, 213, 231, 230, 173, 232, 44, 215, 117, 122, 235, 22, 11, 245, 89, 203, 95, 176, 156, 169, 81, 160, 127, 12, 246, 111, 23, 196, 73, 236, 216, 67,
    31, 45, 164, 118, 123, 183, 204, 187, 62, 90, 251, 96, 177, 134, 59, 82, 161, 108, 170, 85, 41, 157, 151, 178, 135, 144, 97, 190, 220, 252, 188, 149, 207,
    205, 55, 63, 91, 209, 83, 57, 132, 60, 65, 162, 109, 71, 20, 42, 158, 93, 86, 242, 211, 171, 68, 17, 146, 217, 35, 32, 46, 137, 180, 124, 184, 38, 119,
    153, 227, 165, 103, 74, 237, 222, 197, 49, 254, 24, 13, 99, 140, 128, 192, 247, 112, 7,
];

const GF256_EXP_TABLE: [u8; 2 * GF256_ORDER - 2] = [
    1, 3, 5, 15, 17, 51, 85, 255, 26, 46, 114, 150, 161, 248, 19, 53, 95, 225, 56, 72, 216, 115, 149, 164, 247, 2, 6, 10, 30, 34, 102, 170, 229, 52, 92, 228,
    55, 89, 235, 38, 106, 190, 217, 112, 144, 171, 230, 49, 83, 245, 4, 12, 20, 60, 68, 204, 79, 209, 104, 184, 211, 110, 178, 205, 76, 212, 103, 169, 224, 59,
    77, 215, 98, 166, 241, 8, 24, 40, 120, 136, 131, 158, 185, 208, 107, 189, 220, 127, 129, 152, 179, 206, 73, 219, 118, 154, 181, 196, 87, 249, 16, 48, 80,
    240, 11, 29, 39, 105, 187, 214, 97, 163, 254, 25, 43, 125, 135, 146, 173, 236, 47, 113, 147, 174, 233, 32, 96, 160, 251, 22, 58, 78, 210, 109, 183, 194,
    93, 231, 50, 86, 250, 21, 63, 65, 195, 94, 226, 61, 71, 201, 64, 192, 91, 237, 44, 116, 156, 191, 218, 117, 159, 186, 213, 100, 172, 239, 42, 126, 130,
    157, 188, 223, 122, 142, 137, 128, 155, 182, 193, 88, 232, 35, 101, 175, 234, 37, 111, 177, 200, 67, 197, 84, 252, 31, 33, 99, 165, 244, 7, 9, 27, 45, 119,
    153, 176, 203, 70, 202, 69, 207, 74, 222, 121, 139, 134, 145, 168, 227, 62, 66, 198, 81, 243, 14, 18, 54, 90, 238, 41, 123, 141, 140, 143, 138, 133, 148,
    167, 242, 13, 23, 57, 75, 221, 124, 132, 151, 162, 253, 28, 36, 108, 180, 199, 82, 246, 1, 3, 5, 15, 17, 51, 85, 255, 26, 46, 114, 150, 161, 248, 19, 53,
    95, 225, 56, 72, 216, 115, 149, 164, 247, 2, 6, 10, 30, 34, 102, 170, 229, 52, 92, 228, 55, 89, 235, 38, 106, 190, 217, 112, 144, 171, 230, 49, 83, 245, 4,
    12, 20, 60, 68, 204, 79, 209, 104, 184, 211, 110, 178, 205, 76, 212, 103, 169, 224, 59, 77, 215, 98, 166, 241, 8, 24, 40, 120, 136, 131, 158, 185, 208,
    107, 189, 220, 127, 129, 152, 179, 206, 73, 219, 118, 154, 181, 196, 87, 249, 16, 48, 80, 240, 11, 29, 39, 105, 187, 214, 97, 163, 254, 25, 43, 125, 135,
    146, 173, 236, 47, 113, 147, 174, 233, 32, 96, 160, 251, 22, 58, 78, 210, 109, 183, 194, 93, 231, 50, 86, 250, 21, 63, 65, 195, 94, 226, 61, 71, 201, 64,
    192, 91, 237, 44, 116, 156, 191, 218, 117, 159, 186, 213, 100, 172, 239, 42, 126, 130, 157, 188, 223, 122, 142, 137, 128, 155, 182, 193, 88, 232, 35, 101,
    175, 234, 37, 111, 177, 200, 67, 197, 84, 252, 31, 33, 99, 165, 244, 7, 9, 27, 45, 119, 153, 176, 203, 70, 202, 69, 207, 74, 222, 121, 139, 134, 145, 168,
    227, 62, 66, 198, 81, 243, 14, 18, 54, 90, 238, 41, 123, 141, 140, 143, 138, 133, 148, 167, 242, 13, 23, 57, 75, 221, 124, 132, 151, 162, 253, 28, 36, 108,
    180, 199, 82, 246,
];

/// Galois Field GF(2^8) wrapper type.
///
/// This type represents elements of the Galois Field GF(2^8), which is commonly used in coding theory, cryptography, and error correction codes.
/// It supports basic arithmetic operations such as addition, subtraction, multiplication, and division.
/// The operations are defined over the finite field GF(2^8) with the irreducible polynomial x^8 + x^4 + x^3 + x + 1
/// and the primitive element x = 3.
///
/// We assign the `transparent` attribute to ensure that the Rust compiler representation of `Gf256` is the same as its underlying `u8` value,
/// providing a guarantee that it can be used interchangeably with `u8` in contexts where the underlying value is needed.
#[repr(transparent)]
#[derive(Default, Clone, Copy, Debug)]
pub struct Gf256 {
    val: u8,
}

impl Gf256 {
    /// Creates a new Gf256 element from a u8 value.
    pub const fn new(val: u8) -> Self {
        Gf256 { val }
    }

    /// Returns the raw u8 value of the Gf256 element.
    pub const fn get(&self) -> u8 {
        self.val
    }

    /// Returns the additive identity element (0).
    pub const fn zero() -> Self {
        Gf256::new(0)
    }

    /// Returns the multiplicative identity element (1).
    pub const fn one() -> Self {
        Gf256::new(1)
    }

    /// Returns primitive element x + 1, for GF(2^8) field with irreducible polynomial x^8 + x^4 + x^3 + x + 1.
    pub const fn primitive_element() -> Self {
        Gf256::new(3)
    }

    /// Compile-time executable multiplication of two bytes, over GF(2^8).
    pub const fn mul_const(a: u8, b: u8) -> u8 {
        if a == 0 || b == 0 {
            return 0;
        }

        let l = GF256_LOG_TABLE[a as usize] as usize;
        let r = GF256_LOG_TABLE[b as usize] as usize;

        GF256_EXP_TABLE[l + r]
    }

    /// Computes the multiplicative inverse of the element. Returns `None` for the zero element.
    pub const fn inv(self) -> Option<Self> {
        if self.val == 0 {
            return None;
        }

        Some(Gf256 {
            val: GF256_EXP_TABLE[(GF256_ORDER - 1) - GF256_LOG_TABLE[self.val as usize] as usize],
        })
    }
}

impl Add for Gf256 {
    type Output = Self;

    /// Performs addition (XOR) of two Gf256 elements.
    #[allow(clippy::suspicious_arithmetic_impl)]
    fn add(self, rhs: Self) -> Self::Output {
        Gf256 { val: self.val ^ rhs.val }
    }
}

impl AddAssign for Gf256 {
    /// Performs in-place addition i.e. compound addition operation (XOR) of two Gf256 elements.
    #[allow(clippy::suspicious_op_assign_impl)]
    fn add_assign(&mut self, rhs: Self) {
        self.val ^= rhs.val;
    }
}

impl Neg for Gf256 {
    type Output = Self;

    /// Computes the additive inverse (itself, as XOR is self-inverse).
    fn neg(self) -> Self::Output {
        Gf256 { val: self.val }
    }
}

impl Sub for Gf256 {
    type Output = Self;

    /// Performs subtraction (XOR) of two Gf256 elements.
    #[allow(clippy::suspicious_arithmetic_impl)]
    fn sub(self, rhs: Self) -> Self::Output {
        Gf256 { val: self.val ^ rhs.val }
    }
}

impl Mul for Gf256 {
    type Output = Self;

    /// Performs multiplication of two Gf256 elements using logarithm and exponentiation tables.
    fn mul(self, rhs: Self) -> Self::Output {
        Gf256 {
            val: Self::mul_const(self.val, rhs.val),
        }
    }
}

impl Div for Gf256 {
    type Output = Option<Self>;

    /// Performs division of two Gf256 elements using multiplicative inverse. Returns `None` if dividing by zero.
    #[allow(clippy::suspicious_arithmetic_impl)]
    fn div(self, rhs: Self) -> Self::Output {
        rhs.inv().map(|rhs_inv| self * rhs_inv)
    }
}

impl PartialEq for Gf256 {
    /// Checks for equality between two Gf256 elements.
    fn eq(&self, other: &Self) -> bool {
        self.val == other.val
    }
}

impl Distribution<Gf256> for StandardUniform {
    /// Samples a random Gf256 element.
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Gf256 {
        Gf256 { val: rng.random() }
    }
}

#[cfg(test)]
mod test {
    use super::Gf256;
    use rand::Rng;

    #[test]
    fn prop_test_gf256_operations() {
        const NUM_TEST_ITERATIONS: usize = 100_000;

        let mut rng = rand::rng();

        (0..NUM_TEST_ITERATIONS).for_each(|_| {
            let a: Gf256 = rng.random();
            let b: Gf256 = rng.random();

            // Addition, Subtraction, Negation
            let sum = a + b;
            let diff = sum - b;

            assert_eq!(diff, a);

            // Multiplication, Division, Inversion
            let mul = a * b;
            let div = mul / b;

            if b == Gf256::zero() {
                assert_eq!(div, None);
                assert_eq!(mul, Gf256::zero());
            } else {
                assert_eq!(a, div.unwrap());
            }
        });
    }
}