Reed-Solomon Error Correction

This Julia code demonstrates an implementation of the Reed-Solomon error correction algorithm, showcasing the power and flexibility of Julia's ecosystem in cryptographic applications. The implementation leverages the custom finite field arithmetic provided by CryptoGroups, specifically using a Galois Field GF(2^8) constructed from as an instance of generic polynomial extension field, and seamlessly integrates it with the external Polynomials package for polynomial operations.

The code illustrates versatility of CryptoGroups package also for low-level finite field operations with high-level polynomial manipulations. It implements key components of the Reed-Solomon algorithm, including encoding, syndrome calculation, the Berlekamp-Massey algorithm for error locator polynomial generation, and error correction.

HELP WANTED: Currently the implementation has bugs and can't even get correct error positions; To proceed one could write a tests for berklekamp_massey and when it works debug error_positions.

using Test
using CryptoGroups.Fields
using Polynomials

const GF256 = @F2PB{X^8 + X^4 + X^3 + X + 1}

roots(p) = GF256[i for i in 0:255 if iszero(p(GF256(i)))]

function berklekamp_massey(syndromes::Vector{GF256})
    B = Polynomial{GF256}([1])
    C = Polynomial{GF256}([1])

    L, m = 0, 1
    for n in 1:length(syndromes)
        # Adjust for zero-based indexing in Polynomial
        d = syndromes[n] + sum(C[i-1] * syndromes[n - i + 1] for i in 1:L; init = zero(GF256))
        if iszero(d)
            m += 1
        elseif 2*L <= n
            T = C
            # Use zero-based indexing for polynomial multiplication
            C -= d * B * Polynomial{GF256}([0, 1])^(m-1)
            L = n + 1 - L
            B = T
            m = 1
        else
            # Use zero-based indexing for polynomial multiplication
            C -= d * B * Polynomial{GF256}([0, 1])^(m-1)
            m += 1
        end
    end

    return C
end

function rs_encode(message::Vector{UInt8}, g::Polynomial{GF256})

    m = Polynomial{GF256}(message)

    r_poly = m % g
    r_octets = r_poly[:] .|> (x -> octet(x)[1])

    return [message..., r_octets...]
end

function rs_decode(received::Vector{UInt8}, n::Int, k::Int, g::Polynomial{GF256})

    received_poly = Polynomial{GF256}(received)

    syndromes = [received_poly(r) for r in roots(g)]

    if all(iszero(s) for s in syndromes)
        return received[1:k] ## no errors detected
    end

    error_locator = berklekamp_massey(syndromes)

    error_positions = [i for i in 1:n if iszero(error_locator(GF256(2)^i))]
    # @show error_positions

    error_evaluator = Polynomial{GF256}(syndromes) * error_locator % Polynomial{GF256}([zeros(Int, n - k)..., 1])

    errors = zeros(GF256, n)

    for i in error_positions
        x = GF256(2) ^ (n - i)
        y = error_evaluator(x) / derivative(error_locator)(x)
        errors[i] = y
    end

    corrected = GF256[received...] .- errors

    return corrected[1:k] .|> (x -> octet(x)[1])
end

n, k = 15, 11
g = Polynomial{GF256}([1, 25, 6, 8, 14])

message = rand(UInt8, k)
encoded = rs_encode(message, g)

errors = zeros(UInt8, n)
errors[3] = rand(0:255)
errors[7] = rand(0:255)
received = encoded + errors

decoded = rs_decode(received, n, k, g)

# @test message == decoded

11-element Vector{UInt8}:
 0x67
 0x70
 0x91
 0xe7
 0x6e
 0xf6
 0x9d
 0xdb
 0x37
 0x71
 0x09

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