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Aaru/Aaru.Checksums/ReedSolomon.cs

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// /***************************************************************************
// Aaru Data Preservation Suite
// ----------------------------------------------------------------------------
//
// Filename : ReedSolomon.cs
// Author(s) : Natalia Portillo <claunia@claunia.com>
//
// Component : Checksums.
//
// --[ Description ] ----------------------------------------------------------
//
// Calculates a Reed-Solomon.
//
// --[ License ] --------------------------------------------------------------
//
// This program is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as
// published by the Free Software Foundation, either version 3 of the
// License, or (at your option) any later version.
//
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with this program. If not, see <http://www.gnu.org/licenses/>.
//
// ----------------------------------------------------------------------------
// Copyright © 2011-2025 Natalia Portillo
// Copyright (C) 1996 Phil Karn
// Copyright (C) 1995 Robert Morelos-Zaragoza
// Copyright (C) 1995 Hari Thirumoorthy
// ****************************************************************************/
/*
* Reed-Solomon coding and decoding
* Phil Karn (karn at ka9q.ampr.org) September 1996
*
* This file is derived from the program "new_rs_erasures.c" by Robert
* Morelos-Zaragoza (robert at spectra.eng.hawaii.edu) and Hari Thirumoorthy
* (harit at spectra.eng.hawaii.edu), Aug 1995
*
* I've made changes to improve performance, clean up the code and make it
* easier to follow. Data is now passed to the encoding and decoding functions
* through arguments rather than in global arrays. The decode function returns
* the number of corrected symbols, or -1 if the word is uncorrectable.
*
* This code supports a symbol size from 2 bits up to 16 bits,
* implying a block size of 3 2-bit symbols (6 bits) up to 65535
* 16-bit symbols (1,048,560 bits). The code parameters are set in rs.h.
*
* Note that if symbols larger than 8 bits are used, the type of each
* data array element switches from unsigned char to unsigned int. The
* caller must ensure that elements larger than the symbol range are
* not passed to the encoder or decoder.
*
*/
using System;
using System.Diagnostics.CodeAnalysis;
using Aaru.Logging;
namespace Aaru.Checksums;
/// <summary>Implements the Reed-Solomon algorithm</summary>
[SuppressMessage("ReSharper", "UnusedMember.Global")]
[SuppressMessage("ReSharper", "UnusedType.Global")]
public class ReedSolomon
{
/// <summary>Alpha exponent for the first root of the generator polynomial</summary>
const int B0 = 1;
const string MODULE_NAME = "Reed Solomon";
/// <summary>No legal value in index form represents zero, so we need a special value for this purpose</summary>
int _a0;
/// <summary>index->polynomial form conversion table</summary>
int[] _alphaTo;
/// <summary>Generator polynomial g(x) Degree of g(x) = 2*TT has roots @**B0, @**(B0+1), ... ,@^(B0+2*TT-1)</summary>
int[] _gg;
/// <summary>Polynomial->index form conversion table</summary>
int[] _indexOf;
bool _initialized;
int _mm, _kk, _nn;
/// <summary>
/// Primitive polynomials - see Lin & Costello, Error Control Coding Appendix A, and Lee & Messerschmitt, Digital
/// Communication p. 453.
/// </summary>
int[] _pp;
/// <summary>Initializes the Reed-Solomon with RS(n,k) with GF(2^m)</summary>
public void InitRs(int n, int k, int m)
{
_pp = m switch
{
2 => [1, 1, 1],
3 => [1, 1, 0, 1],
4 => [1, 1, 0, 0, 1],
5 => [1, 0, 1, 0, 0, 1],
6 => [1, 1, 0, 0, 0, 0, 1],
7 => [1, 0, 0, 1, 0, 0, 0, 1],
8 => [1, 0, 1, 1, 1, 0, 0, 0, 1],
9 => [1, 0, 0, 0, 1, 0, 0, 0, 0, 1],
10 => [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1],
11 => [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1],
12 => [1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1],
13 => [1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1],
14 => [1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1],
15 => [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
16 => [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1],
_ => throw new ArgumentOutOfRangeException(nameof(m),
Localization.m_must_be_between_2_and_16_inclusive)
};
_mm = m;
_kk = k;
_nn = n;
_a0 = n;
_alphaTo = new int[n + 1];
_indexOf = new int[n + 1];
_gg = new int[_nn - _kk + 1];
generate_gf();
gen_poly();
_initialized = true;
}
int Modnn(int x)
{
while(x >= _nn)
{
x -= _nn;
x = (x >> _mm) + (x & _nn);
}
return x;
}
static int Min(int a, int b) => a < b ? a : b;
static void Clear(ref int[] a, int n)
{
int ci;
for(ci = n - 1; ci >= 0; ci--) a[ci] = 0;
}
static void Copy(ref int[] a, ref int[] b, int n)
{
int ci;
for(ci = n - 1; ci >= 0; ci--) a[ci] = b[ci];
}
static void Copydown(ref int[] a, ref int[] b, int n)
{
int ci;
for(ci = n - 1; ci >= 0; ci--) a[ci] = b[ci];
}
/* generate GF(2**m) from the irreducible polynomial p(X) in p[0]..p[m]
lookup tables: index->polynomial form alpha_to[] contains j=alpha**i;
polynomial form -> index form index_of[j=alpha**i] = i
alpha=2 is the primitive element of GF(2**m)
HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
Let @ represent the primitive element commonly called "alpha" that
is the root of the primitive polynomial p(x). Then in GF(2^m), for any
0 <= i <= 2^m-2,
@^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
example the polynomial representation of @^5 would be given by the binary
representation of the integer "alpha_to[5]".
Similarily, index_of[] can be used as follows:
As above, let @ represent the primitive element of GF(2^m) that is
the root of the primitive polynomial p(x). In order to find the power
of @ (alpha) that has the polynomial representation
a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
we consider the integer "i" whose binary representation with a(0) being LSB
and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
"index_of[i]". Now, @^index_of[i] is that element whose polynomial
representation is (a(0),a(1),a(2),...,a(m-1)).
NOTE:
The element alpha_to[2^m-1] = 0 always signifying that the
representation of "@^infinity" = 0 is (0,0,0,...,0).
Similarily, the element index_of[0] = A0 always signifying
that the power of alpha which has the polynomial representation
(0,0,...,0) is "infinity".
*/
void generate_gf()
{
int i;
var mask = 1;
_alphaTo[_mm] = 0;
for(i = 0; i < _mm; i++)
{
_alphaTo[i] = mask;
_indexOf[_alphaTo[i]] = i;
/* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
if(_pp[i] != 0) _alphaTo[_mm] ^= mask; /* Bit-wise EXOR operation */
mask <<= 1; /* single left-shift */
}
_indexOf[_alphaTo[_mm]] = _mm;
/*
* Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
* poly-repr of @^i shifted left one-bit and accounting for any @^MM
* term that may occur when poly-repr of @^i is shifted.
*/
mask >>= 1;
for(i = _mm + 1; i < _nn; i++)
{
if(_alphaTo[i - 1] >= mask)
_alphaTo[i] = _alphaTo[_mm] ^ (_alphaTo[i - 1] ^ mask) << 1;
else
_alphaTo[i] = _alphaTo[i - 1] << 1;
_indexOf[_alphaTo[i]] = i;
}
_indexOf[0] = _a0;
_alphaTo[_nn] = 0;
}
/*
* Obtain the generator polynomial of the TT-error correcting, length
* NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
* ... ,(2*TT-1)
*
* Examples:
*
* If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
* g(x) = (x+@) (x+@**2)
*
* If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
* g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
*/
void gen_poly()
{
int i;
_gg[0] = _alphaTo[B0];
_gg[1] = 1; /* g(x) = (X+@**B0) initially */
for(i = 2; i <= _nn - _kk; i++)
{
_gg[i] = 1;
/*
* Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
* (@**(B0+i-1) + x)
*/
for(int j = i - 1; j > 0; j--)
{
if(_gg[j] != 0)
_gg[j] = _gg[j - 1] ^ _alphaTo[Modnn(_indexOf[_gg[j]] + B0 + i - 1)];
else
_gg[j] = _gg[j - 1];
}
/* Gg[0] can never be zero */
_gg[0] = _alphaTo[Modnn(_indexOf[_gg[0]] + B0 + i - 1)];
}
/* convert Gg[] to index form for quicker encoding */
for(i = 0; i <= _nn - _kk; i++) _gg[i] = _indexOf[_gg[i]];
}
/*
* take the string of symbols in data[i], i=0..(k-1) and encode
* systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
* is input and bb[] is output in polynomial form. Encoding is done by using
* a feedback shift register with appropriate connections specified by the
* elements of Gg[], which was generated above. Codeword is c(X) =
* data(X)*X**(NN-KK)+ b(X)
*/
/// <summary>Takes the symbols in data to output parity in bb.</summary>
/// <returns>Returns -1 if an illegal symbol is found.</returns>
/// <param name="data">Data symbols.</param>
/// <param name="bb">Outs parity symbols.</param>
public int encode_rs(int[] data, out int[] bb)
{
if(!_initialized)
throw new UnauthorizedAccessException(Localization.Trying_to_calculate_RS_without_initializing);
int i;
bb = new int[_nn - _kk];
Clear(ref bb, _nn - _kk);
for(i = _kk - 1; i >= 0; i--)
{
if(_mm != 8)
if(data[i] > _nn)
return -1; /* Illegal symbol */
int feedback = _indexOf[data[i] ^ bb[_nn - _kk - 1]];
if(feedback != _a0)
{
/* feedback term is non-zero */
for(int j = _nn - _kk - 1; j > 0; j--)
{
if(_gg[j] != _a0)
bb[j] = bb[j - 1] ^ _alphaTo[Modnn(_gg[j] + feedback)];
else
bb[j] = bb[j - 1];
}
bb[0] = _alphaTo[Modnn(_gg[0] + feedback)];
}
else
{
/* feedback term is zero. encoder becomes a
* single-byte shifter */
for(int j = _nn - _kk - 1; j > 0; j--) bb[j] = bb[j - 1];
bb[0] = 0;
}
}
return 0;
}
/*
* Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
* writes the codeword into data[] itself. Otherwise data[] is unaltered.
*
* Return number of symbols corrected, or -1 if codeword is illegal
* or uncorrectable.
*
* First "no_eras" erasures are declared by the calling program. Then, the
* maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
* If the number of channel errors is not greater than "t_after_eras" the
* transmitted codeword will be recovered. Details of algorithm can be found
* in R. Blahut's "Theory ... of Error-Correcting Codes".
*/
/// <summary>Decodes the RS. If decoding is successful outputs corrected data symbols.</summary>
/// <returns>Returns corrected symbols, -1 if illegal or uncorrectable</returns>
/// <param name="data">Data symbols.</param>
/// <param name="erasPos">Position of erasures.</param>
/// <param name="noEras">Number of erasures.</param>
public int eras_dec_rs(ref int[] data, out int[] erasPos, int noEras)
{
if(!_initialized)
throw new UnauthorizedAccessException(Localization.Trying_to_calculate_RS_without_initializing);
erasPos = new int[_nn - _kk];
int i, j;
int q, tmp;
var recd = new int[_nn];
var lambda = new int[_nn - _kk + 1]; /* Err+Eras Locator poly */
var s = new int[_nn - _kk + 1]; /* syndrome poly */
var b = new int[_nn - _kk + 1];
var t = new int[_nn - _kk + 1];
var omega = new int[_nn - _kk + 1];
var root = new int[_nn - _kk];
var reg = new int[_nn - _kk + 1];
var loc = new int[_nn - _kk];
int count;
/* data[] is in polynomial form, copy and convert to index form */
for(i = _nn - 1; i >= 0; i--)
{
if(_mm != 8)
if(data[i] > _nn)
return -1; /* Illegal symbol */
recd[i] = _indexOf[data[i]];
}
/* first form the syndromes; i.e., evaluate recd(x) at roots of g(x)
* namely @**(B0+i), i = 0, ... ,(NN-KK-1)
*/
var synError = 0;
for(i = 1; i <= _nn - _kk; i++)
{
tmp = 0;
for(j = 0; j < _nn; j++)
if(recd[j] != _a0) /* recd[j] in index form */
tmp ^= _alphaTo[Modnn(recd[j] + (B0 + i - 1) * j)];
synError |= tmp; /* set flag if non-zero syndrome =>
* error */
/* store syndrome in index form */
s[i] = _indexOf[tmp];
}
if(synError == 0) return 0;
Clear(ref lambda, _nn - _kk);
lambda[0] = 1;
if(noEras > 0)
{
/* Init lambda to be the erasure locator polynomial */
lambda[1] = _alphaTo[erasPos[0]];
for(i = 1; i < noEras; i++)
{
int u = erasPos[i];
for(j = i + 1; j > 0; j--)
{
tmp = _indexOf[lambda[j - 1]];
if(tmp != _a0) lambda[j] ^= _alphaTo[Modnn(u + tmp)];
}
}
#if DEBUG
/* find roots of the erasure location polynomial */
for(i = 1; i <= noEras; i++) reg[i] = _indexOf[lambda[i]];
count = 0;
for(i = 1; i <= _nn; i++)
{
q = 1;
for(j = 1; j <= noEras; j++)
{
if(reg[j] == _a0) continue;
reg[j] = Modnn(reg[j] + j);
q ^= _alphaTo[reg[j]];
}
if(q != 0) continue;
/* store root and error location
* number indices
*/
root[count] = i;
loc[count] = _nn - i;
count++;
}
if(count != noEras)
{
AaruLogging.Debug(MODULE_NAME, Localization.lambda_is_wrong);
return -1;
}
AaruLogging.Debug(MODULE_NAME, Localization.Erasure_positions_as_determined_by_roots_of_Eras_Loc_Poly);
for(i = 0; i < count; i++) AaruLogging.Debug(MODULE_NAME, "{0} ", loc[i]);
AaruLogging.Debug(MODULE_NAME, "\n");
#endif
}
for(i = 0; i < _nn - _kk + 1; i++) b[i] = _indexOf[lambda[i]];
/*
* Begin Berlekamp-Massey algorithm to determine error+erasure
* locator polynomial
*/
int r = noEras;
int el = noEras;
while(++r <= _nn - _kk)
{
/* r is the step number */
/* Compute discrepancy at the r-th step in poly-form */
var discrR = 0;
for(i = 0; i < r; i++)
if(lambda[i] != 0 && s[r - i] != _a0)
discrR ^= _alphaTo[Modnn(_indexOf[lambda[i]] + s[r - i])];
discrR = _indexOf[discrR]; /* Index form */
if(discrR == _a0)
{
/* 2 lines below: B(x) <-- x*B(x) */
Copydown(ref b, ref b, _nn - _kk);
b[0] = _a0;
}
else
{
/* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
t[0] = lambda[0];
for(i = 0; i < _nn - _kk; i++)
{
if(b[i] != _a0)
t[i + 1] = lambda[i + 1] ^ _alphaTo[Modnn(discrR + b[i])];
else
t[i + 1] = lambda[i + 1];
}
if(2 * el <= r + noEras - 1)
{
el = r + noEras - el;
/*
* 2 lines below: B(x) <-- inv(discr_r) *
* lambda(x)
*/
for(i = 0; i <= _nn - _kk; i++)
b[i] = lambda[i] == 0 ? _a0 : Modnn(_indexOf[lambda[i]] - discrR + _nn);
}
else
{
/* 2 lines below: B(x) <-- x*B(x) */
Copydown(ref b, ref b, _nn - _kk);
b[0] = _a0;
}
Copy(ref lambda, ref t, _nn - _kk + 1);
}
}
/* Convert lambda to index form and compute deg(lambda(x)) */
var degLambda = 0;
for(i = 0; i < _nn - _kk + 1; i++)
{
lambda[i] = _indexOf[lambda[i]];
if(lambda[i] != _a0) degLambda = i;
}
/*
* Find roots of the error+erasure locator polynomial. By Chien
* Search
*/
int temp = reg[0];
Copy(ref reg, ref lambda, _nn - _kk);
reg[0] = temp;
count = 0; /* Number of roots of lambda(x) */
for(i = 1; i <= _nn; i++)
{
q = 1;
for(j = degLambda; j > 0; j--)
{
if(reg[j] == _a0) continue;
reg[j] = Modnn(reg[j] + j);
q ^= _alphaTo[reg[j]];
}
if(q != 0) continue;
/* store root (index-form) and error location number */
root[count] = i;
loc[count] = _nn - i;
count++;
}
#if DEBUG
AaruLogging.Debug(MODULE_NAME, Localization.Final_error_positions);
for(i = 0; i < count; i++) AaruLogging.Debug(MODULE_NAME, "{0} ", loc[i]);
AaruLogging.Debug(MODULE_NAME, "\n");
#endif
if(degLambda != count) return -1;
/*
* Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
* x**(NN-KK)). in index form. Also find deg(omega).
*/
var degOmega = 0;
for(i = 0; i < _nn - _kk; i++)
{
tmp = 0;
j = degLambda < i ? degLambda : i;
for(; j >= 0; j--)
if(s[i + 1 - j] != _a0 && lambda[j] != _a0)
tmp ^= _alphaTo[Modnn(s[i + 1 - j] + lambda[j])];
if(tmp != 0) degOmega = i;
omega[i] = _indexOf[tmp];
}
omega[_nn - _kk] = _a0;
/*
* Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
* inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
*/
for(j = count - 1; j >= 0; j--)
{
var num1 = 0;
for(i = degOmega; i >= 0; i--)
if(omega[i] != _a0)
num1 ^= _alphaTo[Modnn(omega[i] + i * root[j])];
int num2 = _alphaTo[Modnn(root[j] * (B0 - 1) + _nn)];
var den = 0;
/* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
for(i = Min(degLambda, _nn - _kk - 1) & ~1; i >= 0; i -= 2)
if(lambda[i + 1] != _a0)
den ^= _alphaTo[Modnn(lambda[i + 1] + i * root[j])];
if(den == 0)
{
AaruLogging.Debug(MODULE_NAME, Localization.ERROR_denominator_equals_zero);
return -1;
}
/* Apply error to data */
if(num1 != 0) data[loc[j]] ^= _alphaTo[Modnn(_indexOf[num1] + _indexOf[num2] + _nn - _indexOf[den])];
}
return count;
}
}
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