/**
* Licensed to the Apache Software Foundation (ASF) under one
* or more contributor license agreements. See the NOTICE file
* distributed with this work for additional information
* regarding copyright ownership. The ASF licenses this file
* to you under the Apache License, Version 2.0 (the
* "License"); you may not use this file except in compliance
* with the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.hadoop.raid;
import java.util.HashMap;
import java.util.Map;
/**
* Implementation of Galois field arithmetics with 2^p elements.
* The input must be unsigned integers.
*/
public class GaloisField {
private final int[] logTable;
private final int[] powTable;
private final int[][] mulTable;
private final int[][] divTable;
private final int fieldSize;
private final int primitivePeriod;
private final int primitivePolynomial;
// Field size 256 is good for byte based system
private static final int DEFAULT_FIELD_SIZE = 256;
// primitive polynomial 1 + X^2 + X^3 + X^4 + X^8
private static final int DEFAULT_PRIMITIVE_POLYNOMIAL = 285;
static private final Map<Integer, GaloisField> instances =
new HashMap<Integer, GaloisField>();
/**
* Get the object performs Galois field arithmetics
* @param fieldSize size of the field
* @param primitivePolynomial a primitive polynomial corresponds to the size
*/
public static GaloisField getInstance(int fieldSize,
int primitivePolynomial) {
int key = ((fieldSize << 16) & 0xFFFF0000) + (primitivePolynomial & 0x0000FFFF);
GaloisField gf;
synchronized (instances) {
gf = instances.get(key);
if (gf == null) {
gf = new GaloisField(fieldSize, primitivePolynomial);
instances.put(key, gf);
}
}
return gf;
}
/**
* Get the object performs Galois field arithmetics with default setting
*/
public static GaloisField getInstance() {
return getInstance(DEFAULT_FIELD_SIZE, DEFAULT_PRIMITIVE_POLYNOMIAL);
}
private GaloisField(int fieldSize, int primitivePolynomial) {
this.fieldSize = fieldSize;
this.primitivePeriod = fieldSize - 1;
this.primitivePolynomial = primitivePolynomial;
logTable = new int[fieldSize];
powTable = new int[fieldSize];
mulTable = new int[fieldSize][fieldSize];
divTable = new int[fieldSize][fieldSize];
int value = 1;
for (int pow = 0; pow < fieldSize - 1; pow++) {
powTable[pow] = value;
logTable[value] = pow;
value = value * 2;
if (value >= fieldSize) {
value = value ^ primitivePolynomial;
}
}
// building multiplication table
for (int i = 0; i < fieldSize; i++) {
for (int j = 0; j < fieldSize; j++) {
if (i == 0 || j == 0) {
mulTable[i][j] = 0;
continue;
}
int z = logTable[i] + logTable[j];
z = z >= primitivePeriod ? z - primitivePeriod : z;
z = powTable[z];
mulTable[i][j] = z;
}
}
// building division table
for (int i = 0; i < fieldSize; i++) {
for (int j = 1; j < fieldSize; j++) {
if (i == 0) {
divTable[i][j] = 0;
continue;
}
int z = logTable[i] - logTable[j];
z = z < 0 ? z + primitivePeriod : z;
z = powTable[z];
divTable[i][j] = z;
}
}
}
/**
* Return number of elements in the field
* @return number of elements in the field
*/
public int getFieldSize() {
return fieldSize;
}
/**
* Return the primitive polynomial in GF(2)
* @return primitive polynomial as a integer
*/
public int getPrimitivePolynomial() {
return primitivePolynomial;
}
/**
* Compute the sum of two fields
* @param x input field
* @param y input field
* @return result of addition
*/
public int add(int x, int y) {
assert(x >= 0 && x < getFieldSize() && y >= 0 && y < getFieldSize());
return x ^ y;
}
/**
* Compute the multiplication of two fields
* @param x input field
* @param y input field
* @return result of multiplication
*/
public int multiply(int x, int y) {
assert(x >= 0 && x < getFieldSize() && y >= 0 && y < getFieldSize());
return mulTable[x][y];
}
/**
* Compute the division of two fields
* @param x input field
* @param y input field
* @return x/y
*/
public int divide(int x, int y) {
assert(x >= 0 && x < getFieldSize() && y > 0 && y < getFieldSize());
return divTable[x][y];
}
/**
* Compute power n of a field
* @param x input field
* @param n power
* @return x^n
*/
public int power(int x, int n) {
assert(x >= 0 && x < getFieldSize());
if (n == 0) {
return 1;
}
if (x == 0) {
return 0;
}
x = logTable[x] * n;
if (x < primitivePeriod) {
return powTable[x];
}
x = x % primitivePeriod;
return powTable[x];
}
/**
* Given a Vandermonde matrix V[i][j]=x[j]^i and vector y, solve for z such
* that Vz=y. The output z will be placed in y.
* @param x the vector which describe the Vandermonde matrix
* @param y right-hand side of the Vandermonde system equation.
* will be replaced the output in this vector
*/
public void solveVandermondeSystem(int[] x, int[] y) {
solveVandermondeSystem(x, y, x.length);
}
/**
* Given a Vandermonde matrix V[i][j]=x[j]^i and vector y, solve for z such
* that Vz=y. The output z will be placed in y.
* @param x the vector which describe the Vandermonde matrix
* @param y right-hand side of the Vandermonde system equation.
* will be replaced the output in this vector
* @param len consider x and y only from 0...len-1
*/
public void solveVandermondeSystem(int[] x, int[] y, int len) {
assert(x.length <= len && y.length <= len);
for (int i = 0; i < len - 1; i++) {
for (int j = len - 1; j > i; j--) {
y[j] = y[j] ^ mulTable[x[i]][y[j - 1]];
}
}
for (int i = len - 1; i >= 0; i--) {
for (int j = i + 1; j < len; j++) {
y[j] = divTable[y[j]][x[j] ^ x[j - i - 1]];
}
for (int j = i; j < len - 1; j++) {
y[j] = y[j] ^ y[j + 1];
}
}
}
/**
* A "bulk" version of the solveVandermondeSystem
*/
public void solveVandermondeSystem(int[] x, byte[][] y,
int len, int dataStart, int dataLen) {
assert(x.length <= len && y.length <= len);
int dataEnd = dataStart + dataLen;
for (int i = 0; i < len - 1; i++) {
for (int j = len - 1; j > i; j--) {
for (int k = dataStart; k < dataEnd; k++) {
y[j][k] = (byte)(y[j][k] ^ mulTable[x[i]][y[j - 1][k] & 0x000000FF]);
}
}
}
for (int i = len - 1; i >= 0; i--) {
for (int j = i + 1; j < len; j++) {
for (int k = dataStart; k < dataEnd; k++) {
y[j][k] = (byte)(divTable[y[j][k] & 0x000000FF][x[j] ^ x[j - i - 1]]);
}
}
for (int j = i; j < len - 1; j++) {
for (int k = dataStart; k < dataEnd; k++) {
y[j][k] = (byte)(y[j][k] ^ y[j + 1][k]);
}
}
}
}
/**
* Compute the multiplication of two polynomials. The index in the
* array corresponds to the power of the entry. For example p[0] is the
* constant term of the polynomial p.
* @param p input polynomial
* @param q input polynomial
* @return polynomial represents p*q
*/
public int[] multiply(int[] p, int[] q) {
int len = p.length + q.length - 1;
int[] result = new int[len];
for (int i = 0; i < len; i++) {
result[i] = 0;
}
for (int i = 0; i < p.length; i++) {
for (int j = 0; j < q.length; j++) {
result[i + j] = add(result[i + j], multiply(p[i], q[j]));
}
}
return result;
}
/**
* Compute the remainder of a dividend and divisor pair. The index in the
* array corresponds to the power of the entry. For example p[0] is the
* constant term of the polynomial p.
* @param dividend dividend polynomial, the remainder will be placed here when return
* @param divisor divisor polynomial
*/
public void remainder(int[] dividend, int[] divisor) {
for (int i = dividend.length - divisor.length; i >= 0; i--) {
int ratio =
divTable[dividend[i + divisor.length - 1]][divisor[divisor.length - 1]];
for (int j = 0; j < divisor.length; j++) {
int k = j + i;
dividend[k] = dividend[k] ^ mulTable[ratio][divisor[j]];
}
}
}
/**
* The "bulk" version of the remainder.
* Warning: This function will modify the "dividend" inputs.
*/
public void remainder(byte[][] dividend, int[] divisor) {
for (int i = dividend.length - divisor.length; i >= 0; i--) {
for (int j = 0; j < divisor.length; j++) {
for (int k = 0; k < dividend[i].length; k++) {
int ratio =
divTable[dividend[i + divisor.length - 1][k] & 0x00FF][divisor[divisor.length - 1]];
dividend[j + i][k] = (byte)((dividend[j + i][k] & 0x00FF) ^ mulTable[ratio][divisor[j]]);
}
}
}
}
/**
* Compute the sum of two polynomials. The index in the
* array corresponds to the power of the entry. For example p[0] is the
* constant term of the polynomial p.
* @param p input polynomial
* @param q input polynomial
* @return polynomial represents p+q
*/
public int[] add(int[] p, int[] q) {
int len = Math.max(p.length, q.length);
int[] result = new int[len];
for (int i = 0; i < len; i++) {
if (i < p.length && i < q.length) {
result[i] = add(p[i], q[i]);
} else if (i < p.length) {
result[i] = p[i];
} else {
result[i] = q[i];
}
}
return result;
}
/**
* Substitute x into polynomial p(x).
* @param p input polynomial
* @param x input field
* @return p(x)
*/
public int substitute(int[] p, int x) {
int result = 0;
int y = 1;
for (int i = 0; i < p.length; i++) {
result = result ^ mulTable[p[i]][y];
y = mulTable[x][y];
}
return result;
}
public void substitute(byte[][] p, byte[] q, int x,
int dataStart, int dataLen) {
int y = 1;
for (int i = 0; i < p.length; i++) {
byte[] pi = p[i];
for (int j = dataStart; j < dataStart + dataLen; j++) {
int pij = pi[j] & 0x000000FF;
q[j] = (byte)(q[j] ^ mulTable[pij][y]);
}
y = mulTable[x][y];
}
}
/**
* A "bulk" version of the substitute.
* Tends to be 2X faster than the "int" substitute in a loop.
*
* @param p input polynomial
* @param q store the return result
* @param x input field
*/
public void substitute(byte[][] p, byte[] q, int x) {
substitute(p, q, x, 0, p[0].length);
}
/**
* Perform Gaussian elimination on the given matrix. This matrix has to be a
* fat matrix (number of rows > number of columns).
*/
public void gaussianElimination(int[][] matrix) {
assert(matrix != null && matrix.length > 0 && matrix[0].length > 0
&& matrix.length < matrix[0].length);
int height = matrix.length;
int width = matrix[0].length;
for (int i = 0; i < height; i++) {
boolean pivotFound = false;
// scan the column for a nonzero pivot and swap it to the diagonal
for (int j = i; j < height; j++) {
if (matrix[i][j] != 0) {
int[] tmp = matrix[i];
matrix[i] = matrix[j];
matrix[j] = tmp;
pivotFound = true;
break;
}
}
if (!pivotFound) {
continue;
}
int pivot = matrix[i][i];
for (int j = i; j < width; j++) {
matrix[i][j] = divide(matrix[i][j], pivot);
}
for (int j = i + 1; j < height; j++) {
int lead = matrix[j][i];
for (int k = i; k < width; k++) {
matrix[j][k] = add(matrix[j][k], multiply(lead, matrix[i][k]));
}
}
}
for (int i = height - 1; i >=0; i--) {
for (int j = 0; j < i; j++) {
int lead = matrix[j][i];
for (int k = i; k < width; k++) {
matrix[j][k] = add(matrix[j][k], multiply(lead, matrix[i][k]));
}
}
}
}
}