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‎com/backblaze/erasure/Galois.java‎

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/**
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* 8-bit Galois Field
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*
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* Copyright 2015, Backblaze, Inc. All rights reserved.
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*/
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package com.backblaze.erasure;
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import java.util.ArrayList;
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import java.util.List;
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/**
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* 8-bit Galois Field
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*
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* This class implements multiplication, division, addition,
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* subtraction, and exponentiation.
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*
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* The multiplication operation is in the inner loop of
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* erasure coding, so it's been optimized. Having the
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* class be "final" helps a little, and having the EXP_TABLE
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* repeat the data, so there's no need to bound the sum
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* of two logarithms to 255 helps a lot.
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*/
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public final class Galois {
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/**
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* The number of elements in the field.
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*/
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public static final int FIELD_SIZE = 256;
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/**
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* The polynomial used to generate the logarithm table.
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*
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* There are a number of polynomials that work to generate
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* a Galois field of 256 elements. The choice is arbitrary,
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* and we just use the first one.
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*
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* The possibilities are: 29, 43, 45, 77, 95, 99, 101, 105,
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* 113, 135, 141, 169, 195, 207, 231, and 245.
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*/
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public static final int GENERATING_POLYNOMIAL = 29;
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/**
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* Mapping from members of the Galois Field to their
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* integer logarithms. The entry for 0 is meaningless
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* because there is no log of 0.
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*
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* This array is shorts, not bytes, so that they can
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* be used directly to index arrays without casting.
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* The values (except the non-value at index 0) are
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* all really bytes, so they range from 0 to 255.
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*
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* This table was generated by java_tables.py, and the
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* unit tests check it against the Java implementation.
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*/
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static final public short [] LOG_TABLE = new short [] {
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-1, 0, 1, 25, 2, 50, 26, 198,
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3, 223, 51, 238, 27, 104, 199, 75,
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4, 100, 224, 14, 52, 141, 239, 129,
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28, 193, 105, 248, 200, 8, 76, 113,
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5, 138, 101, 47, 225, 36, 15, 33,
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53, 147, 142, 218, 240, 18, 130, 69,
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29, 181, 194, 125, 106, 39, 249, 185,
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201, 154, 9, 120, 77, 228, 114, 166,
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6, 191, 139, 98, 102, 221, 48, 253,
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226, 152, 37, 179, 16, 145, 34, 136,
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54, 208, 148, 206, 143, 150, 219, 189,
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241, 210, 19, 92, 131, 56, 70, 64,
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30, 66, 182, 163, 195, 72, 126, 110,
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107, 58, 40, 84, 250, 133, 186, 61,
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202, 94, 155, 159, 10, 21, 121, 43,
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78, 212, 229, 172, 115, 243, 167, 87,
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7, 112, 192, 247, 140, 128, 99, 13,
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103, 74, 222, 237, 49, 197, 254, 24,
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227, 165, 153, 119, 38, 184, 180, 124,
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17, 68, 146, 217, 35, 32, 137, 46,
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55, 63, 209, 91, 149, 188, 207, 205,
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144, 135, 151, 178, 220, 252, 190, 97,
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242, 86, 211, 171, 20, 42, 93, 158,
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132, 60, 57, 83, 71, 109, 65, 162,
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31, 45, 67, 216, 183, 123, 164, 118,
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196, 23, 73, 236, 127, 12, 111, 246,
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108, 161, 59, 82, 41, 157, 85, 170,
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251, 96, 134, 177, 187, 204, 62, 90,
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203, 89, 95, 176, 156, 169, 160, 81,
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11, 245, 22, 235, 122, 117, 44, 215,
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79, 174, 213, 233, 230, 231, 173, 232,
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116, 214, 244, 234, 168, 80, 88, 175
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};
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/**
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* Inverse of the logarithm table. Maps integer logarithms
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* to members of the field. There is no entry for 255
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* because the highest log is 254.
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*
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* This table was generated by java_tables.py
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*/
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static final byte [] EXP_TABLE = new byte [] {
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1, 2, 4, 8, 16, 32, 64, -128,
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29, 58, 116, -24, -51, -121, 19, 38,
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76, -104, 45, 90, -76, 117, -22, -55,
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-113, 3, 6, 12, 24, 48, 96, -64,
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-99, 39, 78, -100, 37, 74, -108, 53,
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106, -44, -75, 119, -18, -63, -97, 35,
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70, -116, 5, 10, 20, 40, 80, -96,
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93, -70, 105, -46, -71, 111, -34, -95,
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95, -66, 97, -62, -103, 47, 94, -68,
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101, -54, -119, 15, 30, 60, 120, -16,
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-3, -25, -45, -69, 107, -42, -79, 127,
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-2, -31, -33, -93, 91, -74, 113, -30,
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-39, -81, 67, -122, 17, 34, 68, -120,
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13, 26, 52, 104, -48, -67, 103, -50,
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-127, 31, 62, 124, -8, -19, -57, -109,
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59, 118, -20, -59, -105, 51, 102, -52,
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-123, 23, 46, 92, -72, 109, -38, -87,
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79, -98, 33, 66, -124, 21, 42, 84,
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-88, 77, -102, 41, 82, -92, 85, -86,
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73, -110, 57, 114, -28, -43, -73, 115,
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-26, -47, -65, 99, -58, -111, 63, 126,
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-4, -27, -41, -77, 123, -10, -15, -1,
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-29, -37, -85, 75, -106, 49, 98, -60,
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-107, 55, 110, -36, -91, 87, -82, 65,
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-126, 25, 50, 100, -56, -115, 7, 14,
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28, 56, 112, -32, -35, -89, 83, -90,
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81, -94, 89, -78, 121, -14, -7, -17,
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-61, -101, 43, 86, -84, 69, -118, 9,
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18, 36, 72, -112, 61, 122, -12, -11,
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-9, -13, -5, -21, -53, -117, 11, 22,
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44, 88, -80, 125, -6, -23, -49, -125,
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27, 54, 108, -40, -83, 71, -114,
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// Repeat the table a second time, so multiply()
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// does not have to check bounds.
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1, 2, 4, 8, 16, 32, 64, -128,
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29, 58, 116, -24, -51, -121, 19, 38,
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76, -104, 45, 90, -76, 117, -22, -55,
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-113, 3, 6, 12, 24, 48, 96, -64,
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-99, 39, 78, -100, 37, 74, -108, 53,
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106, -44, -75, 119, -18, -63, -97, 35,
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70, -116, 5, 10, 20, 40, 80, -96,
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93, -70, 105, -46, -71, 111, -34, -95,
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95, -66, 97, -62, -103, 47, 94, -68,
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101, -54, -119, 15, 30, 60, 120, -16,
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-3, -25, -45, -69, 107, -42, -79, 127,
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-2, -31, -33, -93, 91, -74, 113, -30,
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-39, -81, 67, -122, 17, 34, 68, -120,
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13, 26, 52, 104, -48, -67, 103, -50,
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-127, 31, 62, 124, -8, -19, -57, -109,
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59, 118, -20, -59, -105, 51, 102, -52,
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-123, 23, 46, 92, -72, 109, -38, -87,
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79, -98, 33, 66, -124, 21, 42, 84,
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-88, 77, -102, 41, 82, -92, 85, -86,
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73, -110, 57, 114, -28, -43, -73, 115,
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-26, -47, -65, 99, -58, -111, 63, 126,
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-4, -27, -41, -77, 123, -10, -15, -1,
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-29, -37, -85, 75, -106, 49, 98, -60,
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-107, 55, 110, -36, -91, 87, -82, 65,
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-126, 25, 50, 100, -56, -115, 7, 14,
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28, 56, 112, -32, -35, -89, 83, -90,
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81, -94, 89, -78, 121, -14, -7, -17,
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-61, -101, 43, 86, -84, 69, -118, 9,
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18, 36, 72, -112, 61, 122, -12, -11,
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-9, -13, -5, -21, -53, -117, 11, 22,
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44, 88, -80, 125, -6, -23, -49, -125,
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27, 54, 108, -40, -83, 71, -114
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};
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/**
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* Adds two elements of the field. If you're in an inner loop,
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* you should inline this function: it's just XOR.
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*/
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public static byte add(byte a, byte b) {
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return (byte) (a ^ b);
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}
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/**
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* Inverse of addition. If you're in an inner loop,
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* you should inline this function: it's just XOR.
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*/
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public static byte subtract(byte a, byte b) {
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return (byte) (a ^ b);
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}
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/**
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* Multiplies to elements of the field.
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*/
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public static byte multiply(byte a, byte b) {
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if (a == 0 || b == 0) {
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return 0;
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}
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else {
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int logA = LOG_TABLE[a & 0xFF];
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int logB = LOG_TABLE[b & 0xFF];
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int logResult = logA + logB;
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return EXP_TABLE[logResult];
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}
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}
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/**
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* Inverse of multiplication.
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*/
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public static byte divide(byte a, byte b) {
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if (a == 0) {
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return 0;
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}
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if (b == 0) {
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throw new IllegalArgumentException("Argument 'divisor' is 0");
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}
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int logA = LOG_TABLE[a & 0xFF];
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int logB = LOG_TABLE[b & 0xFF];
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int logResult = logA - logB;
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if (logResult < 0) {
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logResult += 255;
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}
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return EXP_TABLE[logResult];
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}
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/**
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* Computes a**n.
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*
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* The result will be the same as multiplying a times itself n times.
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*
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* @param a A member of the field.
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* @param n A plain-old integer.
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* @return The result of multiplying a by itself n times.
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*/
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public static byte exp(byte a, int n) {
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if (n == 0) {
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return 1;
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}
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else if (a == 0) {
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return 0;
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}
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else {
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int logA = LOG_TABLE[a & 0xFF];
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int logResult = logA * n;
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while (255 <= logResult) {
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logResult -= 255;
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}
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return EXP_TABLE[logResult];
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}
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}
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/**
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* Generates a logarithm table given a starting polynomial.
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*/
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public static short [] generateLogTable(int polynomial) {
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short [] result = new short[FIELD_SIZE];
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for (int i = 0; i < FIELD_SIZE; i++) {
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result[i] = -1; // -1 means "not set"
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}
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int b = 1;
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for (int log = 0; log < FIELD_SIZE - 1; log++) {
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if (result[b] != -1) {
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throw new RuntimeException("BUG: duplicate logarithm (bad polynomial?)");
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}
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result[b] = (short) log;
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b = (b << 1);
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if (FIELD_SIZE <= b) {
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b = ((b - FIELD_SIZE) ^ polynomial);
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}
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}
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return result;
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}
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/**
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* Generates the inverse log table.
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*/
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public static byte [] generateExpTable(short [] logTable) {
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final byte [] result = new byte [FIELD_SIZE * 2 - 2];
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for (int i = 1; i < FIELD_SIZE; i++) {
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int log = logTable[i];
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result[log] = (byte) i;
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result[log + FIELD_SIZE - 1] = (byte) i;
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}
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return result;
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}
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/**
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* Returns a list of all polynomials that can be used to generate
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* the field.
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*
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* This is never used in the code; it's just here for completeness.
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*/
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public static Integer [] allPossiblePolynomials() {
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List<Integer> result = new ArrayList<Integer>();
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for (int i = 0; i < FIELD_SIZE; i++) {
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try {
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generateLogTable(i);
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result.add(i);
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}
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catch (RuntimeException e) {
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// this one didn't work
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}
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}
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return result.toArray(new Integer [result.size()]);
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}
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}

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