Re-implemented the CRC functions under own copyright
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#define Cdrskin_timestamP "2012.02.13.102837"
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#define Cdrskin_timestamP "2012.02.19.101022"
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@ -1,47 +1,554 @@
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/* -*- indent-tabs-mode: t; tab-width: 8; c-basic-offset: 8; -*- */
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/* Copyright (c) 2012 Thomas Schmitt <scdbackup@gmx.net>
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Provided under GPL version 2 or later.
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Containing disabled code pieces from other GPL programs.
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They are just quotes for reference.
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The activated code uses plain polynomial division and other primitve
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algorithms to build tables of pre-computed CRC values. It then computes
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the CRCs by algorithms which are derived from mathematical considerations
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and from analysing the mathematical meaning of the disabled code pieces.
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The comments here are quite detailed in order to prove my own understanding
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of the topic.
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*/
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#ifdef HAVE_CONFIG_H
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#include "../config.h"
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#endif
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#include "crc.h"
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static unsigned short ccitt_table[256] = {
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0x0000, 0x1021, 0x2042, 0x3063, 0x4084, 0x50A5, 0x60C6, 0x70E7,
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0x8108, 0x9129, 0xA14A, 0xB16B, 0xC18C, 0xD1AD, 0xE1CE, 0xF1EF,
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0x1231, 0x0210, 0x3273, 0x2252, 0x52B5, 0x4294, 0x72F7, 0x62D6,
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0x9339, 0x8318, 0xB37B, 0xA35A, 0xD3BD, 0xC39C, 0xF3FF, 0xE3DE,
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0x2462, 0x3443, 0x0420, 0x1401, 0x64E6, 0x74C7, 0x44A4, 0x5485,
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0xA56A, 0xB54B, 0x8528, 0x9509, 0xE5EE, 0xF5CF, 0xC5AC, 0xD58D,
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0x3653, 0x2672, 0x1611, 0x0630, 0x76D7, 0x66F6, 0x5695, 0x46B4,
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0xB75B, 0xA77A, 0x9719, 0x8738, 0xF7DF, 0xE7FE, 0xD79D, 0xC7BC,
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0x48C4, 0x58E5, 0x6886, 0x78A7, 0x0840, 0x1861, 0x2802, 0x3823,
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0xC9CC, 0xD9ED, 0xE98E, 0xF9AF, 0x8948, 0x9969, 0xA90A, 0xB92B,
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0x5AF5, 0x4AD4, 0x7AB7, 0x6A96, 0x1A71, 0x0A50, 0x3A33, 0x2A12,
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0xDBFD, 0xCBDC, 0xFBBF, 0xEB9E, 0x9B79, 0x8B58, 0xBB3B, 0xAB1A,
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0x6CA6, 0x7C87, 0x4CE4, 0x5CC5, 0x2C22, 0x3C03, 0x0C60, 0x1C41,
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0xEDAE, 0xFD8F, 0xCDEC, 0xDDCD, 0xAD2A, 0xBD0B, 0x8D68, 0x9D49,
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0x7E97, 0x6EB6, 0x5ED5, 0x4EF4, 0x3E13, 0x2E32, 0x1E51, 0x0E70,
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0xFF9F, 0xEFBE, 0xDFDD, 0xCFFC, 0xBF1B, 0xAF3A, 0x9F59, 0x8F78,
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0x9188, 0x81A9, 0xB1CA, 0xA1EB, 0xD10C, 0xC12D, 0xF14E, 0xE16F,
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0x1080, 0x00A1, 0x30C2, 0x20E3, 0x5004, 0x4025, 0x7046, 0x6067,
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0x83B9, 0x9398, 0xA3FB, 0xB3DA, 0xC33D, 0xD31C, 0xE37F, 0xF35E,
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0x02B1, 0x1290, 0x22F3, 0x32D2, 0x4235, 0x5214, 0x6277, 0x7256,
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0xB5EA, 0xA5CB, 0x95A8, 0x8589, 0xF56E, 0xE54F, 0xD52C, 0xC50D,
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0x34E2, 0x24C3, 0x14A0, 0x0481, 0x7466, 0x6447, 0x5424, 0x4405,
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0xA7DB, 0xB7FA, 0x8799, 0x97B8, 0xE75F, 0xF77E, 0xC71D, 0xD73C,
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0x26D3, 0x36F2, 0x0691, 0x16B0, 0x6657, 0x7676, 0x4615, 0x5634,
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0xD94C, 0xC96D, 0xF90E, 0xE92F, 0x99C8, 0x89E9, 0xB98A, 0xA9AB,
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0x5844, 0x4865, 0x7806, 0x6827, 0x18C0, 0x08E1, 0x3882, 0x28A3,
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0xCB7D, 0xDB5C, 0xEB3F, 0xFB1E, 0x8BF9, 0x9BD8, 0xABBB, 0xBB9A,
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0x4A75, 0x5A54, 0x6A37, 0x7A16, 0x0AF1, 0x1AD0, 0x2AB3, 0x3A92,
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0xFD2E, 0xED0F, 0xDD6C, 0xCD4D, 0xBDAA, 0xAD8B, 0x9DE8, 0x8DC9,
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0x7C26, 0x6C07, 0x5C64, 0x4C45, 0x3CA2, 0x2C83, 0x1CE0, 0x0CC1,
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0xEF1F, 0xFF3E, 0xCF5D, 0xDF7C, 0xAF9B, 0xBFBA, 0x8FD9, 0x9FF8,
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0x6E17, 0x7E36, 0x4E55, 0x5E74, 0x2E93, 0x3EB2, 0x0ED1, 0x1EF0
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};
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unsigned long crc32_table[256] = {
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/* Exploration ts B00214 :
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ECMA-130, 22.3.6 "CRC field"
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"This field contains the inverted parity bits. The CRC code word must be
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divisible by the check polynomial. [...]
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The generating polynomial shall be
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G(x) = x^16 + x^12 + x^5 + 1
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"
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Also known as CRC-16-CCITT, CRC-CCITT
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Used in libburn for raw write modes in sector.c.
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There is also disabled code in read.c which would use it.
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ts B11222:
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The same algorithm is prescribed for CD-TEXT in MMC-3 Annex J.
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"CRC Field consists of 2 bytes. Initiator system may use these bytes
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to check errors in the Pack. The polynomial is x^16 + x^12 + x^5 + 1.
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All bits shall be inverted."
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libburn/cdtext.c uses a simple bit shifting function : crc_11021()
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ts B20211:
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Discussion why both are equivalent in respect to their result:
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Both map the bits of the given bytes to a polynomial over the finite field
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of two elements "GF(2)". If bytes 0 .. M are given, then bit n of byte m
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is mapped to the coefficient of x exponent (n + ((M - m) * 8) + 16).
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I.e. they translate the bits into a polynomial with the highest bit
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becomming the coefficient of the highest power of x. Then this polynomial
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is multiplied by (x exp 16).
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The set of all such polynomials forms a commutative ring. Its addition
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corresponds to bitwise exclusive or. Addition and subtraction are identical.
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Multiplication with polynomials of only one single non-zero coefficient
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corresponds to leftward bit shifting by the exponent of that coefficient.
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The same rules apply as with elementary school arithmetics on integer
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numbers, but with surprising results due to the finite nature of the
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coefficient number space.
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Note that multiplication is _not_ an iteration of addition here.
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Function crc_11021() performs a division with residue by the euclidian
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algorithm. I.e. it splits polynomial d into quotient q(d) and residue r(d)
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in respect to the polynomial p = x exp 16 + x exp 12 + x exp 5 + x exp 0
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d = p * q(d) + r(d)
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where r(d) is of a polynomial degree lower than p, i.e. only x exp 15
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or lower have non-zero coefficients.
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The checksum crc(D) is derived by reverse mapping (r(d) * (x exp 16)).
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I.e. by mapping the coefficient of (x exp n) to bit n of the 16 bit word
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crc(D).
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The function result is the bit-wise complement of crc(D).
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Function crc_ccitt uses a table ccitt_table of r(d) values for the
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polynomials d which represent the single byte values 0x00 to 0xff.
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It computes r(d) by computing the residues of an iteratively expanded
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polynomial. The expansion of the processed byte string A by the next byte B
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from the input byte string happens by shifting the string 8 bits to the
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left, and by oring B onto bits 0 to 7.
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In the space of polynomials, the already processed polynomial "a" (image of
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byte string A) gets expanded by polynomial b (the image of byte B) like this
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a * X + b
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where X is (x exp 8), i.e. the single coefficient polynomial of degree 8.
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The following argumentation uses algebra with commutative, associative
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and distributive laws.
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Valid especially with polynomials is this rule:
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(1): r(a + b) = r(a) + r(b)
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because r(a) and r(b) are of degree lower than degree(p) and
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degree(a + b) <= max(degree(a), degree(b))
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Further valid are:
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(2): r(a) = r(r(a))
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(3): r(p * a) = 0
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The residue of this expanded polynomial can be expressed by means of the
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residue r(a) which is known from the previous iteration step, and the
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residue r(b) which may be looked up in ccitt_table.
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r(a * X + b)
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= r(p * q(a) * X + r(a) * X + p * q(b) + r(b))
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Applying rule (1):
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= r(p * q(a) * X) + r(r(a) * X) + r(p * q(b)) + r(r(b))
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Rule (3) and rule (2):
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= r(r(a) * X) + r(b)
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Be h(a) and l(a) chosen so that: r(a) = h(a) * X + l(a),
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and l(a) has zero coefficients above (x exp 7), and h(a) * X has zero
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coefficients below (x exp 8). (They correspond to the high and low byte
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of the 16 bit word crc(A).)
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So the previous statement can be written as:
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= r(h(a) * X * X) + r(l(a) * X) + r(b)
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Since the degree of l(a) is lower than 8, the degree of l(a) * X is lower
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than 16. Thus it cannot be divisible by p which has degree 16.
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So: r(l(a) * X) = l(a) * X
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This yields
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= l(a) * X + r(h(a) * X * X + b)
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h(a) * X * X is the polynomial representation of the high byte of 16 bit
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word crc(A).
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So in the world of bit patterns the iteration step is:
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crc(byte string A expanded by byte B)
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= (low_byte(crc(A)) << 8) ^ crc(high_byte(crc(A)) ^ B)
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And this is what function crc_ccitt() does, modulo swapping the exor
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operants and the final bit inversion which is prescribed by ECMA-130
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and MMC-3 Annex J.
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*/
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/* Plain implementation of polynomial division on a Galois field, where
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addition and subtraction both are binary exor. Euclidian algorithm.
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Divisor is x^16 + x^12 + x^5 + 1 = 0x11021.
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This is about ten times slower than the table driven algorithm.
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*/
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static int crc_11021(unsigned char *data, int count, int flag)
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{
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int acc = 0, i;
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for (i = 0; i < count * 8 + 16; i++) {
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acc = (acc << 1);
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if (i < count * 8)
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acc |= ((data[i / 8] >> (7 - (i % 8))) & 1);
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if (acc & 0x10000)
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acc ^= 0x11021;
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}
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return acc;
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}
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/* This is my own table driven implementation for which i claim copyright.
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Copyright (c) 2012 Thomas Schmitt <scdbackup@gmx.net>
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*/
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unsigned short crc_ccitt(unsigned char *data, int count)
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{
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static unsigned short crc_tab[256], tab_initialized = 0;
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unsigned short acc = 0;
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unsigned char b[1];
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int i;
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if (!tab_initialized) {
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/* Create table of byte residues */
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for (i = 0; i < 256; i++) {
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b[0] = i;
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crc_tab[i] = crc_11021(b, 1, 0);
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}
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tab_initialized = 1;
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}
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/* There seems to be a speed advantage on amd64 if (acc << 8) is the
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second operant of exor, and *(data++) seems faster than data[i].
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*/
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for (i = 0; i < count; i++)
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acc = crc_tab[(acc >> 8) ^ *(data++)] ^ (acc << 8);
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/* ECMA-130 22.3.6 and MMC-3 Annex J (CD-TEXT) want the result with
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inverted bits
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*/
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return ~acc;
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}
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/*
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This was the function inherited with libburn-0.2.
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static unsigned short ccitt_table[256] = {
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0x0000, 0x1021, 0x2042, 0x3063, 0x4084, 0x50A5, 0x60C6, 0x70E7,
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0x8108, 0x9129, 0xA14A, 0xB16B, 0xC18C, 0xD1AD, 0xE1CE, 0xF1EF,
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0x1231, 0x0210, 0x3273, 0x2252, 0x52B5, 0x4294, 0x72F7, 0x62D6,
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0x9339, 0x8318, 0xB37B, 0xA35A, 0xD3BD, 0xC39C, 0xF3FF, 0xE3DE,
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0x2462, 0x3443, 0x0420, 0x1401, 0x64E6, 0x74C7, 0x44A4, 0x5485,
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0xA56A, 0xB54B, 0x8528, 0x9509, 0xE5EE, 0xF5CF, 0xC5AC, 0xD58D,
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0x3653, 0x2672, 0x1611, 0x0630, 0x76D7, 0x66F6, 0x5695, 0x46B4,
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0xB75B, 0xA77A, 0x9719, 0x8738, 0xF7DF, 0xE7FE, 0xD79D, 0xC7BC,
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0x48C4, 0x58E5, 0x6886, 0x78A7, 0x0840, 0x1861, 0x2802, 0x3823,
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0xC9CC, 0xD9ED, 0xE98E, 0xF9AF, 0x8948, 0x9969, 0xA90A, 0xB92B,
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0x5AF5, 0x4AD4, 0x7AB7, 0x6A96, 0x1A71, 0x0A50, 0x3A33, 0x2A12,
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0xDBFD, 0xCBDC, 0xFBBF, 0xEB9E, 0x9B79, 0x8B58, 0xBB3B, 0xAB1A,
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0x6CA6, 0x7C87, 0x4CE4, 0x5CC5, 0x2C22, 0x3C03, 0x0C60, 0x1C41,
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0xEDAE, 0xFD8F, 0xCDEC, 0xDDCD, 0xAD2A, 0xBD0B, 0x8D68, 0x9D49,
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0x7E97, 0x6EB6, 0x5ED5, 0x4EF4, 0x3E13, 0x2E32, 0x1E51, 0x0E70,
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0xFF9F, 0xEFBE, 0xDFDD, 0xCFFC, 0xBF1B, 0xAF3A, 0x9F59, 0x8F78,
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0x9188, 0x81A9, 0xB1CA, 0xA1EB, 0xD10C, 0xC12D, 0xF14E, 0xE16F,
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0x1080, 0x00A1, 0x30C2, 0x20E3, 0x5004, 0x4025, 0x7046, 0x6067,
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0x83B9, 0x9398, 0xA3FB, 0xB3DA, 0xC33D, 0xD31C, 0xE37F, 0xF35E,
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0x02B1, 0x1290, 0x22F3, 0x32D2, 0x4235, 0x5214, 0x6277, 0x7256,
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0xB5EA, 0xA5CB, 0x95A8, 0x8589, 0xF56E, 0xE54F, 0xD52C, 0xC50D,
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0x34E2, 0x24C3, 0x14A0, 0x0481, 0x7466, 0x6447, 0x5424, 0x4405,
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0xA7DB, 0xB7FA, 0x8799, 0x97B8, 0xE75F, 0xF77E, 0xC71D, 0xD73C,
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0x26D3, 0x36F2, 0x0691, 0x16B0, 0x6657, 0x7676, 0x4615, 0x5634,
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0xD94C, 0xC96D, 0xF90E, 0xE92F, 0x99C8, 0x89E9, 0xB98A, 0xA9AB,
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0x5844, 0x4865, 0x7806, 0x6827, 0x18C0, 0x08E1, 0x3882, 0x28A3,
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0xCB7D, 0xDB5C, 0xEB3F, 0xFB1E, 0x8BF9, 0x9BD8, 0xABBB, 0xBB9A,
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0x4A75, 0x5A54, 0x6A37, 0x7A16, 0x0AF1, 0x1AD0, 0x2AB3, 0x3A92,
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0xFD2E, 0xED0F, 0xDD6C, 0xCD4D, 0xBDAA, 0xAD8B, 0x9DE8, 0x8DC9,
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0x7C26, 0x6C07, 0x5C64, 0x4C45, 0x3CA2, 0x2C83, 0x1CE0, 0x0CC1,
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0xEF1F, 0xFF3E, 0xCF5D, 0xDF7C, 0xAF9B, 0xBFBA, 0x8FD9, 0x9FF8,
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0x6E17, 0x7E36, 0x4E55, 0x5E74, 0x2E93, 0x3EB2, 0x0ED1, 0x1EF0
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};
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unsigned short crc_ccitt(unsigned char *q, int len)
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{
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unsigned short crc = 0;
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while (len-- > 0)
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crc = ccitt_table[(crc >> 8 ^ *q++) & 0xff] ^ (crc << 8);
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return ~crc;
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}
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*/
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/* Exploration ts B00214 :
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ECMA-130, 14.3 "EDC field"
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"The EDC field shall consist of 4 bytes recorded in positions 2064 to 2067.
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The error detection code shall be a 32-bit CRC applied on bytes 0 to 2063.
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The least significant bit of a data byte is used first. The EDC codeword
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must be divisible by the check polynomial:
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P(x) = (x^16 + x^15 + x^2 + 1) . (x^16 + x^2 + x + 1)
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The least significant parity bit (x^0) is stored in the most significant
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bit position of byte 2067.
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"
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Used for raw writing in sector.c
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ts B20211:
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Discussion why function crc_32() implements above prescription of ECMA-130.
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See end of this file for the ofunction inherited with libburn-0.2.
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The mentioned polynomial product
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(x^16 + x^15 + x^2 + 1) . (x^16 + x^2 + x + 1)
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yields this sum of x exponents
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32 31 18 16
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18 17 4 2
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17 16 3 1
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16 15 2 0
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======================================
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32 31 16 15 4 3 1 0
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(The number of x^18 and x^17 is divisible by two and thus 0 in GF(2).)
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This yields as 33 bit number:
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0x18001801b
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If above prescription gets implemented straight forward by function
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crc_18001801b(), then its results match the ones of crc_32() with all test
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strings which i could invent.
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The function consists of a conventional polynomial division with reverse
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input order of bits per byte.
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Further it swaps the bits in the resulting 32 bit word. That is because
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sector.c:sector_headers writes the 4 bytes of crc_32() as little endian.
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The ECMA-130 prescription rather demands big endianness and bit swapping
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towards the normal bit order in bytes:
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"The EDC field shall consist of 4 bytes recorded in positions 2064 to 2067.
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[...]
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The least significant parity bit (x^0) is stored in the most
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significant bit position of byte 2067."
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-----------------------------------------------------------------------
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*/
|
||||
|
||||
|
||||
/* Overall bit mirroring of a 32 bit word */
|
||||
unsigned int rfl32(unsigned int acc)
|
||||
{
|
||||
unsigned int inv_acc;
|
||||
int i;
|
||||
|
||||
inv_acc = 0;
|
||||
for (i = 0; i < 32; i++)
|
||||
if (acc & (1 << i))
|
||||
inv_acc |= 1 << (31 - i);
|
||||
return inv_acc;
|
||||
}
|
||||
|
||||
|
||||
/* Plain implementation of polynomial division on a Galois field, where
|
||||
addition and subtraction both are binary exor. Euclidian algorithm.
|
||||
Divisor is (x^16 + x^15 + x^2 + 1) * (x^16 + x^2 + x + 1).
|
||||
|
||||
This is about ten times slower than the table driven algorithm.
|
||||
|
||||
@param flag bit0= do not mirror bits in input bytes and result word
|
||||
(Useful for building the byte indexed CRC table)
|
||||
*/
|
||||
static unsigned int crc_18001801b(unsigned char *data, int count, int flag)
|
||||
{
|
||||
long int acc = 0, i, top;
|
||||
unsigned int inv_acc;
|
||||
|
||||
for (i = 0; i < count * 8 + 32; i++) {
|
||||
top = acc & 0x80000000;
|
||||
acc = (acc << 1);
|
||||
if (i < count * 8) {
|
||||
if (flag & 1)
|
||||
/* Normal bit sequence of input bytes */
|
||||
acc |= ((data[i / 8] >> (7 - (i % 8))) & 1);
|
||||
else
|
||||
/* Bit sequence of input bytes mirrored */
|
||||
acc |= ((data[i / 8] >> (i % 8)) & 1);
|
||||
}
|
||||
if (top)
|
||||
acc ^= 0x18001801b;
|
||||
}
|
||||
|
||||
if (flag & 1)
|
||||
return (unsigned int) acc;
|
||||
|
||||
/* The bits of the whole 32 bit result are mirrored for ECMA-130
|
||||
output compliance and for sector.c habit to store CRC little endian
|
||||
although ECMA-130 prescribes it big endian.
|
||||
*/
|
||||
inv_acc = rfl32((unsigned int) acc);
|
||||
return inv_acc;
|
||||
}
|
||||
|
||||
|
||||
/*
|
||||
-----------------------------------------------------------------------
|
||||
|
||||
Above discussion why crc_ccitt() and crc_11021() yield identical results
|
||||
can be changed from 16 bit to 32 bit by chosing h(a) and l(a) so that:
|
||||
r(a) = h(a) * X * X * X + l(a)
|
||||
h(a) corresponds to the highest byte of crc(A), whereas l(a) corresponds
|
||||
to the lower three bytes of crc(A).
|
||||
|
||||
This yields
|
||||
r(a * X + b)
|
||||
= l(a) * X + r(h(a) * X * X * X * X + b)
|
||||
|
||||
h(a) * X * X * X * X is the polynomial representation of the high byte of
|
||||
32 bit word crc(A).
|
||||
So in the world of bit patterns we have:
|
||||
|
||||
crc(byte string A expanded by byte B)
|
||||
= (lowest_three_bytes(crc(A)) << 8) ^ crc(high_byte(crc(A)) ^ B)
|
||||
|
||||
|
||||
Regrettably this does not yet account for the byte-internal mirroring of
|
||||
bits during the conversion from bit pattern to polynomial, and during
|
||||
conversion from polynomial residue to bit pattern.
|
||||
|
||||
Be rfl8(D) the result of byte-internal mirroring of bit pattern D,
|
||||
and mirr8(d) its corresponding polynom.
|
||||
|
||||
Be now h(a) and l(a) chosen so that: r(mirr8(a)) = h(a) * X * X * X + l(a)
|
||||
This corresponds to highest byte and lower three bytes of crc(A).
|
||||
|
||||
r(mirr8(a) * X + mirr8(b))
|
||||
= r(h(a) * X * X * X * X) + r(l(a) * X) + r(mirr8(b))
|
||||
= l(a)) * X + r(h(a) * X * X * X * X + mirr8(b))
|
||||
|
||||
The corresponding bit pattern operation is
|
||||
|
||||
crc(mirrored byte string A expanded by mirrored byte B)
|
||||
= (lowest_three_bytes(crc(A)) << 8) ^ crc(high_byte(crc(A)) ^ rfl8(B))
|
||||
|
||||
This demands a final result mirroring to meet the ECMA-130 prescription.
|
||||
|
||||
rfl8() can be implemented as lookup table.
|
||||
|
||||
The following function crc32_by_tab() yields the same results as functions
|
||||
crc_18001801b() and crc_32():
|
||||
|
||||
-----------------------------------------------------------------------
|
||||
*/
|
||||
|
||||
|
||||
/* Byte-internal bit mirroring function.
|
||||
*/
|
||||
unsigned int rfl8(unsigned int acc)
|
||||
{
|
||||
unsigned int inv_acc;
|
||||
int i, j;
|
||||
|
||||
inv_acc = 0;
|
||||
for (j = 0; j < 4; j++)
|
||||
for (i = 0; i < 8; i++)
|
||||
if (acc & (1 << (i + 8 * j)))
|
||||
inv_acc |= 1 << ((7 - i) + 8 * j);
|
||||
return inv_acc;
|
||||
}
|
||||
|
||||
|
||||
#ifdef Libburn_with_crc_illustratioN
|
||||
/* Not needed for libburn. The new implementation of function crc_32() is the
|
||||
one that is used.
|
||||
*/
|
||||
|
||||
unsigned int crc32_by_tab(unsigned char *data, int count, int flag)
|
||||
{
|
||||
static unsigned int crc_tab[256], tab_initialized = 0;
|
||||
static unsigned char mirr_tab[256];
|
||||
unsigned int acc, inv_acc;
|
||||
unsigned char b[1];
|
||||
int i;
|
||||
|
||||
if (!tab_initialized) {
|
||||
for (i = 0; i < 256; i++) {
|
||||
b[0] = i;
|
||||
/* Create table of non-mirrored 0x18001801b residues */
|
||||
crc_tab[i] = crc_18001801b(b, 1, 1);
|
||||
/* Create table of mirrored byte values */
|
||||
mirr_tab[i] = rfl8(i);
|
||||
}
|
||||
tab_initialized = 1;
|
||||
}
|
||||
|
||||
acc = 0;
|
||||
for (i = 0; i < count; i++)
|
||||
acc = (acc << 8) ^ crc_tab[(acc >> 24) ^ mirr_tab[data[i]]];
|
||||
|
||||
/* The bits of the whole 32 bit result are mirrored for ECMA-130
|
||||
output compliance and for sector.c habit to store CRC little endian
|
||||
although ECMA-130 prescribes it big endian.
|
||||
*/
|
||||
inv_acc = rfl32((unsigned int) acc);
|
||||
return inv_acc;
|
||||
}
|
||||
|
||||
#endif /* Libburn_with_crc_illustratioN */
|
||||
|
||||
|
||||
/*
|
||||
-----------------------------------------------------------------------
|
||||
|
||||
Above function yields sufficient performance, nevertheless the old function
|
||||
crc_32() (see below) is faster by avoiding the additional mirror table
|
||||
lookup.
|
||||
A test with 10 times 650 MB on 3000 MHz amd64:
|
||||
crc_18001801b : 187 s
|
||||
crc32_by_tab : 27 s
|
||||
crc_32 : 16 s
|
||||
|
||||
So how does crc_32() avoid the application of bit mirroring to B ?.
|
||||
|
||||
Inherited crc_32() performs
|
||||
crc = crc32_table[(crc ^ *data++) & 0xffL] ^ (crc >> 8);
|
||||
|
||||
Above function crc32_by_tab() would be
|
||||
crc = crc_tab[(crc >> 24) ^ mirr_tab[*data++]] ^ (crc << 8);
|
||||
|
||||
The shortcut does not change the polynomial representation of the algorithm
|
||||
or the mapping from and to bit patterns. It only mirrors the bit direction
|
||||
in the bytes and in the 32-bit words which are involved in the bit pattern
|
||||
computation. This affects input (which is desired), intermediate state
|
||||
(which is as good as unmirrored), and final output (which would be slightly
|
||||
undesirable if libburn could not use the mirrored result anyway).
|
||||
|
||||
Instead of the high byte (crc >> 24), the abbreviated algorithm uses
|
||||
the low byte of the mirrored intermediate checksum (crc & 0xffL).
|
||||
Instead of shifting the other three intermediate bytes to the left
|
||||
(crc << 8), the abbreviated algorithm shifts them to the right (crc >> 8).
|
||||
In both cases they overwrite the single byte that was used for computing
|
||||
the table index.
|
||||
|
||||
The byte indexed table of CRC values needs to hold mirrored 32 bit values.
|
||||
The byte index [(crc ^ *data++) & 0xffL] would need to be mirrored, which
|
||||
would eat up the gain of not mirroring the input bytes. But this mirroring
|
||||
can be pre-computed into the table by exchanging each value with the value
|
||||
of its mirrored index.
|
||||
|
||||
So this relation exists between the CRC table crc_tab[] of crc32_by_tab()
|
||||
and the table crc32_table[] of the abbreviated algorithm crc_32():
|
||||
|
||||
crc_tab[i] == rfl32(crc32_table[rfl8(i)])
|
||||
|
||||
for i={0..255}.
|
||||
|
||||
I compared the generated table in crc32_by_tab() by this test
|
||||
for (i = 0; i < 256; i++) {
|
||||
if (rfl32(crc_tab[rfl8(i)]) != crc32_table[i] ||
|
||||
crc_tab[i] != rfl32(crc32_table[rfl8(i)])) {
|
||||
printf("DEVIATION : i = %d\n", i);
|
||||
exit(1);
|
||||
}
|
||||
}
|
||||
No screaming abort happened.
|
||||
|
||||
-----------------------------------------------------------------------
|
||||
*/
|
||||
|
||||
/* This is my own mirrored table implementation for which i claim copyright.
|
||||
With gcc -O2 it shows the same efficiency as the inherited implementation
|
||||
below. With -O3, -O1, or -O0 it is only slightly slower.
|
||||
|
||||
Copyright (c) 2012 Thomas Schmitt <scdbackup@gmx.net>
|
||||
*/
|
||||
unsigned int crc_32(unsigned char *data, int count)
|
||||
{
|
||||
static unsigned int crc_tab[256], tab_initialized = 0;
|
||||
unsigned int acc = 0;
|
||||
unsigned char b[1];
|
||||
int i;
|
||||
|
||||
if (!tab_initialized) {
|
||||
/* Create table of mirrored 0x18001801b residues in
|
||||
bit-mirrored index positions.
|
||||
*/
|
||||
for (i = 0; i < 256; i++) {
|
||||
b[0] = i;
|
||||
crc_tab[rfl8(i)] = rfl32(crc_18001801b(b, 1, 1));
|
||||
}
|
||||
tab_initialized = 1;
|
||||
}
|
||||
for (i = 0; i < count; i++)
|
||||
acc = (acc >> 8) ^ crc_tab[(acc & 0xff) ^ data[i]];
|
||||
|
||||
/* The bits of the whole 32 bit result stay mirrored for ECMA-130
|
||||
output 8-bit mirroring and for sector.c habit to store the CRC
|
||||
little endian although ECMA-130 prescribes it big endian.
|
||||
*/
|
||||
return acc;
|
||||
}
|
||||
|
||||
|
||||
/*
|
||||
-----------------------------------------------------------------------
|
||||
|
||||
This was the function inherited with libburn-0.2 which implements the
|
||||
abbreviated algorithm. Its obscure existence led me to above insights.
|
||||
My compliments to the (unknown) people who invented this.
|
||||
|
||||
unsigned long crc32_table[256] = {
|
||||
0x00000000L, 0x90910101L, 0x91210201L, 0x01B00300L,
|
||||
0x92410401L, 0x02D00500L, 0x03600600L, 0x93F10701L,
|
||||
0x94810801L, 0x04100900L, 0x05A00A00L, 0x95310B01L,
|
||||
@ -106,132 +613,16 @@ unsigned long crc32_table[256] = {
|
||||
0xE541F401L, 0x75D0F500L, 0x7460F600L, 0xE4F1F701L,
|
||||
0xE381F801L, 0x7310F900L, 0x72A0FA00L, 0xE231FB01L,
|
||||
0x71C0FC00L, 0xE151FD01L, 0xE0E1FE01L, 0x7070FF00L
|
||||
};
|
||||
};
|
||||
|
||||
|
||||
/* Exploration ts B00214 :
|
||||
ECMA-130, 22.3.6 "CRC field"
|
||||
Generating polynomial: x^16 + x^12 + x^5 + 1
|
||||
Also known as CRC-16-CCITT, CRC-CCITT
|
||||
|
||||
Used in libburn for raw write modes in sector.c.
|
||||
There is also disabled code in read.c which would use it.
|
||||
|
||||
|
||||
ts B11222:
|
||||
The same algorithm is prescribed for CD-TEXT.
|
||||
libburn/cdtext.c uses a simple bit shifting function : crc_11021()
|
||||
|
||||
ts B20211:
|
||||
Discussion why both are equivalent in respect to their result:
|
||||
|
||||
Both map the bits of the given bytes to a polynomial over the finite field
|
||||
of two elements. If bytes 0 .. M are given, then bit n of byte m is mapped
|
||||
to the coefficient of x exponent (n + ((M - m) * 8) + 16).
|
||||
I.e. they translate the bits into a polynomial with the highest bit
|
||||
becomming the coefficient of the highest power of x. Then this polynomial
|
||||
is multiplied by (x exp 16).
|
||||
|
||||
The set of all such polynomials forms a commutative ring. Its addition
|
||||
corresponds to bitwise exclusive or. Addition and subtraction are identical.
|
||||
Multiplication with polynomials of only one single non-zero coefficient
|
||||
corresponds to leftward bit shifting by the exponent of that coefficient.
|
||||
The same rules apply as with elementary school arithmetics on integer
|
||||
numbers, but with surprising results due to the finite nature of the
|
||||
coefficient number space.
|
||||
Note that multiplication is _not_ an iteration of addition here.
|
||||
|
||||
Function crc_11021() performs a division with residue by the euclidian
|
||||
algorithm. I.e. it splits polynomial d into quotient q(d) and residue r(d)
|
||||
in respect to the polynomial p = x exp 16 + x exp 12 + x exp 5 + x exp 0
|
||||
d = p * q(d) + r(d)
|
||||
where r(d) is of a polynomial rank lower than p, i.e. only x exp 15
|
||||
or lower have non-zero coefficients.
|
||||
The checksum crc(D) is derived by reverse mapping (r(d) * (x exp 16)).
|
||||
I.e. by mapping the coefficient of (x exp n) to bit n of the 16 bit word
|
||||
crc(D).
|
||||
The function result is the bit-wise complement of crc(D).
|
||||
|
||||
Function crc_ccitt uses a table ccitt_table of r(d) values for the
|
||||
polynomials d which represent the single byte values 0x00 to 0xff.
|
||||
It computes r(d) by computing the residues of an iteratively expanded
|
||||
polynomial. The expansion of the processed byte string A by the next byte B
|
||||
from the input byte string happens by shifting the string 8 bits to the
|
||||
left, and by oring B onto bits 0 to 7.
|
||||
In the space of polynomials, the already processed polynomial "a" (image of
|
||||
byte string A) gets expanded by polynomial b (the image of byte B) like this
|
||||
a * X + b
|
||||
where X is (x exp 8), i.e. the single coefficient polynomial of rank 8.
|
||||
|
||||
The following argumentation uses algebra with commutative, associative
|
||||
and distributive laws.
|
||||
Valid especially with polynomials is this rule:
|
||||
(1): r(a + b) = r(a) + r(b)
|
||||
because r(a) and r(b) are of rank lower than rank(p) and
|
||||
rank(a + b) <= max(rank(a), rank(b))
|
||||
Further valid are:
|
||||
(2): r(a) = r(r(a))
|
||||
(3): r(p * a) = 0
|
||||
|
||||
The residue of this expanded polynomials can be expressed by means of the
|
||||
residue r(a) which is known from the previous iteration step, and the
|
||||
residue r(b) which may be looked up in ccitt_table.
|
||||
r(a * X + b)
|
||||
= r(p * q(a) * X + r(a) * X + p * q(b) + r(b))
|
||||
|
||||
Applying rule (1):
|
||||
= r(p * q(a) * X) + r(r(a) * X) + r(p * q(b)) + r(r(b))
|
||||
|
||||
Rule (3) and rule (2):
|
||||
= r(r(a) * X) + r(b)
|
||||
|
||||
Be h(a) and l(a) chosen so that: r(a) = h(a) * X + l(a),
|
||||
and l(a) has zero coefficients above (x exp 7), and h(a) * X has zero
|
||||
coefficients below (x exp 8). (They correspond to the high and low byte
|
||||
of the 16 bit word crc(A).)
|
||||
Now we have:
|
||||
= r(h(a) * X * X) + r(l(a) * X) + r(b)
|
||||
|
||||
Since the rank of l(a) is lower than 8, rank of l(a) * X is lower than 16.
|
||||
Thus it cannot be divisible by p which has rank 16.
|
||||
So: r(l(a) * X) = l(a) * X
|
||||
This yields
|
||||
= l(a) * X + r(h(a) * X * X + b)
|
||||
|
||||
h(a) * X * X is the polynomial representation of the high byte of 16 bit
|
||||
word crc(A).
|
||||
So in the world of bit patterns we have:
|
||||
|
||||
crc(byte string A expanded by byte B)
|
||||
= (low_byte(crc(A)) << 8) ^ crc(high_byte(crc(A)) ^ B)
|
||||
|
||||
And this is what function crc_ccitt() does, modulo swapping the exor
|
||||
operants and some C obfuscation.
|
||||
|
||||
*/
|
||||
unsigned short crc_ccitt(unsigned char *q, int len)
|
||||
{
|
||||
unsigned short crc = 0;
|
||||
|
||||
while (len-- > 0)
|
||||
crc = ccitt_table[(crc >> 8 ^ *q++) & 0xff] ^ (crc << 8);
|
||||
return ~crc;
|
||||
}
|
||||
|
||||
|
||||
/* Exploration ts B00214 :
|
||||
ECMA-130, 14.3 "EDC field"
|
||||
"The EDC codeword must be divisible by the check polynomial:
|
||||
P(x) = (x^16 + x^15 + x^2 + 1) . (x^16 + x^2 + x + 1)
|
||||
"
|
||||
|
||||
Used for raw writing in sector.c
|
||||
*/
|
||||
unsigned int crc_32(unsigned char *data, int len)
|
||||
{
|
||||
unsigned int crc_32(unsigned char *data, int len)
|
||||
{
|
||||
unsigned int crc = 0;
|
||||
|
||||
while (len-- > 0)
|
||||
crc = crc32_table[(crc ^ *data++) & 0xffL] ^ (crc >> 8);
|
||||
return crc;
|
||||
}
|
||||
}
|
||||
*/
|
||||
|
||||
|
||||
|
@ -1,11 +1,21 @@
|
||||
/* -*- indent-tabs-mode: t; tab-width: 8; c-basic-offset: 8; -*- */
|
||||
|
||||
/* Copyright (c) 2004 - 2006 Derek Foreman, Ben Jansens
|
||||
Copyright (c) 2012 Thomas Schmitt <scdbackup@gmx.net>
|
||||
Provided under GPL version 2 or later.
|
||||
*/
|
||||
#ifndef BURN__CRC_H
|
||||
#define BURN__CRC_H
|
||||
|
||||
|
||||
#ifdef Xorriso_standalonE
|
||||
/* Source module crc.c of yet unclear ancestry is excluded from GNU xorriso */
|
||||
/* ts B20219 : The functions have been re-implemented from scratch after
|
||||
studying texts about CRC computation and understanding the
|
||||
meaning of the underlying ECMA-130 specs.
|
||||
Nevertheless, there is no need to include them into xorriso
|
||||
because it does neither CD-TEXT nor raw CD writing.
|
||||
*/
|
||||
#ifndef Libburn_no_crc_C
|
||||
#define Libburn_no_crc_C 1
|
||||
#endif
|
||||
|
Loading…
Reference in New Issue
Block a user