lib/x86/crc32_pclmul_template.h

/*
 * x86/crc32_pclmul_template.h
 *
 * Copyright 2016 Eric Biggers
 *
 * Permission is hereby granted, free of charge, to any person
 * obtaining a copy of this software and associated documentation
 * files (the "Software"), to deal in the Software without
 * restriction, including without limitation the rights to use,
 * copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the
 * Software is furnished to do so, subject to the following
 * conditions:
 *
 * The above copyright notice and this permission notice shall be
 * included in all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
 * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
 * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
 * HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
 * WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
 * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
 * OTHER DEALINGS IN THE SOFTWARE.
 */

#include <wmmintrin.h>

/*
 * CRC-32 folding with PCLMULQDQ.
 *
 * The basic idea is to repeatedly "fold" each 512 bits into the next 512 bits,
 * producing an abbreviated message which is congruent the original message
 * modulo the generator polynomial G(x).
 *
 * Folding each 512 bits is implemented as eight 64-bit folds, each of which
 * uses one carryless multiplication instruction.  It's expected that CPUs may
 * be able to execute some of these multiplications in parallel.
 *
 * Explanation of "folding": let A(x) be 64 bits from the message, and let B(x)
 * be 95 bits from a constant distance D later in the message.  The relevant
 * portion of the message can be written as:
 *
 *	M(x) = A(x)*x^D + B(x)
 *
 * ... where + and * represent addition and multiplication, respectively, of
 * polynomials over GF(2).  Note that when implemented on a computer, these
 * operations are equivalent to XOR and carryless multiplication, respectively.
 *
 * For the purpose of CRC calculation, only the remainder modulo the generator
 * polynomial G(x) matters:
 *
 *	M(x) mod G(x) = (A(x)*x^D + B(x)) mod G(x)
 *
 * Since the modulo operation can be applied anywhere in a sequence of additions
 * and multiplications without affecting the result, this is equivalent to:
 *
 *	M(x) mod G(x) = (A(x)*(x^D mod G(x)) + B(x)) mod G(x)
 *
 * For any D, 'x^D mod G(x)' will be a polynomial with maximum degree 31, i.e.
 * a 32-bit quantity.  So 'A(x) * (x^D mod G(x))' is equivalent to a carryless
 * multiplication of a 64-bit quantity by a 32-bit quantity, producing a 95-bit
 * product.  Then, adding (XOR-ing) the product to B(x) produces a polynomial
 * with the same length as B(x) but with the same remainder as 'A(x)*x^D +
 * B(x)'.  This is the basic fold operation with 64 bits.
 *
 * Note that the carryless multiplication instruction PCLMULQDQ actually takes
 * two 64-bit inputs and produces a 127-bit product in the low-order bits of a
 * 128-bit XMM register.  This works fine, but care must be taken to account for
 * "bit endianness".  With the CRC version implemented here, bits are always
 * ordered such that the lowest-order bit represents the coefficient of highest
 * power of x and the highest-order bit represents the coefficient of the lowest
 * power of x.  This is backwards from the more intuitive order.  Still,
 * carryless multiplication works essentially the same either way.  It just must
 * be accounted for that when we XOR the 95-bit product in the low-order 95 bits
 * of a 128-bit XMM register into 128-bits of later data held in another XMM
 * register, we'll really be XOR-ing the product into the mathematically higher
 * degree end of those later bits, not the lower degree end as may be expected.
 *
 * So given that caveat and the fact that we process 512 bits per iteration, the
 * 'D' values we need for the two 64-bit halves of each 128 bits of data are:
 *
 *	D = (512 + 95) - 64	 for the higher-degree half of each 128 bits,
 *				 i.e. the lower order bits in the XMM register
 *
 *	D = (512 + 95) - 128	 for the lower-degree half of each 128 bits,
 *				 i.e. the higher order bits in the XMM register
 *
 * The required 'x^D mod G(x)' values were precomputed.
 *
 * When <= 512 bits remain in the message, we finish up by folding across
 * smaller distances.  This works similarly; the distance D is just different,
 * so different constant multipliers must be used.  Finally, once the remaining
 * message is just 64 bits, it is is reduced to the CRC-32 using Barrett
 * reduction (explained later).
 *
 * For more information see the original paper from Intel:
 *	"Fast CRC Computation for Generic Polynomials Using PCLMULQDQ Instruction"
 *	December 2009
 *	http://www.intel.com/content/dam/www/public/us/en/documents/white-papers/fast-crc-computation-generic-polynomials-pclmulqdq-paper.pdf
 */
static u32 ATTRIBUTES
FUNCNAME_ALIGNED(u32 remainder, const __m128i *p, size_t nr_segs)
{
	/* Constants precomputed by gen_crc32_multipliers.c.  Do not edit! */
	const __v2di multipliers_4 = (__v2di){ 0x8F352D95, 0x1D9513D7 };
	const __v2di multipliers_2 = (__v2di){ 0xF1DA05AA, 0x81256527 };
	const __v2di multipliers_1 = (__v2di){ 0xAE689191, 0xCCAA009E };
	const __v2di final_multiplier = (__v2di){ 0xB8BC6765 };
	const __m128i mask32 = (__m128i)(__v4si){ 0xFFFFFFFF };
	const __v2di barrett_reduction_constants =
			(__v2di){ 0x00000001F7011641, 0x00000001DB710641 };

	const __m128i * const end = p + nr_segs;
	const __m128i * const end512 = p + (nr_segs & ~3);
	__m128i x0, x1, x2, x3;

	/*
	 * Account for the current 'remainder', i.e. the CRC of the part of the
	 * message already processed.  Explanation: rewrite the message
	 * polynomial M(x) in terms of the first part A(x), the second part
	 * B(x), and the length of the second part in bits |B(x)| >= 32:
	 *
	 *	M(x) = A(x)*x^|B(x)| + B(x)
	 *
	 * Then the CRC of M(x) is:
	 *
	 *	CRC(M(x)) = CRC(A(x)*x^|B(x)| + B(x))
	 *	          = CRC(A(x)*x^32*x^(|B(x)| - 32) + B(x))
	 *	          = CRC(CRC(A(x))*x^(|B(x)| - 32) + B(x))
	 *
	 * Note: all arithmetic is modulo G(x), the generator polynomial; that's
	 * why A(x)*x^32 can be replaced with CRC(A(x)) = A(x)*x^32 mod G(x).
	 *
	 * So the CRC of the full message is the CRC of the second part of the
	 * message where the first 32 bits of the second part of the message
	 * have been XOR'ed with the CRC of the first part of the message.
	 */
	x0 = *p++;
	x0 ^= (__m128i)(__v4si){ remainder };

	if (p > end512) /* only 128, 256, or 384 bits of input? */
		goto _128_bits_at_a_time;
	x1 = *p++;
	x2 = *p++;
	x3 = *p++;

	/* Fold 512 bits at a time */
	for (; p != end512; p += 4) {
		__m128i y0, y1, y2, y3;

		y0 = p[0];
		y1 = p[1];
		y2 = p[2];
		y3 = p[3];

		/*
		 * Note: the immediate constant for PCLMULQDQ specifies which
		 * 64-bit halves of the 128-bit vectors to multiply:
		 *
		 * 0x00 means low halves (higher degree polynomial terms for us)
		 * 0x11 means high halves (lower degree polynomial terms for us)
		 */
		y0 ^= _mm_clmulepi64_si128(x0, multipliers_4, 0x00);
		y1 ^= _mm_clmulepi64_si128(x1, multipliers_4, 0x00);
		y2 ^= _mm_clmulepi64_si128(x2, multipliers_4, 0x00);
		y3 ^= _mm_clmulepi64_si128(x3, multipliers_4, 0x00);
		y0 ^= _mm_clmulepi64_si128(x0, multipliers_4, 0x11);
		y1 ^= _mm_clmulepi64_si128(x1, multipliers_4, 0x11);
		y2 ^= _mm_clmulepi64_si128(x2, multipliers_4, 0x11);
		y3 ^= _mm_clmulepi64_si128(x3, multipliers_4, 0x11);

		x0 = y0;
		x1 = y1;
		x2 = y2;
		x3 = y3;
	}

	/* Fold 512 bits => 128 bits */
	x2 ^= _mm_clmulepi64_si128(x0, multipliers_2, 0x00);
	x3 ^= _mm_clmulepi64_si128(x1, multipliers_2, 0x00);
	x2 ^= _mm_clmulepi64_si128(x0, multipliers_2, 0x11);
	x3 ^= _mm_clmulepi64_si128(x1, multipliers_2, 0x11);
	x3 ^= _mm_clmulepi64_si128(x2, multipliers_1, 0x00);
	x3 ^= _mm_clmulepi64_si128(x2, multipliers_1, 0x11);
	x0 = x3;

_128_bits_at_a_time:
	while (p != end) {
		/* Fold 128 bits into next 128 bits */
		x1 = *p++;
		x1 ^= _mm_clmulepi64_si128(x0, multipliers_1, 0x00);
		x1 ^= _mm_clmulepi64_si128(x0, multipliers_1, 0x11);
		x0 = x1;
	}

	/* Now there are just 128 bits left, stored in 'x0'. */

	/*
	 * Fold 128 => 96 bits.  This also implicitly appends 32 zero bits,
	 * which is equivalent to multiplying by x^32.  This is needed because
	 * the CRC is defined as M(x)*x^32 mod G(x), not just M(x) mod G(x).
	 */
	x0 = _mm_srli_si128(x0, 8) ^
	     _mm_clmulepi64_si128(x0, multipliers_1, 0x10);

	/* Fold 96 => 64 bits */
	x0 = _mm_srli_si128(x0, 4) ^
	     _mm_clmulepi64_si128(x0 & mask32, final_multiplier, 0x00);

        /*
	 * Finally, reduce 64 => 32 bits using Barrett reduction.
	 *
	 * Let M(x) = A(x)*x^32 + B(x) be the remaining message.  The goal is to
	 * compute R(x) = M(x) mod G(x).  Since degree(B(x)) < degree(G(x)):
	 *
	 *	R(x) = (A(x)*x^32 + B(x)) mod G(x)
	 *	     = (A(x)*x^32) mod G(x) + B(x)
	 *
	 * Then, by the Division Algorithm there exists a unique q(x) such that:
	 *
	 *	A(x)*x^32 mod G(x) = A(x)*x^32 - q(x)*G(x)
	 *
	 * Since the left-hand side is of maximum degree 31, the right-hand side
	 * must be too.  This implies that we can apply 'mod x^32' to the
	 * right-hand side without changing its value:
	 *
	 *	(A(x)*x^32 - q(x)*G(x)) mod x^32 = q(x)*G(x) mod x^32
	 *
	 * Note that '+' is equivalent to '-' in polynomials over GF(2).
	 *
	 * We also know that:
	 *
	 *	              / A(x)*x^32 \
	 *	q(x) = floor (  ---------  )
	 *	              \    G(x)   /
	 *
	 * To compute this efficiently, we can multiply the top and bottom by
	 * x^32 and move the division by G(x) to the top:
	 *
	 *	              / A(x) * floor(x^64 / G(x)) \
	 *	q(x) = floor (  -------------------------  )
	 *	              \           x^32            /
	 *
	 * Note that floor(x^64 / G(x)) is a constant.
	 *
	 * So finally we have:
	 *
	 *	                          / A(x) * floor(x^64 / G(x)) \
	 *	R(x) = B(x) + G(x)*floor (  -------------------------  )
	 *	                          \           x^32            /
	 */
	x1 = x0;
	x0 = _mm_clmulepi64_si128(x0 & mask32, barrett_reduction_constants, 0x00);
	x0 = _mm_clmulepi64_si128(x0 & mask32, barrett_reduction_constants, 0x10);
	return _mm_cvtsi128_si32(_mm_srli_si128(x0 ^ x1, 4));
}

#define IMPL_ALIGNMENT		16
#define IMPL_SEGMENT_SIZE	16
#include "../crc32_vec_template.h"