Introduce new page checksum algorithm and module.

Isolate checksum calculation to its own module, so that bufpage
knows little if anything about the details of the calculation.

This implementation is a modified FNV-1a hash checksum, details
of which are given in the new checksum.c header comments.

Basic implementation only, so we fix the output value.

Later related commits will add version numbers to pg_control,
compiler optimization flags and memory barriers.

Ants Aasma, reviewed by Jeff Davis and Simon Riggs

src/backend/storage/page/Makefile

@@ -12,6 +12,6 @@ subdir = src/backend/storage/page
 top_builddir = ../../../..
 include $(top_builddir)/src/Makefile.global
 
-OBJS =  bufpage.o itemptr.o
+OBJS =  bufpage.o checksum.o itemptr.o
 
 include $(top_srcdir)/src/backend/common.mk

src/backend/storage/page/bufpage.c

@@ -16,6 +16,7 @@
 
 #include "access/htup_details.h"
 #include "access/xlog.h"
+#include "storage/checksum.h"
 
 bool ignore_checksum_failure = false;
 
@@ -948,33 +949,30 @@ PageSetChecksumInplace(Page page, BlockNumber blkno)
 static uint16
 PageCalcChecksum16(Page page, BlockNumber blkno)
 {
-	pg_crc32    		crc;
-	PageHeader	p = (PageHeader) page;
+	PageHeader	phdr   = (PageHeader) page;
+	uint16		save_checksum;
+	uint32		checksum;
 
 	/* only calculate the checksum for properly-initialized pages */
 	Assert(!PageIsNew(page));
 
-	INIT_CRC32(crc);
-
 	/*
-	 * Initialize the checksum calculation with the block number. This helps
-	 * catch corruption from whole blocks being transposed with other whole
-	 * blocks.
+	 * Save pd_checksum and set it to zero, so that the checksum calculation
+	 * isn't affected by the checksum stored on the page. We do this to
+	 * allow optimization of the checksum calculation on the whole block
+	 * in one go.
 	 */
-	COMP_CRC32(crc, &blkno, sizeof(blkno));
+	save_checksum = phdr->pd_checksum;
+	phdr->pd_checksum = 0;
+	checksum = checksum_block(page, BLCKSZ);
+	phdr->pd_checksum = save_checksum;
 
-	/*
-	 * Now add in the LSN, which is always the first field on the page.
-	 */
-	COMP_CRC32(crc, page, sizeof(p->pd_lsn));
+	/* mix in the block number to detect transposed pages */
+	checksum ^= blkno;
 
 	/*
-	 * Now add the rest of the page, skipping the pd_checksum field.
+	 * Reduce to a uint16 (to fit in the pd_checksum field) with an offset of
+	 * one. That avoids checksums of zero, which seems like a good idea.
 	 */
-	COMP_CRC32(crc, page + sizeof(p->pd_lsn) + sizeof(p->pd_checksum),
-				  BLCKSZ - sizeof(p->pd_lsn) - sizeof(p->pd_checksum));
-
-	FIN_CRC32(crc);
-
-	return (uint16) crc;
+	return (checksum % 65535) + 1;
 }

src/backend/storage/page/checksum.c

+/*-------------------------------------------------------------------------
+ *
+ * checksum.c
+ *	  Checksum implementation for data pages.
+ *
+ * Portions Copyright (c) 1996-2013, PostgreSQL Global Development Group
+ * Portions Copyright (c) 1994, Regents of the University of California
+ *
+ *
+ * IDENTIFICATION
+ *	  src/backend/storage/page/checksum.c
+ *
+ *-------------------------------------------------------------------------
+ *
+ * Checksum algorithm
+ *
+ * The algorithm used to checksum pages is chosen for very fast calculation.
+ * Workloads where the database working set fits into OS file cache but not
+ * into shared buffers can read in pages at a very fast pace and the checksum
+ * algorithm itself can become the largest bottleneck.
+ *
+ * The checksum algorithm itself is based on the FNV-1a hash (FNV is shorthand
+ * for Fowler/Noll/Vo) The primitive of a plain FNV-1a hash folds in data 1
+ * byte at a time according to the formula:
+ *
+ *     hash = (hash ^ value) * FNV_PRIME
+ *
+ * FNV-1a algorithm is described at http://www.isthe.com/chongo/tech/comp/fnv/
+ *
+ * PostgreSQL doesn't use FNV-1a hash directly because it has bad mixing of
+ * high bits - high order bits in input data only affect high order bits in
+ * output data. To resolve this we xor in the value prior to multiplication
+ * shifted right by 17 bits. The number 17 was chosen because it doesn't
+ * have common denominator with set bit positions in FNV_PRIME and empirically
+ * provides the fastest mixing for high order bits of final iterations quickly
+ * avalanche into lower positions. For performance reasons we choose to combine
+ * 4 bytes at a time. The actual hash formula used as the basis is:
+ *
+ *     hash = (hash ^ value) * FNV_PRIME ^ ((hash ^ value) >> 17)
+ *
+ * The main bottleneck in this calculation is the multiplication latency. To
+ * hide the latency and to make use of SIMD parallelism multiple hash values
+ * are calculated in parallel. The page is treated as a 32 column two
+ * dimensional array of 32 bit values. Each column is aggregated separately
+ * into a partial checksum. Each partial checksum uses a different initial
+ * value (offset basis in FNV terminology). The initial values actually used
+ * were chosen randomly, as the values themselves don't matter as much as that
+ * they are different and don't match anything in real data. After initializing
+ * partial checksums each value in the column is aggregated according to the
+ * above formula. Finally two more iterations of the formula are performed with
+ * value 0 to mix the bits of the last value added.
+ *
+ * The partial checksums are then folded together using xor to form a single
+ * 32-bit checksum. The caller can safely reduce the value to 16 bits
+ * using modulo 2^16-1. That will cause a very slight bias towards lower
+ * values but this is not significant for the performance of the
+ * checksum.
+ *
+ * The algorithm choice was based on what instructions are available in SIMD
+ * instruction sets. This meant that a fast and good algorithm needed to use
+ * multiplication as the main mixing operator. The simplest multiplication
+ * based checksum primitive is the one used by FNV. The prime used is chosen
+ * for good dispersion of values. It has no known simple patterns that result
+ * in collisions. Test of 5-bit differentials of the primitive over 64bit keys
+ * reveals no differentials with 3 or more values out of 100000 random keys
+ * colliding. Avalanche test shows that only high order bits of the last word
+ * have a bias. Tests of 1-4 uncorrelated bit errors, stray 0 and 0xFF bytes,
+ * overwriting page from random position to end with 0 bytes, and overwriting
+ * random segments of page with 0x00, 0xFF and random data all show optimal
+ * 2e-16 false positive rate within margin of error.
+ *
+ * Vectorization of the algorithm requires 32bit x 32bit -> 32bit integer
+ * multiplication instruction. As of 2013 the corresponding instruction is
+ * available on x86 SSE4.1 extensions (pmulld) and ARM NEON (vmul.i32).
+ * Vectorization requires a compiler to do the vectorization for us. For recent
+ * GCC versions the flags -msse4.1 -funroll-loops -ftree-vectorize are enough
+ * to achieve vectorization.
+ *
+ * The optimal amount of parallelism to use depends on CPU specific instruction
+ * latency, SIMD instruction width, throughput and the amount of registers
+ * available to hold intermediate state. Generally, more parallelism is better
+ * up to the point that state doesn't fit in registers and extra load-store
+ * instructions are needed to swap values in/out. The number chosen is a fixed
+ * part of the algorithm because changing the parallelism changes the checksum
+ * result.
+ *
+ * The parallelism number 32 was chosen based on the fact that it is the
+ * largest state that fits into architecturally visible x86 SSE registers while
+ * leaving some free registers for intermediate values. For future processors
+ * with 256bit vector registers this will leave some performance on the table.
+ * When vectorization is not available it might be beneficial to restructure
+ * the computation to calculate a subset of the columns at a time and perform
+ * multiple passes to avoid register spilling. This optimization opportunity
+ * is not used. Current coding also assumes that the compiler has the ability
+ * to unroll the inner loop to avoid loop overhead and minimize register
+ * spilling. For less sophisticated compilers it might be beneficial to manually
+ * unroll the inner loop.
+ */
+#include "postgres.h"
+
+#include "storage/checksum.h"
+
+/* number of checksums to calculate in parallel */
+#define N_SUMS 32
+/* prime multiplier of FNV-1a hash */
+#define FNV_PRIME 16777619
+
+/*
+ * Base offsets to initialize each of the parallel FNV hashes into a
+ * different initial state.
+ */
+static const uint32 checksumBaseOffsets[N_SUMS] = {
+	0x5B1F36E9, 0xB8525960, 0x02AB50AA, 0x1DE66D2A,
+	0x79FF467A, 0x9BB9F8A3, 0x217E7CD2, 0x83E13D2C,
+	0xF8D4474F, 0xE39EB970, 0x42C6AE16, 0x993216FA,
+	0x7B093B5D, 0x98DAFF3C, 0xF718902A, 0x0B1C9CDB,
+	0xE58F764B, 0x187636BC, 0x5D7B3BB1, 0xE73DE7DE,
+	0x92BEC979, 0xCCA6C0B2, 0x304A0979, 0x85AA43D4,
+	0x783125BB, 0x6CA8EAA2, 0xE407EAC6, 0x4B5CFC3E,
+	0x9FBF8C76, 0x15CA20BE, 0xF2CA9FD3, 0x959BD756
+};
+
+/*
+ * Calculate one round of the checksum.
+ */
+#define CHECKSUM_COMP(checksum, value) do {\
+	uint32 __tmp = (checksum) ^ (value);\
+	(checksum) = __tmp * FNV_PRIME ^ (__tmp >> 17);\
+} while (0)
+
+uint32
+checksum_block(char *data, uint32 size)
+{
+	uint32 sums[N_SUMS];
+	uint32 (*dataArr)[N_SUMS] = (uint32 (*)[N_SUMS]) data;
+	uint32 result = 0;
+	int i, j;
+
+	/* ensure that the size is compatible with the algorithm */
+	Assert((size % (sizeof(uint32)*N_SUMS)) == 0);
+
+	/* initialize partial checksums to their corresponding offsets */
+	memcpy(sums, checksumBaseOffsets, sizeof(checksumBaseOffsets));
+
+	/* main checksum calculation */
+	for (i = 0; i < size/sizeof(uint32)/N_SUMS; i++)
+		for (j = 0; j < N_SUMS; j++)
+			CHECKSUM_COMP(sums[j], dataArr[i][j]);
+
+	/* finally add in two rounds of zeroes for additional mixing */
+	for (i = 0; i < 2; i++)
+		for (j = 0; j < N_SUMS; j++)
+			CHECKSUM_COMP(sums[j], 0);
+
+	/* xor fold partial checksums together */
+	for (i = 0; i < N_SUMS; i++)
+		result ^= sums[i];
+
+	return result;
+}

src/include/storage/checksum.h

+/*-------------------------------------------------------------------------
+ *
+ * checksum.h
+ *	  Checksum implementation for data pages.
+ *
+ *
+ * Portions Copyright (c) 1996-2013, PostgreSQL Global Development Group
+ * Portions Copyright (c) 1994, Regents of the University of California
+ *
+ * src/include/storage/checksum.h
+ *
+ *-------------------------------------------------------------------------
+ */
+#ifndef CHECKSUM_H
+#define CHECKSUM_H
+
+/*
+ * Fowler-Noll-Vo 1a block checksum algorithm. The data argument should be
+ * aligned on a 4-byte boundary.
+ */
+extern uint32 checksum_block(char *data, uint32 size);
+
+#endif   /* CHECKSUM_H */