* CDDL HEADER END
*/
/*
- * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
+ * Copyright 2009 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
-#pragma ident "%Z%%M% %I% %E% SMI"
+/*
+ * Fletcher Checksums
+ * ------------------
+ *
+ * ZFS's 2nd and 4th order Fletcher checksums are defined by the following
+ * recurrence relations:
+ *
+ * a = a + f
+ * i i-1 i-1
+ *
+ * b = b + a
+ * i i-1 i
+ *
+ * c = c + b (fletcher-4 only)
+ * i i-1 i
+ *
+ * d = d + c (fletcher-4 only)
+ * i i-1 i
+ *
+ * Where
+ * a_0 = b_0 = c_0 = d_0 = 0
+ * and
+ * f_0 .. f_(n-1) are the input data.
+ *
+ * Using standard techniques, these translate into the following series:
+ *
+ * __n_ __n_
+ * \ | \ |
+ * a = > f b = > i * f
+ * n /___| n - i n /___| n - i
+ * i = 1 i = 1
+ *
+ *
+ * __n_ __n_
+ * \ | i*(i+1) \ | i*(i+1)*(i+2)
+ * c = > ------- f d = > ------------- f
+ * n /___| 2 n - i n /___| 6 n - i
+ * i = 1 i = 1
+ *
+ * For fletcher-2, the f_is are 64-bit, and [ab]_i are 64-bit accumulators.
+ * Since the additions are done mod (2^64), errors in the high bits may not
+ * be noticed. For this reason, fletcher-2 is deprecated.
+ *
+ * For fletcher-4, the f_is are 32-bit, and [abcd]_i are 64-bit accumulators.
+ * A conservative estimate of how big the buffer can get before we overflow
+ * can be estimated using f_i = 0xffffffff for all i:
+ *
+ * % bc
+ * f=2^32-1;d=0; for (i = 1; d<2^64; i++) { d += f*i*(i+1)*(i+2)/6 }; (i-1)*4
+ * 2264
+ * quit
+ * %
+ *
+ * So blocks of up to 2k will not overflow. Our largest block size is
+ * 128k, which has 32k 4-byte words, so we can compute the largest possible
+ * accumulators, then divide by 2^64 to figure the max amount of overflow:
+ *
+ * % bc
+ * a=b=c=d=0; f=2^32-1; for (i=1; i<=32*1024; i++) { a+=f; b+=a; c+=b; d+=c }
+ * a/2^64;b/2^64;c/2^64;d/2^64
+ * 0
+ * 0
+ * 1365
+ * 11186858
+ * quit
+ * %
+ *
+ * So a and b cannot overflow. To make sure each bit of input has some
+ * effect on the contents of c and d, we can look at what the factors of
+ * the coefficients in the equations for c_n and d_n are. The number of 2s
+ * in the factors determines the lowest set bit in the multiplier. Running
+ * through the cases for n*(n+1)/2 reveals that the highest power of 2 is
+ * 2^14, and for n*(n+1)*(n+2)/6 it is 2^15. So while some data may overflow
+ * the 64-bit accumulators, every bit of every f_i effects every accumulator,
+ * even for 128k blocks.
+ *
+ * If we wanted to make a stronger version of fletcher4 (fletcher4c?),
+ * we could do our calculations mod (2^32 - 1) by adding in the carries
+ * periodically, and store the number of carries in the top 32-bits.
+ *
+ * --------------------
+ * Checksum Performance
+ * --------------------
+ *
+ * There are two interesting components to checksum performance: cached and
+ * uncached performance. With cached data, fletcher-2 is about four times
+ * faster than fletcher-4. With uncached data, the performance difference is
+ * negligible, since the cost of a cache fill dominates the processing time.
+ * Even though fletcher-4 is slower than fletcher-2, it is still a pretty
+ * efficient pass over the data.
+ *
+ * In normal operation, the data which is being checksummed is in a buffer
+ * which has been filled either by:
+ *
+ * 1. a compression step, which will be mostly cached, or
+ * 2. a bcopy() or copyin(), which will be uncached (because the
+ * copy is cache-bypassing).
+ *
+ * For both cached and uncached data, both fletcher checksums are much faster
+ * than sha-256, and slower than 'off', which doesn't touch the data at all.
+ */
#include <sys/types.h>
#include <sys/sysmacros.h>
git://git.camperquake.de / zfs.git / blobdiff