+++ /dev/null
-/*
- * Copyright (C) 2013 Andrea Mazzoleni
- *
- * This program is free software: you can redistribute it and/or modify
- * it under the terms of the GNU General Public License as published by
- * the Free Software Foundation, either version 2 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU General Public License for more details.
- */
-
-#include "internal.h"
-#include "gf.h"
-
-/*
- * This is a RAID implementation working in the Galois Field GF(2^8) with
- * the primitive polynomial x^8 + x^4 + x^3 + x^2 + 1 (285 decimal), and
- * supporting up to six parity levels.
- *
- * For RAID5 and RAID6 it works as as described in the H. Peter Anvin's
- * paper "The mathematics of RAID-6" [1]. Please refer to this paper for a
- * complete explanation.
- *
- * To support triple parity, it was first evaluated and then dropped, an
- * extension of the same approach, with additional parity coefficients set
- * as powers of 2^-1, with equations:
- *
- * P = sum(Di)
- * Q = sum(2^i * Di)
- * R = sum(2^-i * Di) with 0<=i<N
- *
- * This approach works well for triple parity and it's very efficient,
- * because we can implement very fast parallel multiplications and
- * divisions by 2 in GF(2^8).
- *
- * It's also similar at the approach used by ZFS RAIDZ3, with the
- * difference that ZFS uses powers of 4 instead of 2^-1.
- *
- * Unfortunately it doesn't work beyond triple parity, because whatever
- * value we choose to generate the power coefficients to compute other
- * parities, the resulting equations are not solvable for some
- * combinations of missing disks.
- *
- * This is expected, because the Vandermonde matrix used to compute the
- * parity has no guarantee to have all submatrices not singular
- * [2, Chap 11, Problem 7] and this is a requirement to have
- * a MDS (Maximum Distance Separable) code [2, Chap 11, Theorem 8].
- *
- * To overcome this limitation, we use a Cauchy matrix [3][4] to compute
- * the parity. A Cauchy matrix has the property to have all the square
- * submatrices not singular, resulting in always solvable equations,
- * for any combination of missing disks.
- *
- * The problem of this approach is that it requires the use of
- * generic multiplications, and not only by 2 or 2^-1, potentially
- * affecting badly the performance.
- *
- * Hopefully there is a method to implement parallel multiplications
- * using SSSE3 or AVX2 instructions [1][5]. Method competitive with the
- * computation of triple parity using power coefficients.
- *
- * Another important property of the Cauchy matrix is that we can setup
- * the first two rows with coeffients equal at the RAID5 and RAID6 approach
- * decribed, resulting in a compatible extension, and requiring SSSE3
- * or AVX2 instructions only if triple parity or beyond is used.
- *
- * The matrix is also adjusted, multipling each row by a constant factor
- * to make the first column of all 1, to optimize the computation for
- * the first disk.
- *
- * This results in the matrix A[row,col] defined as:
- *
- * 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01...
- * 01 02 04 08 10 20 40 80 1d 3a 74 e8 cd 87 13 26 4c 98 2d 5a b4 75...
- * 01 f5 d2 c4 9a 71 f1 7f fc 87 c1 c6 19 2f 40 55 3d ba 53 04 9c 61...
- * 01 bb a6 d7 c7 07 ce 82 4a 2f a5 9b b6 60 f1 ad e7 f4 06 d2 df 2e...
- * 01 97 7f 9c 7c 18 bd a2 58 1a da 74 70 a3 e5 47 29 07 f5 80 23 e9...
- * 01 2b 3f cf 73 2c d6 ed cb 74 15 78 8a c1 17 c9 89 68 21 ab 76 3b...
- *
- * This matrix supports 6 level of parity, one for each row, for up to 251
- * data disks, one for each column, with all the 377,342,351,231 square
- * submatrices not singular, verified also with brute-force.
- *
- * This matrix can be extended to support any number of parities, just
- * adding additional rows, and removing one column for each new row.
- * (see mktables.c for more details in how the matrix is generated)
- *
- * In details, parity is computed as:
- *
- * P = sum(Di)
- * Q = sum(2^i * Di)
- * R = sum(A[2,i] * Di)
- * S = sum(A[3,i] * Di)
- * T = sum(A[4,i] * Di)
- * U = sum(A[5,i] * Di) with 0<=i<N
- *
- * To recover from a failure of six disks at indexes x,y,z,h,v,w,
- * with 0<=x<y<z<h<v<w<N, we compute the parity of the available N-6
- * disks as:
- *
- * Pa = sum(Di)
- * Qa = sum(2^i * Di)
- * Ra = sum(A[2,i] * Di)
- * Sa = sum(A[3,i] * Di)
- * Ta = sum(A[4,i] * Di)
- * Ua = sum(A[5,i] * Di) with 0<=i<N,i!=x,i!=y,i!=z,i!=h,i!=v,i!=w.
- *
- * And if we define:
- *
- * Pd = Pa + P
- * Qd = Qa + Q
- * Rd = Ra + R
- * Sd = Sa + S
- * Td = Ta + T
- * Ud = Ua + U
- *
- * we can sum these two sets of equations, obtaining:
- *
- * Pd = Dx + Dy + Dz + Dh + Dv + Dw
- * Qd = 2^x * Dx + 2^y * Dy + 2^z * Dz + 2^h * Dh + 2^v * Dv + 2^w * Dw
- * Rd = A[2,x] * Dx + A[2,y] * Dy + A[2,z] * Dz + A[2,h] * Dh + A[2,v] * Dv + A[2,w] * Dw
- * Sd = A[3,x] * Dx + A[3,y] * Dy + A[3,z] * Dz + A[3,h] * Dh + A[3,v] * Dv + A[3,w] * Dw
- * Td = A[4,x] * Dx + A[4,y] * Dy + A[4,z] * Dz + A[4,h] * Dh + A[4,v] * Dv + A[4,w] * Dw
- * Ud = A[5,x] * Dx + A[5,y] * Dy + A[5,z] * Dz + A[5,h] * Dh + A[5,v] * Dv + A[5,w] * Dw
- *
- * A linear system always solvable because the coefficients matrix is
- * always not singular due the properties of the matrix A[].
- *
- * Resulting speed in x64, with 8 data disks, using a stripe of 256 KiB,
- * for a Core i5-4670K Haswell Quad-Core 3.4GHz is:
- *
- * int8 int32 int64 sse2 ssse3 avx2
- * gen1 13339 25438 45438 50588
- * gen2 4115 6514 21840 32201
- * gen3 814 10154 18613
- * gen4 620 7569 14229
- * gen5 496 5149 10051
- * gen6 413 4239 8190
- *
- * Values are in MiB/s of data processed by a single thread, not counting
- * generated parity.
- *
- * You can replicate these results in your machine using the
- * "raid/test/speedtest.c" program.
- *
- * For comparison, the triple parity computation using the power
- * coeffients "1,2,2^-1" is only a little faster than the one based on
- * the Cauchy matrix if SSSE3 or AVX2 is present.
- *
- * int8 int32 int64 sse2 ssse3 avx2
- * genz 2337 2874 10920 18944
- *
- * In conclusion, the use of power coefficients, and specifically powers
- * of 1,2,2^-1, is the best option to implement triple parity in CPUs
- * without SSSE3 and AVX2.
- * But if a modern CPU with SSSE3 or AVX2 is available, the Cauchy
- * matrix is the best option because it provides a fast and general
- * approach working for any number of parities.
- *
- * References:
- * [1] Anvin, "The mathematics of RAID-6", 2004
- * [2] MacWilliams, Sloane, "The Theory of Error-Correcting Codes", 1977
- * [3] Blomer, "An XOR-Based Erasure-Resilient Coding Scheme", 1995
- * [4] Roth, "Introduction to Coding Theory", 2006
- * [5] Plank, "Screaming Fast Galois Field Arithmetic Using Intel SIMD Instructions", 2013
- */
-
-/**
- * Generator matrix currently used.
- */
-const uint8_t (*raid_gfgen)[256];
-
-void raid_mode(int mode)
-{
- if (mode == RAID_MODE_VANDERMONDE) {
- raid_gen_ptr[2] = raid_genz_ptr;
- raid_gfgen = gfvandermonde;
- } else {
- raid_gen_ptr[2] = raid_gen3_ptr;
- raid_gfgen = gfcauchy;
- }
-}
-
-/**
- * Buffer filled with 0 used in recovering.
- */
-static void *raid_zero_block;
-
-void raid_zero(void *zero)
-{
- raid_zero_block = zero;
-}
-
-/*
- * Forwarders for parity computation.
- *
- * These functions compute the parity blocks from the provided data.
- *
- * The number of parities to compute is implicit in the position in the
- * forwarder vector. Position at index #i, computes (#i+1) parities.
- *
- * All these functions give the guarantee that parities are written
- * in order. First parity P, then parity Q, and so on.
- * This allows to specify the same memory buffer for multiple parities
- * knowning that you'll get the latest written one.
- * This characteristic is used by the raid_delta_gen() function to
- * avoid to damage unused parities in recovering.
- *
- * @nd Number of data blocks
- * @size Size of the blocks pointed by @v. It must be a multipler of 64.
- * @v Vector of pointers to the blocks of data and parity.
- * It has (@nd + #parities) elements. The starting elements are the blocks
- * for data, following with the parity blocks.
- * Each block has @size bytes.
- */
-void (*raid_gen_ptr[RAID_PARITY_MAX])(int nd, size_t size, void **vv);
-void (*raid_gen3_ptr)(int nd, size_t size, void **vv);
-void (*raid_genz_ptr)(int nd, size_t size, void **vv);
-
-void raid_gen(int nd, int np, size_t size, void **v)
-{
- /* enforce limit on size */
- BUG_ON(size % 64 != 0);
-
- /* enforce limit on number of failures */
- BUG_ON(np < 1);
- BUG_ON(np > RAID_PARITY_MAX);
-
- raid_gen_ptr[np - 1](nd, size, v);
-}
-
-/**
- * Inverts the square matrix M of size nxn into V.
- *
- * This is not a general matrix inversion because we assume the matrix M
- * to have all the square submatrix not singular.
- * We use Gauss elimination to invert.
- *
- * @M Matrix to invert with @n rows and @n columns.
- * @V Destination matrix where the result is put.
- * @n Number of rows and columns of the matrix.
- */
-void raid_invert(uint8_t *M, uint8_t *V, int n)
-{
- int i, j, k;
-
- /* set the identity matrix in V */
- for (i = 0; i < n; ++i)
- for (j = 0; j < n; ++j)
- V[i * n + j] = i == j;
-
- /* for each element in the diagonal */
- for (k = 0; k < n; ++k) {
- uint8_t f;
-
- /* the diagonal element cannot be 0 because */
- /* we are inverting matrices with all the square */
- /* submatrices not singular */
- BUG_ON(M[k * n + k] == 0);
-
- /* make the diagonal element to be 1 */
- f = inv(M[k * n + k]);
- for (j = 0; j < n; ++j) {
- M[k * n + j] = mul(f, M[k * n + j]);
- V[k * n + j] = mul(f, V[k * n + j]);
- }
-
- /* make all the elements over and under the diagonal */
- /* to be zero */
- for (i = 0; i < n; ++i) {
- if (i == k)
- continue;
- f = M[i * n + k];
- for (j = 0; j < n; ++j) {
- M[i * n + j] ^= mul(f, M[k * n + j]);
- V[i * n + j] ^= mul(f, V[k * n + j]);
- }
- }
- }
-}
-
-/**
- * Computes the parity without the missing data blocks
- * and store it in the buffers of such data blocks.
- *
- * This is the parity expressed as Pa,Qa,Ra,Sa,Ta,Ua in the equations.
- */
-void raid_delta_gen(int nr, int *id, int *ip, int nd, size_t size, void **v)
-{
- void *p[RAID_PARITY_MAX];
- void *pa[RAID_PARITY_MAX];
- int i, j;
- int np;
- void *latest;
-
- /* total number of parities we are going to process */
- /* they are both the used and the unused ones */
- np = ip[nr - 1] + 1;
-
- /* latest missing data block */
- latest = v[id[nr - 1]];
-
- /* setup pointers for delta computation */
- for (i = 0, j = 0; i < np; ++i) {
- /* keep a copy of the original parity vector */
- p[i] = v[nd + i];
-
- if (ip[j] == i) {
- /*
- * Set used parities to point to the missing
- * data blocks.
- *
- * The related data blocks are instead set
- * to point to the "zero" buffer.
- */
-
- /* the latest parity to use ends the for loop and */
- /* then it cannot happen to process more of them */
- BUG_ON(j >= nr);
-
- /* buffer for missing data blocks */
- pa[j] = v[id[j]];
-
- /* set at zero the missing data blocks */
- v[id[j]] = raid_zero_block;
-
- /* compute the parity over the missing data blocks */
- v[nd + i] = pa[j];
-
- /* check for the next used entry */
- ++j;
- } else {
- /*
- * Unused parities are going to be rewritten with
- * not significative data, becase we don't have
- * functions able to compute only a subset of
- * parities.
- *
- * To avoid this, we reuse parity buffers,
- * assuming that all the parity functions write
- * parities in order.
- *
- * We assign the unused parity block to the same
- * block of the latest used parity that we know it
- * will be written.
- *
- * This means that this block will be written
- * multiple times and only the latest write will
- * contain the correct data.
- */
- v[nd + i] = latest;
- }
- }
-
- /* all the parities have to be processed */
- BUG_ON(j != nr);
-
- /* recompute the parity, note that np may be smaller than the */
- /* total number of parities available */
- raid_gen(nd, np, size, v);
-
- /* restore data buffers as before */
- for (j = 0; j < nr; ++j)
- v[id[j]] = pa[j];
-
- /* restore parity buffers as before */
- for (i = 0; i < np; ++i)
- v[nd + i] = p[i];
-}
-
-/**
- * Recover failure of one data block for PAR1.
- *
- * Starting from the equation:
- *
- * Pd = Dx
- *
- * and solving we get:
- *
- * Dx = Pd
- */
-void raid_rec1of1(int *id, int nd, size_t size, void **v)
-{
- void *p;
- void *pa;
-
- /* for PAR1 we can directly compute the missing block */
- /* and we don't need to use the zero buffer */
- p = v[nd];
- pa = v[id[0]];
-
- /* use the parity as missing data block */
- v[id[0]] = p;
-
- /* compute the parity over the missing data block */
- v[nd] = pa;
-
- /* compute */
- raid_gen(nd, 1, size, v);
-
- /* restore as before */
- v[id[0]] = pa;
- v[nd] = p;
-}
-
-/**
- * Recover failure of two data blocks for PAR2.
- *
- * Starting from the equations:
- *
- * Pd = Dx + Dy
- * Qd = 2^id[0] * Dx + 2^id[1] * Dy
- *
- * and solving we get:
- *
- * 1 2^(-id[0])
- * Dy = ------------------- * Pd + ------------------- * Qd
- * 2^(id[1]-id[0]) + 1 2^(id[1]-id[0]) + 1
- *
- * Dx = Dy + Pd
- *
- * with conditions:
- *
- * 2^id[0] != 0
- * 2^(id[1]-id[0]) + 1 != 0
- *
- * That are always satisfied for any 0<=id[0]<id[1]<255.
- */
-void raid_rec2of2_int8(int *id, int *ip, int nd, size_t size, void **vv)
-{
- uint8_t **v = (uint8_t **)vv;
- size_t i;
- uint8_t *p;
- uint8_t *pa;
- uint8_t *q;
- uint8_t *qa;
- const uint8_t *T[2];
-
- /* get multiplication tables */
- T[0] = table(inv(pow2(id[1] - id[0]) ^ 1));
- T[1] = table(inv(pow2(id[0]) ^ pow2(id[1])));
-
- /* compute delta parity */
- raid_delta_gen(2, id, ip, nd, size, vv);
-
- p = v[nd];
- q = v[nd + 1];
- pa = v[id[0]];
- qa = v[id[1]];
-
- for (i = 0; i < size; ++i) {
- /* delta */
- uint8_t Pd = p[i] ^ pa[i];
- uint8_t Qd = q[i] ^ qa[i];
-
- /* reconstruct */
- uint8_t Dy = T[0][Pd] ^ T[1][Qd];
- uint8_t Dx = Pd ^ Dy;
-
- /* set */
- pa[i] = Dx;
- qa[i] = Dy;
- }
-}
-
-/*
- * Forwarders for data recovery.
- *
- * These functions recover data blocks using the specified parity
- * to recompute the missing data.
- *
- * Note that the format of vectors @id/@ip is different than raid_rec().
- * For example, in the vector @ip the first parity is represented with the
- * value 0 and not @nd.
- *
- * @nr Number of failed data blocks to recover.
- * @id[] Vector of @nr indexes of the data blocks to recover.
- * The indexes start from 0. They must be in order.
- * @ip[] Vector of @nr indexes of the parity blocks to use in the recovering.
- * The indexes start from 0. They must be in order.
- * @nd Number of data blocks.
- * @np Number of parity blocks.
- * @size Size of the blocks pointed by @v. It must be a multipler of 64.
- * @v Vector of pointers to the blocks of data and parity.
- * It has (@nd + @np) elements. The starting elements are the blocks
- * for data, following with the parity blocks.
- * Each block has @size bytes.
- */
-void (*raid_rec_ptr[RAID_PARITY_MAX])(
- int nr, int *id, int *ip, int nd, size_t size, void **vv);
-
-void raid_rec(int nr, int *ir, int nd, int np, size_t size, void **v)
-{
- int nrd; /* number of data blocks to recover */
- int nrp; /* number of parity blocks to recover */
-
- /* enforce limit on size */
- BUG_ON(size % 64 != 0);
-
- /* enforce limit on number of failures */
- BUG_ON(nr > np);
- BUG_ON(np > RAID_PARITY_MAX);
-
- /* enforce order in index vector */
- BUG_ON(nr >= 2 && ir[0] >= ir[1]);
- BUG_ON(nr >= 3 && ir[1] >= ir[2]);
- BUG_ON(nr >= 4 && ir[2] >= ir[3]);
- BUG_ON(nr >= 5 && ir[3] >= ir[4]);
- BUG_ON(nr >= 6 && ir[4] >= ir[5]);
-
- /* enforce limit on index vector */
- BUG_ON(nr > 0 && ir[nr-1] >= nd + np);
-
- /* count the number of data blocks to recover */
- nrd = 0;
- while (nrd < nr && ir[nrd] < nd)
- ++nrd;
-
- /* all the remaining are parity */
- nrp = nr - nrd;
-
- /* enforce limit on number of failures */
- BUG_ON(nrd > nd);
- BUG_ON(nrp > np);
-
- /* if failed data is present */
- if (nrd != 0) {
- int ip[RAID_PARITY_MAX];
- int i, j, k;
-
- /* setup the vector of parities to use */
- for (i = 0, j = 0, k = 0; i < np; ++i) {
- if (j < nrp && ir[nrd + j] == nd + i) {
- /* this parity has to be recovered */
- ++j;
- } else {
- /* this parity is used for recovering */
- ip[k] = i;
- ++k;
- }
- }
-
- /* recover the nrd data blocks specified in ir[], */
- /* using the first nrd parity in ip[] for recovering */
- raid_rec_ptr[nrd - 1](nrd, ir, ip, nd, size, v);
- }
-
- /* recompute all the parities up to the last bad one */
- if (nrp != 0)
- raid_gen(nd, ir[nr - 1] - nd + 1, size, v);
-}
-
-void raid_data(int nr, int *id, int *ip, int nd, size_t size, void **v)
-{
- /* enforce limit on size */
- BUG_ON(size % 64 != 0);
-
- /* enforce limit on number of failures */
- BUG_ON(nr > nd);
- BUG_ON(nr > RAID_PARITY_MAX);
-
- /* enforce order in index vector for data */
- BUG_ON(nr >= 2 && id[0] >= id[1]);
- BUG_ON(nr >= 3 && id[1] >= id[2]);
- BUG_ON(nr >= 4 && id[2] >= id[3]);
- BUG_ON(nr >= 5 && id[3] >= id[4]);
- BUG_ON(nr >= 6 && id[4] >= id[5]);
-
- /* enforce limit on index vector for data */
- BUG_ON(nr > 0 && id[nr-1] >= nd);
-
- /* enforce order in index vector for parity */
- BUG_ON(nr >= 2 && ip[0] >= ip[1]);
- BUG_ON(nr >= 3 && ip[1] >= ip[2]);
- BUG_ON(nr >= 4 && ip[2] >= ip[3]);
- BUG_ON(nr >= 5 && ip[3] >= ip[4]);
- BUG_ON(nr >= 6 && ip[4] >= ip[5]);
-
- /* if failed data is present */
- if (nr != 0)
- raid_rec_ptr[nr - 1](nr, id, ip, nd, size, v);
-}
-
projects / bcachefs-tools-debian / blobdiff