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+/*
+ * 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);
+}
+
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