/*
* 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);
}