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Andrea Mazzoleni 359d943ad8 Auto update
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<pre class="sourceHeading"> Line data Source code</pre>
<pre class="source">
<a name="1"><span class="lineNum"> 1 </span> : // SPDX-License-Identifier: GPL-2.0-or-later</a>
<a name="2"><span class="lineNum"> 2 </span> : // Copyright (C) 2013 Andrea Mazzoleni</a>
<a name="3"><span class="lineNum"> 3 </span> : </a>
<a name="4"><span class="lineNum"> 4 </span> : #include &quot;internal.h&quot;</a>
<a name="5"><span class="lineNum"> 5 </span> : #include &quot;gf.h&quot;</a>
<a name="6"><span class="lineNum"> 6 </span> : </a>
<a name="7"><span class="lineNum"> 7 </span> : /*</a>
<a name="8"><span class="lineNum"> 8 </span> : * Cauchy Matrix Construction for MDS RAID with Arbitrary Parity Levels in GF(2^8)</a>
<a name="9"><span class="lineNum"> 9 </span> : *</a>
<a name="10"><span class="lineNum"> 10 </span> : * This is a RAID implementation operating in the Galois Field GF(2^8) with</a>
<a name="11"><span class="lineNum"> 11 </span> : * the primitive polynomial x^8 + x^4 + x^3 + x^2 + 1 (285 decimal), supporting</a>
<a name="12"><span class="lineNum"> 12 </span> : * up to six parity levels.</a>
<a name="13"><span class="lineNum"> 13 </span> : *</a>
<a name="14"><span class="lineNum"> 14 </span> : * For RAID5 and RAID6, it follows the method described in H. Peter Anvin's</a>
<a name="15"><span class="lineNum"> 15 </span> : * paper &quot;The mathematics of RAID-6&quot; [1]. Please refer to this paper for a</a>
<a name="16"><span class="lineNum"> 16 </span> : * complete explanation.</a>
<a name="17"><span class="lineNum"> 17 </span> : *</a>
<a name="18"><span class="lineNum"> 18 </span> : * Triple parity was first evaluated using an extension of the same approach,</a>
<a name="19"><span class="lineNum"> 19 </span> : * with additional parity coefficients set as powers of 2^-1, defined as:</a>
<a name="20"><span class="lineNum"> 20 </span> : *</a>
<a name="21"><span class="lineNum"> 21 </span> : * P = sum(Di)</a>
<a name="22"><span class="lineNum"> 22 </span> : * Q = sum(2^i * Di)</a>
<a name="23"><span class="lineNum"> 23 </span> : * R = sum(2^-i * Di) for 0 &lt;= i &lt; N</a>
<a name="24"><span class="lineNum"> 24 </span> : *</a>
<a name="25"><span class="lineNum"> 25 </span> : * This approach works efficiently for triple parity because multiplications</a>
<a name="26"><span class="lineNum"> 26 </span> : * and divisions by 2 in GF(2^8) are very fast. It is similar to ZFS RAIDZ3,</a>
<a name="27"><span class="lineNum"> 27 </span> : * which uses powers of 4 instead of 2^-1.</a>
<a name="28"><span class="lineNum"> 28 </span> : *</a>
<a name="29"><span class="lineNum"> 29 </span> : * Unfortunately, this method does not extend beyond triple parity because</a>
<a name="30"><span class="lineNum"> 30 </span> : * no choice of power coefficients guarantees solvable equations for all</a>
<a name="31"><span class="lineNum"> 31 </span> : * combinations of missing disks. This is expected: a Vandermonde matrix,</a>
<a name="32"><span class="lineNum"> 32 </span> : * used in this method, does not ensure that all submatrices are nonsingular</a>
<a name="33"><span class="lineNum"> 33 </span> : * [2, Chap. 11, Problem 7], which is required for an MDS (Maximum Distance</a>
<a name="34"><span class="lineNum"> 34 </span> : * Separable) code [2, Chap. 11, Theorem 8].</a>
<a name="35"><span class="lineNum"> 35 </span> : *</a>
<a name="36"><span class="lineNum"> 36 </span> : * To overcome this limitation, a Cauchy matrix [3][4] is used for parity</a>
<a name="37"><span class="lineNum"> 37 </span> : * computation. A Cauchy matrix ensures that all square submatrices are</a>
<a name="38"><span class="lineNum"> 38 </span> : * nonsingular, so the linear system is always solvable for any combination</a>
<a name="39"><span class="lineNum"> 39 </span> : * of missing disks. This guarantees an MDS code.</a>
<a name="40"><span class="lineNum"> 40 </span> : *</a>
<a name="41"><span class="lineNum"> 41 </span> : * How the matrix is obtained:</a>
<a name="42"><span class="lineNum"> 42 </span> : *</a>
<a name="43"><span class="lineNum"> 43 </span> : * The matrix is constructed to match Linux RAID coefficients for the</a>
<a name="44"><span class="lineNum"> 44 </span> : * first two rows, while ensuring that all square submatrices are nonsingular.</a>
<a name="45"><span class="lineNum"> 45 </span> : *</a>
<a name="46"><span class="lineNum"> 46 </span> : * 1. Start by forming a Cauchy matrix with elements defined as 1/(x_i + y_j),</a>
<a name="47"><span class="lineNum"> 47 </span> : * where all x_i and y_j are distinct elements (the textbook definition</a>
<a name="48"><span class="lineNum"> 48 </span> : * of a Cauchy matrix).</a>
<a name="49"><span class="lineNum"> 49 </span> : *</a>
<a name="50"><span class="lineNum"> 50 </span> : * 2. For the first row (j=0), set x_i = 2^-i and y_0 = 0, resulting in:</a>
<a name="51"><span class="lineNum"> 51 </span> : *</a>
<a name="52"><span class="lineNum"> 52 </span> : * row j=0 -&gt; 1/(x_i + y_0) = 1/(2^-i + 0) = 2^i</a>
<a name="53"><span class="lineNum"> 53 </span> : *</a>
<a name="54"><span class="lineNum"> 54 </span> : * which reproduces the RAID-6 coefficients.</a>
<a name="55"><span class="lineNum"> 55 </span> : *</a>
<a name="56"><span class="lineNum"> 56 </span> : * 3. For subsequent rows (j&gt;0), set y_j = 2^j, yielding:</a>
<a name="57"><span class="lineNum"> 57 </span> : *</a>
<a name="58"><span class="lineNum"> 58 </span> : * rows j&gt;0 -&gt; 1/(x_i + y_j) = 1/(2^-i + 2^j)</a>
<a name="59"><span class="lineNum"> 59 </span> : *</a>
<a name="60"><span class="lineNum"> 60 </span> : * ensuring x_i != y_j for any i &gt;= 0, j &gt;= 1, and i + j &lt; 255.</a>
<a name="61"><span class="lineNum"> 61 </span> : *</a>
<a name="62"><span class="lineNum"> 62 </span> : * 4. Place a row filled with 1 at the top of the matrix, transforming it</a>
<a name="63"><span class="lineNum"> 63 </span> : * into an Extended Cauchy Matrix. This preserves the nonsingularity</a>
<a name="64"><span class="lineNum"> 64 </span> : * of all square submatrices.</a>
<a name="65"><span class="lineNum"> 65 </span> : *</a>
<a name="66"><span class="lineNum"> 66 </span> : * This first row reproduces the RAID-5 coefficients.</a>
<a name="67"><span class="lineNum"> 67 </span> : *</a>
<a name="68"><span class="lineNum"> 68 </span> : * 5. Adjust each row with a factor so that the first column contains all 1s.</a>
<a name="69"><span class="lineNum"> 69 </span> : * This transformation also preserves the nonsingularity property,</a>
<a name="70"><span class="lineNum"> 70 </span> : * maintaining the MDS code property.</a>
<a name="71"><span class="lineNum"> 71 </span> : *</a>
<a name="72"><span class="lineNum"> 72 </span> : * Concrete example in GF(256) for k=6, m=4:</a>
<a name="73"><span class="lineNum"> 73 </span> : *</a>
<a name="74"><span class="lineNum"> 74 </span> : * First, create a 3x6 Cauchy matrix using x_i = 2^-i and y_0 = 0, y_j = 2^j for j&gt;0:</a>
<a name="75"><span class="lineNum"> 75 </span> : *</a>
<a name="76"><span class="lineNum"> 76 </span> : * x = { 1, 142, 71, 173, 216, 108 }</a>
<a name="77"><span class="lineNum"> 77 </span> : * y = { 0, 2, 4 }</a>
<a name="78"><span class="lineNum"> 78 </span> : *</a>
<a name="79"><span class="lineNum"> 79 </span> : * The Cauchy matrix is:</a>
<a name="80"><span class="lineNum"> 80 </span> : *</a>
<a name="81"><span class="lineNum"> 81 </span> : * 1 2 4 8 16 32</a>
<a name="82"><span class="lineNum"> 82 </span> : * 244 83 78 183 118 47</a>
<a name="83"><span class="lineNum"> 83 </span> : * 167 39 213 59 153 82</a>
<a name="84"><span class="lineNum"> 84 </span> : *</a>
<a name="85"><span class="lineNum"> 85 </span> : * Divide row 2 by 244 and row 3 by 167. Then extend it with a row of ones</a>
<a name="86"><span class="lineNum"> 86 </span> : * on top, and it remains MDS. This forms the code for m=4, with RAID-6 as</a>
<a name="87"><span class="lineNum"> 87 </span> : * a subset:</a>
<a name="88"><span class="lineNum"> 88 </span> : *</a>
<a name="89"><span class="lineNum"> 89 </span> : * 1 1 1 1 1 1 </a>
<a name="90"><span class="lineNum"> 90 </span> : * 1 2 4 8 16 32</a>
<a name="91"><span class="lineNum"> 91 </span> : * 1 245 210 196 154 113</a>
<a name="92"><span class="lineNum"> 92 </span> : * 1 187 166 215 199 7</a>
<a name="93"><span class="lineNum"> 93 </span> : *</a>
<a name="94"><span class="lineNum"> 94 </span> : * Extending the same computation to k=251 and m=6 gives:</a>
<a name="95"><span class="lineNum"> 95 </span> : *</a>
<a name="96"><span class="lineNum"> 96 </span> : * This results in the matrix A[row, col] defined as:</a>
<a name="97"><span class="lineNum"> 97 </span> : *</a>
<a name="98"><span class="lineNum"> 98 </span> : * 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01...</a>
<a name="99"><span class="lineNum"> 99 </span> : * 01 02 04 08 10 20 40 80 1d 3a 74 e8 cd 87 13 26 4c 98 2d 5a b4 75...</a>
<a name="100"><span class="lineNum"> 100 </span> : * 01 f5 d2 c4 9a 71 f1 7f fc 87 c1 c6 19 2f 40 55 3d ba 53 04 9c 61...</a>
<a name="101"><span class="lineNum"> 101 </span> : * 01 bb a6 d7 c7 07 ce 82 4a 2f a5 9b b6 60 f1 ad e7 f4 06 d2 df 2e...</a>
<a name="102"><span class="lineNum"> 102 </span> : * 01 97 7f 9c 7c 18 bd a2 58 1a da 74 70 a3 e5 47 29 07 f5 80 23 e9...</a>
<a name="103"><span class="lineNum"> 103 </span> : * 01 2b 3f cf 73 2c d6 ed cb 74 15 78 8a c1 17 c9 89 68 21 ab 76 3b...</a>
<a name="104"><span class="lineNum"> 104 </span> : *</a>
<a name="105"><span class="lineNum"> 105 </span> : * This matrix supports six levels of parity, one for each row, for up to 251</a>
<a name="106"><span class="lineNum"> 106 </span> : * data disks, one for each column. All the 377,342,351,231 square submatrices</a>
<a name="107"><span class="lineNum"> 107 </span> : * are nonsingular, verified also by brute-force testing.</a>
<a name="108"><span class="lineNum"> 108 </span> : *</a>
<a name="109"><span class="lineNum"> 109 </span> : * This matrix can be extended to support any number of parities by simply</a>
<a name="110"><span class="lineNum"> 110 </span> : * adding additional rows and removing one column for each new row to maintain</a>
<a name="111"><span class="lineNum"> 111 </span> : * the condition i + j &lt; 255.</a>
<a name="112"><span class="lineNum"> 112 </span> : * (See mktables.c for more details on how the matrix is generated.)</a>
<a name="113"><span class="lineNum"> 113 </span> : *</a>
<a name="114"><span class="lineNum"> 114 </span> : * The downside is that generic multiplications are required to compute the</a>
<a name="115"><span class="lineNum"> 115 </span> : * parity, not only fast multiplications by 2^n or 2^-n, which negatively</a>
<a name="116"><span class="lineNum"> 116 </span> : * affect performance. However, optimized parallel multiplications using SSSE3</a>
<a name="117"><span class="lineNum"> 117 </span> : * or AVX2 instructions [1][5] make this approach competitive with the triple</a>
<a name="118"><span class="lineNum"> 118 </span> : * parity computation using power coefficients.</a>
<a name="119"><span class="lineNum"> 119 </span> : *</a>
<a name="120"><span class="lineNum"> 120 </span> : * Another advantage of the Cauchy matrix is that the first two rows can</a>
<a name="121"><span class="lineNum"> 121 </span> : * replicate the RAID5 and RAID6 approach, resulting in a compatible extension.</a>
<a name="122"><span class="lineNum"> 122 </span> : * SSSE3 or AVX2 instructions are only needed for triple parity or beyond.</a>
<a name="123"><span class="lineNum"> 123 </span> : *</a>
<a name="124"><span class="lineNum"> 124 </span> : * Parity computation is as follows:</a>
<a name="125"><span class="lineNum"> 125 </span> : *</a>
<a name="126"><span class="lineNum"> 126 </span> : * P = sum(Di)</a>
<a name="127"><span class="lineNum"> 127 </span> : * Q = sum(2^i * Di)</a>
<a name="128"><span class="lineNum"> 128 </span> : * R = sum(A[2,i] * Di)</a>
<a name="129"><span class="lineNum"> 129 </span> : * S = sum(A[3,i] * Di)</a>
<a name="130"><span class="lineNum"> 130 </span> : * T = sum(A[4,i] * Di)</a>
<a name="131"><span class="lineNum"> 131 </span> : * U = sum(A[5,i] * Di) for 0 &lt;= i &lt; N</a>
<a name="132"><span class="lineNum"> 132 </span> : *</a>
<a name="133"><span class="lineNum"> 133 </span> : * Recovery from six disk failures at indices x, y, z, h, v, w (0 &lt;= x &lt; y &lt; z &lt; h &lt; v &lt; w &lt; N)</a>
<a name="134"><span class="lineNum"> 134 </span> : * involves computing parity of the remaining N-6 disks:</a>
<a name="135"><span class="lineNum"> 135 </span> : *</a>
<a name="136"><span class="lineNum"> 136 </span> : * Pa = sum(Di)</a>
<a name="137"><span class="lineNum"> 137 </span> : * Qa = sum(2^i * Di)</a>
<a name="138"><span class="lineNum"> 138 </span> : * Ra = sum(A[2,i] * Di)</a>
<a name="139"><span class="lineNum"> 139 </span> : * Sa = sum(A[3,i] * Di)</a>
<a name="140"><span class="lineNum"> 140 </span> : * Ta = sum(A[4,i] * Di)</a>
<a name="141"><span class="lineNum"> 141 </span> : * Ua = sum(A[5,i] * Di) for i != x,y,z,h,v,w</a>
<a name="142"><span class="lineNum"> 142 </span> : *</a>
<a name="143"><span class="lineNum"> 143 </span> : * Defining:</a>
<a name="144"><span class="lineNum"> 144 </span> : *</a>
<a name="145"><span class="lineNum"> 145 </span> : * Pd = Pa + P</a>
<a name="146"><span class="lineNum"> 146 </span> : * Qd = Qa + Q</a>
<a name="147"><span class="lineNum"> 147 </span> : * Rd = Ra + R</a>
<a name="148"><span class="lineNum"> 148 </span> : * Sd = Sa + S</a>
<a name="149"><span class="lineNum"> 149 </span> : * Td = Ta + T</a>
<a name="150"><span class="lineNum"> 150 </span> : * Ud = Ua + U</a>
<a name="151"><span class="lineNum"> 151 </span> : *</a>
<a name="152"><span class="lineNum"> 152 </span> : * yields:</a>
<a name="153"><span class="lineNum"> 153 </span> : *</a>
<a name="154"><span class="lineNum"> 154 </span> : * Pd = Dx + Dy + Dz + Dh + Dv + Dw</a>
<a name="155"><span class="lineNum"> 155 </span> : * Qd = 2^x * Dx + 2^y * Dy + 2^z * Dz + 2^h * Dh + 2^v * Dv + 2^w * Dw</a>
<a name="156"><span class="lineNum"> 156 </span> : * 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</a>
<a name="157"><span class="lineNum"> 157 </span> : * 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</a>
<a name="158"><span class="lineNum"> 158 </span> : * 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</a>
<a name="159"><span class="lineNum"> 159 </span> : * 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>
<a name="160"><span class="lineNum"> 160 </span> : *</a>
<a name="161"><span class="lineNum"> 161 </span> : * This linear system is always solvable since the coefficient matrix is</a>
<a name="162"><span class="lineNum"> 162 </span> : * nonsingular due to the properties of A[].</a>
<a name="163"><span class="lineNum"> 163 </span> : *</a>
<a name="164"><span class="lineNum"> 164 </span> : * Example performance on a Intel Core i7-10700 CPU @ 2.90GHz, stripe 256 KiB, 8 data disks:</a>
<a name="165"><span class="lineNum"> 165 </span> : *</a>
<a name="166"><span class="lineNum"> 166 </span> : * int8 int32 int64 sse2 sse2e ssse3 ssse3e avx2 avx2e</a>
<a name="167"><span class="lineNum"> 167 </span> : * gen1 24995 46562 62683 77069</a>
<a name="168"><span class="lineNum"> 168 </span> : * gen2 8243 15879 25788 27292 44907</a>
<a name="169"><span class="lineNum"> 169 </span> : * genz 5509 10588 14485 13920 25175</a>
<a name="170"><span class="lineNum"> 170 </span> : * gen3 1190 13201 14307 27268</a>
<a name="171"><span class="lineNum"> 171 </span> : * gen4 914 10061 11006 21328</a>
<a name="172"><span class="lineNum"> 172 </span> : * gen5 746 7970 8828 17040</a>
<a name="173"><span class="lineNum"> 173 </span> : * gen6 630 6627 7516 14584</a>
<a name="174"><span class="lineNum"> 174 </span> : *</a>
<a name="175"><span class="lineNum"> 175 </span> : * Values are in MiB/s of data processed by a single thread.</a>
<a name="176"><span class="lineNum"> 176 </span> : * Results can be reproduced using &quot;raid/test/speedtest.c&quot;.</a>
<a name="177"><span class="lineNum"> 177 </span> : *</a>
<a name="178"><span class="lineNum"> 178 </span> : * Triple parity with power coefficients &quot;1, 2, 2^-1&quot; is slightly faster than</a>
<a name="179"><span class="lineNum"> 179 </span> : * the Cauchy matrix computation if SSSE3 or AVX2 are available:</a>
<a name="180"><span class="lineNum"> 180 </span> : *</a>
<a name="181"><span class="lineNum"> 181 </span> : * int8 int32 int64 sse2 ssse3 avx2</a>
<a name="182"><span class="lineNum"> 182 </span> : * genz 2337 2874 10920 18944</a>
<a name="183"><span class="lineNum"> 183 </span> : *</a>
<a name="184"><span class="lineNum"> 184 </span> : * In conclusion, the use of power coefficients, specifically powers</a>
<a name="185"><span class="lineNum"> 185 </span> : * of 1, 2, and 2^-1, is the best option to implement triple parity on CPUs</a>
<a name="186"><span class="lineNum"> 186 </span> : * without SSSE3 and AVX2.</a>
<a name="187"><span class="lineNum"> 187 </span> : * On modern CPUs with SSSE3 or AVX2, the Cauchy matrix is the best</a>
<a name="188"><span class="lineNum"> 188 </span> : * option because it provides a fast and general approach that works for any</a>
<a name="189"><span class="lineNum"> 189 </span> : * number of parities.</a>
<a name="190"><span class="lineNum"> 190 </span> : *</a>
<a name="191"><span class="lineNum"> 191 </span> : * References:</a>
<a name="192"><span class="lineNum"> 192 </span> : * [1] Anvin, &quot;The mathematics of RAID-6&quot;, 2004</a>
<a name="193"><span class="lineNum"> 193 </span> : * [2] MacWilliams, Sloane, &quot;The Theory of Error-Correcting Codes&quot;, 1977</a>
<a name="194"><span class="lineNum"> 194 </span> : * [3] Blomer, &quot;An XOR-Based Erasure-Resilient Coding Scheme&quot;, 1995</a>
<a name="195"><span class="lineNum"> 195 </span> : * [4] Roth, &quot;Introduction to Coding Theory&quot;, 2006</a>
<a name="196"><span class="lineNum"> 196 </span> : * [5] Plank, &quot;Screaming Fast Galois Field Arithmetic Using Intel SIMD Instructions&quot;, 2013</a>
<a name="197"><span class="lineNum"> 197 </span> : */</a>
<a name="198"><span class="lineNum"> 198 </span> : </a>
<a name="199"><span class="lineNum"> 199 </span> : /**</a>
<a name="200"><span class="lineNum"> 200 </span> : * Buffer filled with 0 used in recovering.</a>
<a name="201"><span class="lineNum"> 201 </span> : */</a>
<a name="202"><span class="lineNum"> 202 </span> : static void *raid_zero_block;</a>
<a name="203"><span class="lineNum"> 203 </span> : </a>
<a name="204"><span class="lineNum"> 204 </span><span class="lineCov"> 240 : void raid_zero(void *zero)</span></a>
<a name="205"><span class="lineNum"> 205 </span> : {</a>
<a name="206"><span class="lineNum"> 206 </span><span class="lineCov"> 240 : raid_zero_block = zero;</span></a>
<a name="207"><span class="lineNum"> 207 </span><span class="lineCov"> 240 : }</span></a>
<a name="208"><span class="lineNum"> 208 </span> : </a>
<a name="209"><span class="lineNum"> 209 </span> : /**</a>
<a name="210"><span class="lineNum"> 210 </span> : * Inverts the square matrix M of size nxn into V.</a>
<a name="211"><span class="lineNum"> 211 </span> : *</a>
<a name="212"><span class="lineNum"> 212 </span> : * This is not a general matrix inversion because we assume the matrix M</a>
<a name="213"><span class="lineNum"> 213 </span> : * to have all the square submatrix not singular.</a>
<a name="214"><span class="lineNum"> 214 </span> : * We use Gauss elimination to invert.</a>
<a name="215"><span class="lineNum"> 215 </span> : *</a>
<a name="216"><span class="lineNum"> 216 </span> : * @M Matrix to invert with @n rows and @n columns.</a>
<a name="217"><span class="lineNum"> 217 </span> : * @V Destination matrix where the result is put.</a>
<a name="218"><span class="lineNum"> 218 </span> : * @n Number of rows and columns of the matrix.</a>
<a name="219"><span class="lineNum"> 219 </span> : */</a>
<a name="220"><span class="lineNum"> 220 </span><span class="lineCov"> 204206 : void raid_invert(uint8_t *M, uint8_t *V, int n)</span></a>
<a name="221"><span class="lineNum"> 221 </span> : {</a>
<a name="222"><span class="lineNum"> 222 </span> : int i, j, k;</a>
<a name="223"><span class="lineNum"> 223 </span> : </a>
<a name="224"><span class="lineNum"> 224 </span> : /* set the identity matrix in V */</a>
<a name="225"><span class="lineNum"> 225 </span><span class="lineCov"> 942501 : for (i = 0; i &lt; n; ++i)</span></a>
<a name="226"><span class="lineNum"> 226 </span><span class="lineCov"> 3760858 : for (j = 0; j &lt; n; ++j)</span></a>
<a name="227"><span class="lineNum"> 227 </span><span class="lineCov"> 3022563 : V[i * n + j] = i == j;</span></a>
<a name="228"><span class="lineNum"> 228 </span> : </a>
<a name="229"><span class="lineNum"> 229 </span> : /* for each element in the diagonal */</a>
<a name="230"><span class="lineNum"> 230 </span><span class="lineCov"> 942501 : for (k = 0; k &lt; n; ++k) {</span></a>
<a name="231"><span class="lineNum"> 231 </span> : uint8_t f;</a>
<a name="232"><span class="lineNum"> 232 </span> : </a>
<a name="233"><span class="lineNum"> 233 </span> : /*</a>
<a name="234"><span class="lineNum"> 234 </span> : * The diagonal element cannot be 0 because</a>
<a name="235"><span class="lineNum"> 235 </span> : * we are inverting matrices with all the square</a>
<a name="236"><span class="lineNum"> 236 </span> : * submatrices not singular</a>
<a name="237"><span class="lineNum"> 237 </span> : */</a>
<a name="238"><span class="lineNum"> 238 </span><span class="lineCov"> 738295 : BUG_ON(M[k * n + k] == 0);</span></a>
<a name="239"><span class="lineNum"> 239 </span> : </a>
<a name="240"><span class="lineNum"> 240 </span> : /* make the diagonal element to be 1 */</a>
<a name="241"><span class="lineNum"> 241 </span><span class="lineCov"> 738295 : f = inv(M[k * n + k]);</span></a>
<a name="242"><span class="lineNum"> 242 </span><span class="lineCov"> 3760858 : for (j = 0; j &lt; n; ++j) {</span></a>
<a name="243"><span class="lineNum"> 243 </span><span class="lineCov"> 3022563 : M[k * n + j] = mul(f, M[k * n + j]);</span></a>
<a name="244"><span class="lineNum"> 244 </span><span class="lineCov"> 6045126 : V[k * n + j] = mul(f, V[k * n + j]);</span></a>
<a name="245"><span class="lineNum"> 245 </span> : }</a>
<a name="246"><span class="lineNum"> 246 </span> : </a>
<a name="247"><span class="lineNum"> 247 </span> : /*</a>
<a name="248"><span class="lineNum"> 248 </span> : * Make all the elements over and under the diagonal</a>
<a name="249"><span class="lineNum"> 249 </span> : * to be zero</a>
<a name="250"><span class="lineNum"> 250 </span> : */</a>
<a name="251"><span class="lineNum"> 251 </span><span class="lineCov"> 3760858 : for (i = 0; i &lt; n; ++i) {</span></a>
<a name="252"><span class="lineNum"> 252 </span><span class="lineCov"> 3022563 : if (i == k)</span></a>
<a name="253"><span class="lineNum"> 253 </span><span class="lineCov"> 738295 : continue;</span></a>
<a name="254"><span class="lineNum"> 254 </span><span class="lineCov"> 2284268 : f = M[i * n + k];</span></a>
<a name="255"><span class="lineNum"> 255 </span><span class="lineCov"> 12876408 : for (j = 0; j &lt; n; ++j) {</span></a>
<a name="256"><span class="lineNum"> 256 </span><span class="lineCov"> 10592140 : M[i * n + j] ^= mul(f, M[k * n + j]);</span></a>
<a name="257"><span class="lineNum"> 257 </span><span class="lineCov"> 21184280 : V[i * n + j] ^= mul(f, V[k * n + j]);</span></a>
<a name="258"><span class="lineNum"> 258 </span> : }</a>
<a name="259"><span class="lineNum"> 259 </span> : }</a>
<a name="260"><span class="lineNum"> 260 </span> : }</a>
<a name="261"><span class="lineNum"> 261 </span><span class="lineCov"> 204206 : }</span></a>
<a name="262"><span class="lineNum"> 262 </span> : </a>
<a name="263"><span class="lineNum"> 263 </span> : /**</a>
<a name="264"><span class="lineNum"> 264 </span> : * Computes the parity without the missing data blocks</a>
<a name="265"><span class="lineNum"> 265 </span> : * and store it in the buffers of such data blocks.</a>
<a name="266"><span class="lineNum"> 266 </span> : *</a>
<a name="267"><span class="lineNum"> 267 </span> : * This is the parity expressed as Pa,Qa,Ra,Sa,Ta,Ua in the equations.</a>
<a name="268"><span class="lineNum"> 268 </span> : */</a>
<a name="269"><span class="lineNum"> 269 </span><span class="lineCov"> 203530 : void raid_delta_gen(int nr, int *id, int *ip, int nd, size_t size, void **v)</span></a>
<a name="270"><span class="lineNum"> 270 </span> : {</a>
<a name="271"><span class="lineNum"> 271 </span> : void *p[RAID_PARITY_MAX];</a>
<a name="272"><span class="lineNum"> 272 </span> : void *pa[RAID_PARITY_MAX];</a>
<a name="273"><span class="lineNum"> 273 </span> : int i, j;</a>
<a name="274"><span class="lineNum"> 274 </span> : int np;</a>
<a name="275"><span class="lineNum"> 275 </span> : void *latest;</a>
<a name="276"><span class="lineNum"> 276 </span> : </a>
<a name="277"><span class="lineNum"> 277 </span> : /*</a>
<a name="278"><span class="lineNum"> 278 </span> : * Total number of parities we are going to process</a>
<a name="279"><span class="lineNum"> 279 </span> : * they are both the used and the unused ones</a>
<a name="280"><span class="lineNum"> 280 </span> : */</a>
<a name="281"><span class="lineNum"> 281 </span><span class="lineCov"> 203530 : np = ip[nr - 1] + 1;</span></a>
<a name="282"><span class="lineNum"> 282 </span> : </a>
<a name="283"><span class="lineNum"> 283 </span> : /* latest missing data block */</a>
<a name="284"><span class="lineNum"> 284 </span><span class="lineCov"> 203530 : latest = v[id[nr - 1]];</span></a>
<a name="285"><span class="lineNum"> 285 </span> : </a>
<a name="286"><span class="lineNum"> 286 </span> : /* setup pointers for delta computation */</a>
<a name="287"><span class="lineNum"> 287 </span><span class="lineCov"> 1047537 : for (i = 0, j = 0; i &lt; np; ++i) {</span></a>
<a name="288"><span class="lineNum"> 288 </span> : /* keep a copy of the original parity vector */</a>
<a name="289"><span class="lineNum"> 289 </span><span class="lineCov"> 844007 : p[i] = v[nd + i];</span></a>
<a name="290"><span class="lineNum"> 290 </span> : </a>
<a name="291"><span class="lineNum"> 291 </span><span class="lineCov"> 844007 : if (ip[j] == i) {</span></a>
<a name="292"><span class="lineNum"> 292 </span> : /*</a>
<a name="293"><span class="lineNum"> 293 </span> : * Set used parities to point to the missing</a>
<a name="294"><span class="lineNum"> 294 </span> : * data blocks.</a>
<a name="295"><span class="lineNum"> 295 </span> : *</a>
<a name="296"><span class="lineNum"> 296 </span> : * The related data blocks are instead set</a>
<a name="297"><span class="lineNum"> 297 </span> : * to point to the &quot;zero&quot; buffer.</a>
<a name="298"><span class="lineNum"> 298 </span> : */</a>
<a name="299"><span class="lineNum"> 299 </span> : </a>
<a name="300"><span class="lineNum"> 300 </span> : /*</a>
<a name="301"><span class="lineNum"> 301 </span> : * The latest parity to use ends the for loop and</a>
<a name="302"><span class="lineNum"> 302 </span> : * then it cannot happen to process more of them</a>
<a name="303"><span class="lineNum"> 303 </span> : */</a>
<a name="304"><span class="lineNum"> 304 </span><span class="lineCov"> 718945 : BUG_ON(j &gt;= nr);</span></a>
<a name="305"><span class="lineNum"> 305 </span> : </a>
<a name="306"><span class="lineNum"> 306 </span> : /* buffer for missing data blocks */</a>
<a name="307"><span class="lineNum"> 307 </span><span class="lineCov"> 718945 : pa[j] = v[id[j]];</span></a>
<a name="308"><span class="lineNum"> 308 </span> : </a>
<a name="309"><span class="lineNum"> 309 </span> : /* set at zero the missing data blocks */</a>
<a name="310"><span class="lineNum"> 310 </span><span class="lineCov"> 718945 : v[id[j]] = raid_zero_block;</span></a>
<a name="311"><span class="lineNum"> 311 </span> : </a>
<a name="312"><span class="lineNum"> 312 </span> : /* compute the parity over the missing data blocks */</a>
<a name="313"><span class="lineNum"> 313 </span><span class="lineCov"> 718945 : v[nd + i] = pa[j];</span></a>
<a name="314"><span class="lineNum"> 314 </span> : </a>
<a name="315"><span class="lineNum"> 315 </span> : /* check for the next used entry */</a>
<a name="316"><span class="lineNum"> 316 </span><span class="lineCov"> 718945 : ++j;</span></a>
<a name="317"><span class="lineNum"> 317 </span> : } else {</a>
<a name="318"><span class="lineNum"> 318 </span> : /*</a>
<a name="319"><span class="lineNum"> 319 </span> : * Unused parities are going to be rewritten with</a>
<a name="320"><span class="lineNum"> 320 </span> : * not significant data, because we don't have</a>
<a name="321"><span class="lineNum"> 321 </span> : * functions able to compute only a subset of</a>
<a name="322"><span class="lineNum"> 322 </span> : * parities.</a>
<a name="323"><span class="lineNum"> 323 </span> : *</a>
<a name="324"><span class="lineNum"> 324 </span> : * To avoid this, we reuse parity buffers,</a>
<a name="325"><span class="lineNum"> 325 </span> : * assuming that all the parity functions write</a>
<a name="326"><span class="lineNum"> 326 </span> : * parities in order.</a>
<a name="327"><span class="lineNum"> 327 </span> : *</a>
<a name="328"><span class="lineNum"> 328 </span> : * We assign the unused parity block to the same</a>
<a name="329"><span class="lineNum"> 329 </span> : * block of the latest used parity that we know it</a>
<a name="330"><span class="lineNum"> 330 </span> : * will be written.</a>
<a name="331"><span class="lineNum"> 331 </span> : *</a>
<a name="332"><span class="lineNum"> 332 </span> : * This means that this block will be written</a>
<a name="333"><span class="lineNum"> 333 </span> : * multiple times and only the latest write will</a>
<a name="334"><span class="lineNum"> 334 </span> : * contain the correct data.</a>
<a name="335"><span class="lineNum"> 335 </span> : */</a>
<a name="336"><span class="lineNum"> 336 </span><span class="lineCov"> 125062 : v[nd + i] = latest;</span></a>
<a name="337"><span class="lineNum"> 337 </span> : }</a>
<a name="338"><span class="lineNum"> 338 </span> : }</a>
<a name="339"><span class="lineNum"> 339 </span> : </a>
<a name="340"><span class="lineNum"> 340 </span> : /* all the parities have to be processed */</a>
<a name="341"><span class="lineNum"> 341 </span><span class="lineCov"> 203530 : BUG_ON(j != nr);</span></a>
<a name="342"><span class="lineNum"> 342 </span> : </a>
<a name="343"><span class="lineNum"> 343 </span> : /*</a>
<a name="344"><span class="lineNum"> 344 </span> : * Recompute the parity, note that np may be smaller than the</a>
<a name="345"><span class="lineNum"> 345 </span> : * total number of parities available</a>
<a name="346"><span class="lineNum"> 346 </span> : */</a>
<a name="347"><span class="lineNum"> 347 </span><span class="lineCov"> 203530 : raid_gen(nd, np, size, v);</span></a>
<a name="348"><span class="lineNum"> 348 </span> : </a>
<a name="349"><span class="lineNum"> 349 </span> : /* restore data buffers as before */</a>
<a name="350"><span class="lineNum"> 350 </span><span class="lineCov"> 922475 : for (j = 0; j &lt; nr; ++j)</span></a>
<a name="351"><span class="lineNum"> 351 </span><span class="lineCov"> 718945 : v[id[j]] = pa[j];</span></a>
<a name="352"><span class="lineNum"> 352 </span> : </a>
<a name="353"><span class="lineNum"> 353 </span> : /* restore parity buffers as before */</a>
<a name="354"><span class="lineNum"> 354 </span><span class="lineCov"> 1047537 : for (i = 0; i &lt; np; ++i)</span></a>
<a name="355"><span class="lineNum"> 355 </span><span class="lineCov"> 844007 : v[nd + i] = p[i];</span></a>
<a name="356"><span class="lineNum"> 356 </span><span class="lineCov"> 203530 : }</span></a>
<a name="357"><span class="lineNum"> 357 </span> : </a>
<a name="358"><span class="lineNum"> 358 </span> : /**</a>
<a name="359"><span class="lineNum"> 359 </span> : * Recover failure of one data block for PAR1.</a>
<a name="360"><span class="lineNum"> 360 </span> : *</a>
<a name="361"><span class="lineNum"> 361 </span> : * Starting from the equation:</a>
<a name="362"><span class="lineNum"> 362 </span> : *</a>
<a name="363"><span class="lineNum"> 363 </span> : * Pd = Dx</a>
<a name="364"><span class="lineNum"> 364 </span> : *</a>
<a name="365"><span class="lineNum"> 365 </span> : * and solving we get:</a>
<a name="366"><span class="lineNum"> 366 </span> : *</a>
<a name="367"><span class="lineNum"> 367 </span> : * Dx = Pd</a>
<a name="368"><span class="lineNum"> 368 </span> : */</a>
<a name="369"><span class="lineNum"> 369 </span><span class="lineCov"> 101008 : void raid_rec1of1(int *id, int nd, size_t size, void **v)</span></a>
<a name="370"><span class="lineNum"> 370 </span> : {</a>
<a name="371"><span class="lineNum"> 371 </span> : void *p;</a>
<a name="372"><span class="lineNum"> 372 </span> : void *pa;</a>
<a name="373"><span class="lineNum"> 373 </span> : </a>
<a name="374"><span class="lineNum"> 374 </span> : /*</a>
<a name="375"><span class="lineNum"> 375 </span> : * For PAR1 we can directly compute the missing block</a>
<a name="376"><span class="lineNum"> 376 </span> : * and we don't need to use the zero buffer</a>
<a name="377"><span class="lineNum"> 377 </span> : */</a>
<a name="378"><span class="lineNum"> 378 </span><span class="lineCov"> 101008 : p = v[nd];</span></a>
<a name="379"><span class="lineNum"> 379 </span><span class="lineCov"> 101008 : pa = v[id[0]];</span></a>
<a name="380"><span class="lineNum"> 380 </span> : </a>
<a name="381"><span class="lineNum"> 381 </span> : /* use the parity as missing data block */</a>
<a name="382"><span class="lineNum"> 382 </span><span class="lineCov"> 101008 : v[id[0]] = p;</span></a>
<a name="383"><span class="lineNum"> 383 </span> : </a>
<a name="384"><span class="lineNum"> 384 </span> : /* compute the parity over the missing data block */</a>
<a name="385"><span class="lineNum"> 385 </span><span class="lineCov"> 101008 : v[nd] = pa;</span></a>
<a name="386"><span class="lineNum"> 386 </span> : </a>
<a name="387"><span class="lineNum"> 387 </span> : /* compute */</a>
<a name="388"><span class="lineNum"> 388 </span><span class="lineCov"> 101008 : raid_gen(nd, 1, size, v);</span></a>
<a name="389"><span class="lineNum"> 389 </span> : </a>
<a name="390"><span class="lineNum"> 390 </span> : /* restore as before */</a>
<a name="391"><span class="lineNum"> 391 </span><span class="lineCov"> 101008 : v[id[0]] = pa;</span></a>
<a name="392"><span class="lineNum"> 392 </span><span class="lineCov"> 101008 : v[nd] = p;</span></a>
<a name="393"><span class="lineNum"> 393 </span><span class="lineCov"> 101008 : }</span></a>
<a name="394"><span class="lineNum"> 394 </span> : </a>
<a name="395"><span class="lineNum"> 395 </span> : /**</a>
<a name="396"><span class="lineNum"> 396 </span> : * Recover failure of two data blocks for PAR2.</a>
<a name="397"><span class="lineNum"> 397 </span> : *</a>
<a name="398"><span class="lineNum"> 398 </span> : * Starting from the equations:</a>
<a name="399"><span class="lineNum"> 399 </span> : *</a>
<a name="400"><span class="lineNum"> 400 </span> : * Pd = Dx + Dy</a>
<a name="401"><span class="lineNum"> 401 </span> : * Qd = 2^id[0] * Dx + 2^id[1] * Dy</a>
<a name="402"><span class="lineNum"> 402 </span> : *</a>
<a name="403"><span class="lineNum"> 403 </span> : * and solving we get:</a>
<a name="404"><span class="lineNum"> 404 </span> : *</a>
<a name="405"><span class="lineNum"> 405 </span> : * 1 2^(-id[0])</a>
<a name="406"><span class="lineNum"> 406 </span> : * Dy = ------------------- * Pd + ------------------- * Qd</a>
<a name="407"><span class="lineNum"> 407 </span> : * 2^(id[1]-id[0]) + 1 2^(id[1]-id[0]) + 1</a>
<a name="408"><span class="lineNum"> 408 </span> : *</a>
<a name="409"><span class="lineNum"> 409 </span> : * Dx = Dy + Pd</a>
<a name="410"><span class="lineNum"> 410 </span> : *</a>
<a name="411"><span class="lineNum"> 411 </span> : * with conditions:</a>
<a name="412"><span class="lineNum"> 412 </span> : *</a>
<a name="413"><span class="lineNum"> 413 </span> : * 2^id[0] != 0</a>
<a name="414"><span class="lineNum"> 414 </span> : * 2^(id[1]-id[0]) + 1 != 0</a>
<a name="415"><span class="lineNum"> 415 </span> : *</a>
<a name="416"><span class="lineNum"> 416 </span> : * That are always satisfied for any 0&lt;=id[0]&lt;id[1]&lt;255.</a>
<a name="417"><span class="lineNum"> 417 </span> : */</a>
<a name="418"><span class="lineNum"> 418 </span><span class="lineCov"> 132 : void raid_rec2of2_int8(int *id, int *ip, int nd, size_t size, void **vv)</span></a>
<a name="419"><span class="lineNum"> 419 </span> : {</a>
<a name="420"><span class="lineNum"> 420 </span><span class="lineCov"> 132 : uint8_t **v = (uint8_t **)vv;</span></a>
<a name="421"><span class="lineNum"> 421 </span> : size_t i;</a>
<a name="422"><span class="lineNum"> 422 </span> : uint8_t *p;</a>
<a name="423"><span class="lineNum"> 423 </span> : uint8_t *pa;</a>
<a name="424"><span class="lineNum"> 424 </span> : uint8_t *q;</a>
<a name="425"><span class="lineNum"> 425 </span> : uint8_t *qa;</a>
<a name="426"><span class="lineNum"> 426 </span> : const uint8_t *T[2];</a>
<a name="427"><span class="lineNum"> 427 </span> : </a>
<a name="428"><span class="lineNum"> 428 </span> : /* get multiplication tables */</a>
<a name="429"><span class="lineNum"> 429 </span><span class="lineCov"> 396 : T[0] = table(inv(pow2(id[1] - id[0]) ^ 1));</span></a>
<a name="430"><span class="lineNum"> 430 </span><span class="lineCov"> 528 : T[1] = table(inv(pow2(id[0]) ^ pow2(id[1])));</span></a>
<a name="431"><span class="lineNum"> 431 </span> : </a>
<a name="432"><span class="lineNum"> 432 </span> : /* compute delta parity */</a>
<a name="433"><span class="lineNum"> 433 </span><span class="lineCov"> 132 : raid_delta_gen(2, id, ip, nd, size, vv);</span></a>
<a name="434"><span class="lineNum"> 434 </span> : </a>
<a name="435"><span class="lineNum"> 435 </span><span class="lineCov"> 132 : p = v[nd];</span></a>
<a name="436"><span class="lineNum"> 436 </span><span class="lineCov"> 132 : q = v[nd + 1];</span></a>
<a name="437"><span class="lineNum"> 437 </span><span class="lineCov"> 132 : pa = v[id[0]];</span></a>
<a name="438"><span class="lineNum"> 438 </span><span class="lineCov"> 132 : qa = v[id[1]];</span></a>
<a name="439"><span class="lineNum"> 439 </span> : </a>
<a name="440"><span class="lineNum"> 440 </span><span class="lineCov"> 33924 : for (i = 0; i &lt; size; ++i) {</span></a>
<a name="441"><span class="lineNum"> 441 </span> : /* delta */</a>
<a name="442"><span class="lineNum"> 442 </span><span class="lineCov"> 33792 : uint8_t Pd = p[i] ^ pa[i];</span></a>
<a name="443"><span class="lineNum"> 443 </span><span class="lineCov"> 33792 : uint8_t Qd = q[i] ^ qa[i];</span></a>
<a name="444"><span class="lineNum"> 444 </span> : </a>
<a name="445"><span class="lineNum"> 445 </span> : /* reconstruct */</a>
<a name="446"><span class="lineNum"> 446 </span><span class="lineCov"> 33792 : uint8_t Dy = T[0][Pd] ^ T[1][Qd];</span></a>
<a name="447"><span class="lineNum"> 447 </span><span class="lineCov"> 33792 : uint8_t Dx = Pd ^ Dy;</span></a>
<a name="448"><span class="lineNum"> 448 </span> : </a>
<a name="449"><span class="lineNum"> 449 </span> : /* set */</a>
<a name="450"><span class="lineNum"> 450 </span><span class="lineCov"> 33792 : pa[i] = Dx;</span></a>
<a name="451"><span class="lineNum"> 451 </span><span class="lineCov"> 33792 : qa[i] = Dy;</span></a>
<a name="452"><span class="lineNum"> 452 </span> : }</a>
<a name="453"><span class="lineNum"> 453 </span><span class="lineCov"> 132 : }</span></a>
</pre>
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