MFC r296519: MFV r296518: 5027 zfs large block support (add copyright)

[FreeBSD/stable/10.git] / sys / cddl / contrib / opensolaris / uts / common / fs / zfs / vdev_raidz.c

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License (the "License").
 * You may not use this file except in compliance with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
 * information: Portions Copyright [yyyy] [name of copyright owner]
 *
 * CDDL HEADER END
 */

/*
 * Copyright (c) 2005, 2010, Oracle and/or its affiliates. All rights reserved.
 * Copyright (c) 2012, 2014 by Delphix. All rights reserved.
 * Copyright (c) 2013, Joyent, Inc. All rights reserved.
 * Copyright (c) 2014 Integros [integros.com]
 */

#include <sys/zfs_context.h>
#include <sys/spa.h>
#include <sys/vdev_impl.h>
#ifdef illumos
#include <sys/vdev_disk.h>
#endif
#include <sys/vdev_file.h>
#include <sys/vdev_raidz.h>
#include <sys/zio.h>
#include <sys/zio_checksum.h>
#include <sys/fs/zfs.h>
#include <sys/fm/fs/zfs.h>
#include <sys/bio.h>

/*
 * Virtual device vector for RAID-Z.
 *
 * This vdev supports single, double, and triple parity. For single parity,
 * we use a simple XOR of all the data columns. For double or triple parity,
 * we use a special case of Reed-Solomon coding. This extends the
 * technique described in "The mathematics of RAID-6" by H. Peter Anvin by
 * drawing on the system described in "A Tutorial on Reed-Solomon Coding for
 * Fault-Tolerance in RAID-like Systems" by James S. Plank on which the
 * former is also based. The latter is designed to provide higher performance
 * for writes.
 *
 * Note that the Plank paper claimed to support arbitrary N+M, but was then
 * amended six years later identifying a critical flaw that invalidates its
 * claims. Nevertheless, the technique can be adapted to work for up to
 * triple parity. For additional parity, the amendment "Note: Correction to
 * the 1997 Tutorial on Reed-Solomon Coding" by James S. Plank and Ying Ding
 * is viable, but the additional complexity means that write performance will
 * suffer.
 *
 * All of the methods above operate on a Galois field, defined over the
 * integers mod 2^N. In our case we choose N=8 for GF(8) so that all elements
 * can be expressed with a single byte. Briefly, the operations on the
 * field are defined as follows:
 *
 *   o addition (+) is represented by a bitwise XOR
 *   o subtraction (-) is therefore identical to addition: A + B = A - B
 *   o multiplication of A by 2 is defined by the following bitwise expression:
 *
 *      (A * 2)_7 = A_6
 *      (A * 2)_6 = A_5
 *      (A * 2)_5 = A_4
 *      (A * 2)_4 = A_3 + A_7
 *      (A * 2)_3 = A_2 + A_7
 *      (A * 2)_2 = A_1 + A_7
 *      (A * 2)_1 = A_0
 *      (A * 2)_0 = A_7
 *
 * In C, multiplying by 2 is therefore ((a << 1) ^ ((a & 0x80) ? 0x1d : 0)).
 * As an aside, this multiplication is derived from the error correcting
 * primitive polynomial x^8 + x^4 + x^3 + x^2 + 1.
 *
 * Observe that any number in the field (except for 0) can be expressed as a
 * power of 2 -- a generator for the field. We store a table of the powers of
 * 2 and logs base 2 for quick look ups, and exploit the fact that A * B can
 * be rewritten as 2^(log_2(A) + log_2(B)) (where '+' is normal addition rather
 * than field addition). The inverse of a field element A (A^-1) is therefore
 * A ^ (255 - 1) = A^254.
 *
 * The up-to-three parity columns, P, Q, R over several data columns,
 * D_0, ... D_n-1, can be expressed by field operations:
 *
 *      P = D_0 + D_1 + ... + D_n-2 + D_n-1
 *      Q = 2^n-1 * D_0 + 2^n-2 * D_1 + ... + 2^1 * D_n-2 + 2^0 * D_n-1
 *        = ((...((D_0) * 2 + D_1) * 2 + ...) * 2 + D_n-2) * 2 + D_n-1
 *      R = 4^n-1 * D_0 + 4^n-2 * D_1 + ... + 4^1 * D_n-2 + 4^0 * D_n-1
 *        = ((...((D_0) * 4 + D_1) * 4 + ...) * 4 + D_n-2) * 4 + D_n-1
 *
 * We chose 1, 2, and 4 as our generators because 1 corresponds to the trival
 * XOR operation, and 2 and 4 can be computed quickly and generate linearly-
 * independent coefficients. (There are no additional coefficients that have
 * this property which is why the uncorrected Plank method breaks down.)
 *
 * See the reconstruction code below for how P, Q and R can used individually
 * or in concert to recover missing data columns.
 */

typedef struct raidz_col {
        uint64_t rc_devidx;             /* child device index for I/O */
        uint64_t rc_offset;             /* device offset */
        uint64_t rc_size;               /* I/O size */
        void *rc_data;                  /* I/O data */
        void *rc_gdata;                 /* used to store the "good" version */
        int rc_error;                   /* I/O error for this device */
        uint8_t rc_tried;               /* Did we attempt this I/O column? */
        uint8_t rc_skipped;             /* Did we skip this I/O column? */
} raidz_col_t;

typedef struct raidz_map {
        uint64_t rm_cols;               /* Regular column count */
        uint64_t rm_scols;              /* Count including skipped columns */
        uint64_t rm_bigcols;            /* Number of oversized columns */
        uint64_t rm_asize;              /* Actual total I/O size */
        uint64_t rm_missingdata;        /* Count of missing data devices */
        uint64_t rm_missingparity;      /* Count of missing parity devices */
        uint64_t rm_firstdatacol;       /* First data column/parity count */
        uint64_t rm_nskip;              /* Skipped sectors for padding */
        uint64_t rm_skipstart;          /* Column index of padding start */
        void *rm_datacopy;              /* rm_asize-buffer of copied data */
        uintptr_t rm_reports;           /* # of referencing checksum reports */
        uint8_t rm_freed;               /* map no longer has referencing ZIO */
        uint8_t rm_ecksuminjected;      /* checksum error was injected */
        raidz_col_t rm_col[1];          /* Flexible array of I/O columns */
} raidz_map_t;

#define VDEV_RAIDZ_P            0
#define VDEV_RAIDZ_Q            1
#define VDEV_RAIDZ_R            2

#define VDEV_RAIDZ_MUL_2(x)     (((x) << 1) ^ (((x) & 0x80) ? 0x1d : 0))
#define VDEV_RAIDZ_MUL_4(x)     (VDEV_RAIDZ_MUL_2(VDEV_RAIDZ_MUL_2(x)))

/*
 * We provide a mechanism to perform the field multiplication operation on a
 * 64-bit value all at once rather than a byte at a time. This works by
 * creating a mask from the top bit in each byte and using that to
 * conditionally apply the XOR of 0x1d.
 */
#define VDEV_RAIDZ_64MUL_2(x, mask) \
{ \
        (mask) = (x) & 0x8080808080808080ULL; \
        (mask) = ((mask) << 1) - ((mask) >> 7); \
        (x) = (((x) << 1) & 0xfefefefefefefefeULL) ^ \
            ((mask) & 0x1d1d1d1d1d1d1d1d); \
}

#define VDEV_RAIDZ_64MUL_4(x, mask) \
{ \
        VDEV_RAIDZ_64MUL_2((x), mask); \
        VDEV_RAIDZ_64MUL_2((x), mask); \
}

#define VDEV_LABEL_OFFSET(x)    (x + VDEV_LABEL_START_SIZE)

/*
 * Force reconstruction to use the general purpose method.
 */
int vdev_raidz_default_to_general;

/* Powers of 2 in the Galois field defined above. */
static const uint8_t vdev_raidz_pow2[256] = {
        0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80,
        0x1d, 0x3a, 0x74, 0xe8, 0xcd, 0x87, 0x13, 0x26,
        0x4c, 0x98, 0x2d, 0x5a, 0xb4, 0x75, 0xea, 0xc9,
        0x8f, 0x03, 0x06, 0x0c, 0x18, 0x30, 0x60, 0xc0,
        0x9d, 0x27, 0x4e, 0x9c, 0x25, 0x4a, 0x94, 0x35,
        0x6a, 0xd4, 0xb5, 0x77, 0xee, 0xc1, 0x9f, 0x23,
        0x46, 0x8c, 0x05, 0x0a, 0x14, 0x28, 0x50, 0xa0,
        0x5d, 0xba, 0x69, 0xd2, 0xb9, 0x6f, 0xde, 0xa1,
        0x5f, 0xbe, 0x61, 0xc2, 0x99, 0x2f, 0x5e, 0xbc,
        0x65, 0xca, 0x89, 0x0f, 0x1e, 0x3c, 0x78, 0xf0,
        0xfd, 0xe7, 0xd3, 0xbb, 0x6b, 0xd6, 0xb1, 0x7f,
        0xfe, 0xe1, 0xdf, 0xa3, 0x5b, 0xb6, 0x71, 0xe2,
        0xd9, 0xaf, 0x43, 0x86, 0x11, 0x22, 0x44, 0x88,
        0x0d, 0x1a, 0x34, 0x68, 0xd0, 0xbd, 0x67, 0xce,
        0x81, 0x1f, 0x3e, 0x7c, 0xf8, 0xed, 0xc7, 0x93,
        0x3b, 0x76, 0xec, 0xc5, 0x97, 0x33, 0x66, 0xcc,
        0x85, 0x17, 0x2e, 0x5c, 0xb8, 0x6d, 0xda, 0xa9,
        0x4f, 0x9e, 0x21, 0x42, 0x84, 0x15, 0x2a, 0x54,
        0xa8, 0x4d, 0x9a, 0x29, 0x52, 0xa4, 0x55, 0xaa,
        0x49, 0x92, 0x39, 0x72, 0xe4, 0xd5, 0xb7, 0x73,
        0xe6, 0xd1, 0xbf, 0x63, 0xc6, 0x91, 0x3f, 0x7e,
        0xfc, 0xe5, 0xd7, 0xb3, 0x7b, 0xf6, 0xf1, 0xff,
        0xe3, 0xdb, 0xab, 0x4b, 0x96, 0x31, 0x62, 0xc4,
        0x95, 0x37, 0x6e, 0xdc, 0xa5, 0x57, 0xae, 0x41,
        0x82, 0x19, 0x32, 0x64, 0xc8, 0x8d, 0x07, 0x0e,
        0x1c, 0x38, 0x70, 0xe0, 0xdd, 0xa7, 0x53, 0xa6,
        0x51, 0xa2, 0x59, 0xb2, 0x79, 0xf2, 0xf9, 0xef,
        0xc3, 0x9b, 0x2b, 0x56, 0xac, 0x45, 0x8a, 0x09,
        0x12, 0x24, 0x48, 0x90, 0x3d, 0x7a, 0xf4, 0xf5,
        0xf7, 0xf3, 0xfb, 0xeb, 0xcb, 0x8b, 0x0b, 0x16,
        0x2c, 0x58, 0xb0, 0x7d, 0xfa, 0xe9, 0xcf, 0x83,
        0x1b, 0x36, 0x6c, 0xd8, 0xad, 0x47, 0x8e, 0x01
};
/* Logs of 2 in the Galois field defined above. */
static const uint8_t vdev_raidz_log2[256] = {
        0x00, 0x00, 0x01, 0x19, 0x02, 0x32, 0x1a, 0xc6,
        0x03, 0xdf, 0x33, 0xee, 0x1b, 0x68, 0xc7, 0x4b,
        0x04, 0x64, 0xe0, 0x0e, 0x34, 0x8d, 0xef, 0x81,
        0x1c, 0xc1, 0x69, 0xf8, 0xc8, 0x08, 0x4c, 0x71,
        0x05, 0x8a, 0x65, 0x2f, 0xe1, 0x24, 0x0f, 0x21,
        0x35, 0x93, 0x8e, 0xda, 0xf0, 0x12, 0x82, 0x45,
        0x1d, 0xb5, 0xc2, 0x7d, 0x6a, 0x27, 0xf9, 0xb9,
        0xc9, 0x9a, 0x09, 0x78, 0x4d, 0xe4, 0x72, 0xa6,
        0x06, 0xbf, 0x8b, 0x62, 0x66, 0xdd, 0x30, 0xfd,
        0xe2, 0x98, 0x25, 0xb3, 0x10, 0x91, 0x22, 0x88,
        0x36, 0xd0, 0x94, 0xce, 0x8f, 0x96, 0xdb, 0xbd,
        0xf1, 0xd2, 0x13, 0x5c, 0x83, 0x38, 0x46, 0x40,
        0x1e, 0x42, 0xb6, 0xa3, 0xc3, 0x48, 0x7e, 0x6e,
        0x6b, 0x3a, 0x28, 0x54, 0xfa, 0x85, 0xba, 0x3d,
        0xca, 0x5e, 0x9b, 0x9f, 0x0a, 0x15, 0x79, 0x2b,
        0x4e, 0xd4, 0xe5, 0xac, 0x73, 0xf3, 0xa7, 0x57,
        0x07, 0x70, 0xc0, 0xf7, 0x8c, 0x80, 0x63, 0x0d,
        0x67, 0x4a, 0xde, 0xed, 0x31, 0xc5, 0xfe, 0x18,
        0xe3, 0xa5, 0x99, 0x77, 0x26, 0xb8, 0xb4, 0x7c,
        0x11, 0x44, 0x92, 0xd9, 0x23, 0x20, 0x89, 0x2e,
        0x37, 0x3f, 0xd1, 0x5b, 0x95, 0xbc, 0xcf, 0xcd,
        0x90, 0x87, 0x97, 0xb2, 0xdc, 0xfc, 0xbe, 0x61,
        0xf2, 0x56, 0xd3, 0xab, 0x14, 0x2a, 0x5d, 0x9e,
        0x84, 0x3c, 0x39, 0x53, 0x47, 0x6d, 0x41, 0xa2,
        0x1f, 0x2d, 0x43, 0xd8, 0xb7, 0x7b, 0xa4, 0x76,
        0xc4, 0x17, 0x49, 0xec, 0x7f, 0x0c, 0x6f, 0xf6,
        0x6c, 0xa1, 0x3b, 0x52, 0x29, 0x9d, 0x55, 0xaa,
        0xfb, 0x60, 0x86, 0xb1, 0xbb, 0xcc, 0x3e, 0x5a,
        0xcb, 0x59, 0x5f, 0xb0, 0x9c, 0xa9, 0xa0, 0x51,
        0x0b, 0xf5, 0x16, 0xeb, 0x7a, 0x75, 0x2c, 0xd7,
        0x4f, 0xae, 0xd5, 0xe9, 0xe6, 0xe7, 0xad, 0xe8,
        0x74, 0xd6, 0xf4, 0xea, 0xa8, 0x50, 0x58, 0xaf,
};

static void vdev_raidz_generate_parity(raidz_map_t *rm);

/*
 * Multiply a given number by 2 raised to the given power.
 */
static uint8_t
vdev_raidz_exp2(uint_t a, int exp)
{
        if (a == 0)
                return (0);

        ASSERT(exp >= 0);
        ASSERT(vdev_raidz_log2[a] > 0 || a == 1);

        exp += vdev_raidz_log2[a];
        if (exp > 255)
                exp -= 255;

        return (vdev_raidz_pow2[exp]);
}

static void
vdev_raidz_map_free(raidz_map_t *rm)
{
        int c;
        size_t size;

        for (c = 0; c < rm->rm_firstdatacol; c++) {
                if (rm->rm_col[c].rc_data != NULL)
                        zio_buf_free(rm->rm_col[c].rc_data,
                            rm->rm_col[c].rc_size);

                if (rm->rm_col[c].rc_gdata != NULL)
                        zio_buf_free(rm->rm_col[c].rc_gdata,
                            rm->rm_col[c].rc_size);
        }

        size = 0;
        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++)
                size += rm->rm_col[c].rc_size;

        if (rm->rm_datacopy != NULL)
                zio_buf_free(rm->rm_datacopy, size);

        kmem_free(rm, offsetof(raidz_map_t, rm_col[rm->rm_scols]));
}

static void
vdev_raidz_map_free_vsd(zio_t *zio)
{
        raidz_map_t *rm = zio->io_vsd;

        ASSERT0(rm->rm_freed);
        rm->rm_freed = 1;

        if (rm->rm_reports == 0)
                vdev_raidz_map_free(rm);
}

/*ARGSUSED*/
static void
vdev_raidz_cksum_free(void *arg, size_t ignored)
{
        raidz_map_t *rm = arg;

        ASSERT3U(rm->rm_reports, >, 0);

        if (--rm->rm_reports == 0 && rm->rm_freed != 0)
                vdev_raidz_map_free(rm);
}

static void
vdev_raidz_cksum_finish(zio_cksum_report_t *zcr, const void *good_data)
{
        raidz_map_t *rm = zcr->zcr_cbdata;
        size_t c = zcr->zcr_cbinfo;
        size_t x;

        const char *good = NULL;
        const char *bad = rm->rm_col[c].rc_data;

        if (good_data == NULL) {
                zfs_ereport_finish_checksum(zcr, NULL, NULL, B_FALSE);
                return;
        }

        if (c < rm->rm_firstdatacol) {
                /*
                 * The first time through, calculate the parity blocks for
                 * the good data (this relies on the fact that the good
                 * data never changes for a given logical ZIO)
                 */
                if (rm->rm_col[0].rc_gdata == NULL) {
                        char *bad_parity[VDEV_RAIDZ_MAXPARITY];
                        char *buf;

                        /*
                         * Set up the rm_col[]s to generate the parity for
                         * good_data, first saving the parity bufs and
                         * replacing them with buffers to hold the result.
                         */
                        for (x = 0; x < rm->rm_firstdatacol; x++) {
                                bad_parity[x] = rm->rm_col[x].rc_data;
                                rm->rm_col[x].rc_data = rm->rm_col[x].rc_gdata =
                                    zio_buf_alloc(rm->rm_col[x].rc_size);
                        }

                        /* fill in the data columns from good_data */
                        buf = (char *)good_data;
                        for (; x < rm->rm_cols; x++) {
                                rm->rm_col[x].rc_data = buf;
                                buf += rm->rm_col[x].rc_size;
                        }

                        /*
                         * Construct the parity from the good data.
                         */
                        vdev_raidz_generate_parity(rm);

                        /* restore everything back to its original state */
                        for (x = 0; x < rm->rm_firstdatacol; x++)
                                rm->rm_col[x].rc_data = bad_parity[x];

                        buf = rm->rm_datacopy;
                        for (x = rm->rm_firstdatacol; x < rm->rm_cols; x++) {
                                rm->rm_col[x].rc_data = buf;
                                buf += rm->rm_col[x].rc_size;
                        }
                }

                ASSERT3P(rm->rm_col[c].rc_gdata, !=, NULL);
                good = rm->rm_col[c].rc_gdata;
        } else {
                /* adjust good_data to point at the start of our column */
                good = good_data;

                for (x = rm->rm_firstdatacol; x < c; x++)
                        good += rm->rm_col[x].rc_size;
        }

        /* we drop the ereport if it ends up that the data was good */
        zfs_ereport_finish_checksum(zcr, good, bad, B_TRUE);
}

/*
 * Invoked indirectly by zfs_ereport_start_checksum(), called
 * below when our read operation fails completely.  The main point
 * is to keep a copy of everything we read from disk, so that at
 * vdev_raidz_cksum_finish() time we can compare it with the good data.
 */
static void
vdev_raidz_cksum_report(zio_t *zio, zio_cksum_report_t *zcr, void *arg)
{
        size_t c = (size_t)(uintptr_t)arg;
        caddr_t buf;

        raidz_map_t *rm = zio->io_vsd;
        size_t size;

        /* set up the report and bump the refcount  */
        zcr->zcr_cbdata = rm;
        zcr->zcr_cbinfo = c;
        zcr->zcr_finish = vdev_raidz_cksum_finish;
        zcr->zcr_free = vdev_raidz_cksum_free;

        rm->rm_reports++;
        ASSERT3U(rm->rm_reports, >, 0);

        if (rm->rm_datacopy != NULL)
                return;

        /*
         * It's the first time we're called for this raidz_map_t, so we need
         * to copy the data aside; there's no guarantee that our zio's buffer
         * won't be re-used for something else.
         *
         * Our parity data is already in separate buffers, so there's no need
         * to copy them.
         */

        size = 0;
        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++)
                size += rm->rm_col[c].rc_size;

        buf = rm->rm_datacopy = zio_buf_alloc(size);

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                raidz_col_t *col = &rm->rm_col[c];

                bcopy(col->rc_data, buf, col->rc_size);
                col->rc_data = buf;

                buf += col->rc_size;
        }
        ASSERT3P(buf - (caddr_t)rm->rm_datacopy, ==, size);
}

static const zio_vsd_ops_t vdev_raidz_vsd_ops = {
        vdev_raidz_map_free_vsd,
        vdev_raidz_cksum_report
};

/*
 * Divides the IO evenly across all child vdevs; usually, dcols is
 * the number of children in the target vdev.
 */
static raidz_map_t *
vdev_raidz_map_alloc(caddr_t data, uint64_t size, uint64_t offset, boolean_t dofree,
    uint64_t unit_shift, uint64_t dcols, uint64_t nparity)
{
        raidz_map_t *rm;
        /* The starting RAIDZ (parent) vdev sector of the block. */
        uint64_t b = offset >> unit_shift;
        /* The zio's size in units of the vdev's minimum sector size. */
        uint64_t s = size >> unit_shift;
        /* The first column for this stripe. */
        uint64_t f = b % dcols;
        /* The starting byte offset on each child vdev. */
        uint64_t o = (b / dcols) << unit_shift;
        uint64_t q, r, c, bc, col, acols, scols, coff, devidx, asize, tot;

        /*
         * "Quotient": The number of data sectors for this stripe on all but
         * the "big column" child vdevs that also contain "remainder" data.
         */
        q = s / (dcols - nparity);

        /*
         * "Remainder": The number of partial stripe data sectors in this I/O.
         * This will add a sector to some, but not all, child vdevs.
         */
        r = s - q * (dcols - nparity);

        /* The number of "big columns" - those which contain remainder data. */
        bc = (r == 0 ? 0 : r + nparity);

        /*
         * The total number of data and parity sectors associated with
         * this I/O.
         */
        tot = s + nparity * (q + (r == 0 ? 0 : 1));

        /* acols: The columns that will be accessed. */
        /* scols: The columns that will be accessed or skipped. */
        if (q == 0) {
                /* Our I/O request doesn't span all child vdevs. */
                acols = bc;
                scols = MIN(dcols, roundup(bc, nparity + 1));
        } else {
                acols = dcols;
                scols = dcols;
        }

        ASSERT3U(acols, <=, scols);

        rm = kmem_alloc(offsetof(raidz_map_t, rm_col[scols]), KM_SLEEP);

        rm->rm_cols = acols;
        rm->rm_scols = scols;
        rm->rm_bigcols = bc;
        rm->rm_skipstart = bc;
        rm->rm_missingdata = 0;
        rm->rm_missingparity = 0;
        rm->rm_firstdatacol = nparity;
        rm->rm_datacopy = NULL;
        rm->rm_reports = 0;
        rm->rm_freed = 0;
        rm->rm_ecksuminjected = 0;

        asize = 0;

        for (c = 0; c < scols; c++) {
                col = f + c;
                coff = o;
                if (col >= dcols) {
                        col -= dcols;
                        coff += 1ULL << unit_shift;
                }
                rm->rm_col[c].rc_devidx = col;
                rm->rm_col[c].rc_offset = coff;
                rm->rm_col[c].rc_data = NULL;
                rm->rm_col[c].rc_gdata = NULL;
                rm->rm_col[c].rc_error = 0;
                rm->rm_col[c].rc_tried = 0;
                rm->rm_col[c].rc_skipped = 0;

                if (c >= acols)
                        rm->rm_col[c].rc_size = 0;
                else if (c < bc)
                        rm->rm_col[c].rc_size = (q + 1) << unit_shift;
                else
                        rm->rm_col[c].rc_size = q << unit_shift;

                asize += rm->rm_col[c].rc_size;
        }

        ASSERT3U(asize, ==, tot << unit_shift);
        rm->rm_asize = roundup(asize, (nparity + 1) << unit_shift);
        rm->rm_nskip = roundup(tot, nparity + 1) - tot;
        ASSERT3U(rm->rm_asize - asize, ==, rm->rm_nskip << unit_shift);
        ASSERT3U(rm->rm_nskip, <=, nparity);

        if (!dofree) {
                for (c = 0; c < rm->rm_firstdatacol; c++) {
                        rm->rm_col[c].rc_data =
                            zio_buf_alloc(rm->rm_col[c].rc_size);
                }

                rm->rm_col[c].rc_data = data;

                for (c = c + 1; c < acols; c++) {
                        rm->rm_col[c].rc_data =
                            (char *)rm->rm_col[c - 1].rc_data +
                            rm->rm_col[c - 1].rc_size;
                }
        }

        /*
         * If all data stored spans all columns, there's a danger that parity
         * will always be on the same device and, since parity isn't read
         * during normal operation, that that device's I/O bandwidth won't be
         * used effectively. We therefore switch the parity every 1MB.
         *
         * ... at least that was, ostensibly, the theory. As a practical
         * matter unless we juggle the parity between all devices evenly, we
         * won't see any benefit. Further, occasional writes that aren't a
         * multiple of the LCM of the number of children and the minimum
         * stripe width are sufficient to avoid pessimal behavior.
         * Unfortunately, this decision created an implicit on-disk format
         * requirement that we need to support for all eternity, but only
         * for single-parity RAID-Z.
         *
         * If we intend to skip a sector in the zeroth column for padding
         * we must make sure to note this swap. We will never intend to
         * skip the first column since at least one data and one parity
         * column must appear in each row.
         */
        ASSERT(rm->rm_cols >= 2);
        ASSERT(rm->rm_col[0].rc_size == rm->rm_col[1].rc_size);

        if (rm->rm_firstdatacol == 1 && (offset & (1ULL << 20))) {
                devidx = rm->rm_col[0].rc_devidx;
                o = rm->rm_col[0].rc_offset;
                rm->rm_col[0].rc_devidx = rm->rm_col[1].rc_devidx;
                rm->rm_col[0].rc_offset = rm->rm_col[1].rc_offset;
                rm->rm_col[1].rc_devidx = devidx;
                rm->rm_col[1].rc_offset = o;

                if (rm->rm_skipstart == 0)
                        rm->rm_skipstart = 1;
        }

        return (rm);
}

static void
vdev_raidz_generate_parity_p(raidz_map_t *rm)
{
        uint64_t *p, *src, pcount, ccount, i;
        int c;

        pcount = rm->rm_col[VDEV_RAIDZ_P].rc_size / sizeof (src[0]);

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                src = rm->rm_col[c].rc_data;
                p = rm->rm_col[VDEV_RAIDZ_P].rc_data;
                ccount = rm->rm_col[c].rc_size / sizeof (src[0]);

                if (c == rm->rm_firstdatacol) {
                        ASSERT(ccount == pcount);
                        for (i = 0; i < ccount; i++, src++, p++) {
                                *p = *src;
                        }
                } else {
                        ASSERT(ccount <= pcount);
                        for (i = 0; i < ccount; i++, src++, p++) {
                                *p ^= *src;
                        }
                }
        }
}

static void
vdev_raidz_generate_parity_pq(raidz_map_t *rm)
{
        uint64_t *p, *q, *src, pcnt, ccnt, mask, i;
        int c;

        pcnt = rm->rm_col[VDEV_RAIDZ_P].rc_size / sizeof (src[0]);
        ASSERT(rm->rm_col[VDEV_RAIDZ_P].rc_size ==
            rm->rm_col[VDEV_RAIDZ_Q].rc_size);

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                src = rm->rm_col[c].rc_data;
                p = rm->rm_col[VDEV_RAIDZ_P].rc_data;
                q = rm->rm_col[VDEV_RAIDZ_Q].rc_data;

                ccnt = rm->rm_col[c].rc_size / sizeof (src[0]);

                if (c == rm->rm_firstdatacol) {
                        ASSERT(ccnt == pcnt || ccnt == 0);
                        for (i = 0; i < ccnt; i++, src++, p++, q++) {
                                *p = *src;
                                *q = *src;
                        }
                        for (; i < pcnt; i++, src++, p++, q++) {
                                *p = 0;
                                *q = 0;
                        }
                } else {
                        ASSERT(ccnt <= pcnt);

                        /*
                         * Apply the algorithm described above by multiplying
                         * the previous result and adding in the new value.
                         */
                        for (i = 0; i < ccnt; i++, src++, p++, q++) {
                                *p ^= *src;

                                VDEV_RAIDZ_64MUL_2(*q, mask);
                                *q ^= *src;
                        }

                        /*
                         * Treat short columns as though they are full of 0s.
                         * Note that there's therefore nothing needed for P.
                         */
                        for (; i < pcnt; i++, q++) {
                                VDEV_RAIDZ_64MUL_2(*q, mask);
                        }
                }
        }
}

static void
vdev_raidz_generate_parity_pqr(raidz_map_t *rm)
{
        uint64_t *p, *q, *r, *src, pcnt, ccnt, mask, i;
        int c;

        pcnt = rm->rm_col[VDEV_RAIDZ_P].rc_size / sizeof (src[0]);
        ASSERT(rm->rm_col[VDEV_RAIDZ_P].rc_size ==
            rm->rm_col[VDEV_RAIDZ_Q].rc_size);
        ASSERT(rm->rm_col[VDEV_RAIDZ_P].rc_size ==
            rm->rm_col[VDEV_RAIDZ_R].rc_size);

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                src = rm->rm_col[c].rc_data;
                p = rm->rm_col[VDEV_RAIDZ_P].rc_data;
                q = rm->rm_col[VDEV_RAIDZ_Q].rc_data;
                r = rm->rm_col[VDEV_RAIDZ_R].rc_data;

                ccnt = rm->rm_col[c].rc_size / sizeof (src[0]);

                if (c == rm->rm_firstdatacol) {
                        ASSERT(ccnt == pcnt || ccnt == 0);
                        for (i = 0; i < ccnt; i++, src++, p++, q++, r++) {
                                *p = *src;
                                *q = *src;
                                *r = *src;
                        }
                        for (; i < pcnt; i++, src++, p++, q++, r++) {
                                *p = 0;
                                *q = 0;
                                *r = 0;
                        }
                } else {
                        ASSERT(ccnt <= pcnt);

                        /*
                         * Apply the algorithm described above by multiplying
                         * the previous result and adding in the new value.
                         */
                        for (i = 0; i < ccnt; i++, src++, p++, q++, r++) {
                                *p ^= *src;

                                VDEV_RAIDZ_64MUL_2(*q, mask);
                                *q ^= *src;

                                VDEV_RAIDZ_64MUL_4(*r, mask);
                                *r ^= *src;
                        }

                        /*
                         * Treat short columns as though they are full of 0s.
                         * Note that there's therefore nothing needed for P.
                         */
                        for (; i < pcnt; i++, q++, r++) {
                                VDEV_RAIDZ_64MUL_2(*q, mask);
                                VDEV_RAIDZ_64MUL_4(*r, mask);
                        }
                }
        }
}

/*
 * Generate RAID parity in the first virtual columns according to the number of
 * parity columns available.
 */
static void
vdev_raidz_generate_parity(raidz_map_t *rm)
{
        switch (rm->rm_firstdatacol) {
        case 1:
                vdev_raidz_generate_parity_p(rm);
                break;
        case 2:
                vdev_raidz_generate_parity_pq(rm);
                break;
        case 3:
                vdev_raidz_generate_parity_pqr(rm);
                break;
        default:
                cmn_err(CE_PANIC, "invalid RAID-Z configuration");
        }
}

static int
vdev_raidz_reconstruct_p(raidz_map_t *rm, int *tgts, int ntgts)
{
        uint64_t *dst, *src, xcount, ccount, count, i;
        int x = tgts[0];
        int c;

        ASSERT(ntgts == 1);
        ASSERT(x >= rm->rm_firstdatacol);
        ASSERT(x < rm->rm_cols);

        xcount = rm->rm_col[x].rc_size / sizeof (src[0]);
        ASSERT(xcount <= rm->rm_col[VDEV_RAIDZ_P].rc_size / sizeof (src[0]));
        ASSERT(xcount > 0);

        src = rm->rm_col[VDEV_RAIDZ_P].rc_data;
        dst = rm->rm_col[x].rc_data;
        for (i = 0; i < xcount; i++, dst++, src++) {
                *dst = *src;
        }

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                src = rm->rm_col[c].rc_data;
                dst = rm->rm_col[x].rc_data;

                if (c == x)
                        continue;

                ccount = rm->rm_col[c].rc_size / sizeof (src[0]);
                count = MIN(ccount, xcount);

                for (i = 0; i < count; i++, dst++, src++) {
                        *dst ^= *src;
                }
        }

        return (1 << VDEV_RAIDZ_P);
}

static int
vdev_raidz_reconstruct_q(raidz_map_t *rm, int *tgts, int ntgts)
{
        uint64_t *dst, *src, xcount, ccount, count, mask, i;
        uint8_t *b;
        int x = tgts[0];
        int c, j, exp;

        ASSERT(ntgts == 1);

        xcount = rm->rm_col[x].rc_size / sizeof (src[0]);
        ASSERT(xcount <= rm->rm_col[VDEV_RAIDZ_Q].rc_size / sizeof (src[0]));

        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                src = rm->rm_col[c].rc_data;
                dst = rm->rm_col[x].rc_data;

                if (c == x)
                        ccount = 0;
                else
                        ccount = rm->rm_col[c].rc_size / sizeof (src[0]);

                count = MIN(ccount, xcount);

                if (c == rm->rm_firstdatacol) {
                        for (i = 0; i < count; i++, dst++, src++) {
                                *dst = *src;
                        }
                        for (; i < xcount; i++, dst++) {
                                *dst = 0;
                        }

                } else {
                        for (i = 0; i < count; i++, dst++, src++) {
                                VDEV_RAIDZ_64MUL_2(*dst, mask);
                                *dst ^= *src;
                        }

                        for (; i < xcount; i++, dst++) {
                                VDEV_RAIDZ_64MUL_2(*dst, mask);
                        }
                }
        }

        src = rm->rm_col[VDEV_RAIDZ_Q].rc_data;
        dst = rm->rm_col[x].rc_data;
        exp = 255 - (rm->rm_cols - 1 - x);

        for (i = 0; i < xcount; i++, dst++, src++) {
                *dst ^= *src;
                for (j = 0, b = (uint8_t *)dst; j < 8; j++, b++) {
                        *b = vdev_raidz_exp2(*b, exp);
                }
        }

        return (1 << VDEV_RAIDZ_Q);
}

static int
vdev_raidz_reconstruct_pq(raidz_map_t *rm, int *tgts, int ntgts)
{
        uint8_t *p, *q, *pxy, *qxy, *xd, *yd, tmp, a, b, aexp, bexp;
        void *pdata, *qdata;
        uint64_t xsize, ysize, i;
        int x = tgts[0];
        int y = tgts[1];

        ASSERT(ntgts == 2);
        ASSERT(x < y);
        ASSERT(x >= rm->rm_firstdatacol);
        ASSERT(y < rm->rm_cols);

        ASSERT(rm->rm_col[x].rc_size >= rm->rm_col[y].rc_size);

        /*
         * Move the parity data aside -- we're going to compute parity as
         * though columns x and y were full of zeros -- Pxy and Qxy. We want to
         * reuse the parity generation mechanism without trashing the actual
         * parity so we make those columns appear to be full of zeros by
         * setting their lengths to zero.
         */
        pdata = rm->rm_col[VDEV_RAIDZ_P].rc_data;
        qdata = rm->rm_col[VDEV_RAIDZ_Q].rc_data;
        xsize = rm->rm_col[x].rc_size;
        ysize = rm->rm_col[y].rc_size;

        rm->rm_col[VDEV_RAIDZ_P].rc_data =
            zio_buf_alloc(rm->rm_col[VDEV_RAIDZ_P].rc_size);
        rm->rm_col[VDEV_RAIDZ_Q].rc_data =
            zio_buf_alloc(rm->rm_col[VDEV_RAIDZ_Q].rc_size);
        rm->rm_col[x].rc_size = 0;
        rm->rm_col[y].rc_size = 0;

        vdev_raidz_generate_parity_pq(rm);

        rm->rm_col[x].rc_size = xsize;
        rm->rm_col[y].rc_size = ysize;

        p = pdata;
        q = qdata;
        pxy = rm->rm_col[VDEV_RAIDZ_P].rc_data;
        qxy = rm->rm_col[VDEV_RAIDZ_Q].rc_data;
        xd = rm->rm_col[x].rc_data;
        yd = rm->rm_col[y].rc_data;

        /*
         * We now have:
         *      Pxy = P + D_x + D_y
         *      Qxy = Q + 2^(ndevs - 1 - x) * D_x + 2^(ndevs - 1 - y) * D_y
         *
         * We can then solve for D_x:
         *      D_x = A * (P + Pxy) + B * (Q + Qxy)
         * where
         *      A = 2^(x - y) * (2^(x - y) + 1)^-1
         *      B = 2^(ndevs - 1 - x) * (2^(x - y) + 1)^-1
         *
         * With D_x in hand, we can easily solve for D_y:
         *      D_y = P + Pxy + D_x
         */

        a = vdev_raidz_pow2[255 + x - y];
        b = vdev_raidz_pow2[255 - (rm->rm_cols - 1 - x)];
        tmp = 255 - vdev_raidz_log2[a ^ 1];

        aexp = vdev_raidz_log2[vdev_raidz_exp2(a, tmp)];
        bexp = vdev_raidz_log2[vdev_raidz_exp2(b, tmp)];

        for (i = 0; i < xsize; i++, p++, q++, pxy++, qxy++, xd++, yd++) {
                *xd = vdev_raidz_exp2(*p ^ *pxy, aexp) ^
                    vdev_raidz_exp2(*q ^ *qxy, bexp);

                if (i < ysize)
                        *yd = *p ^ *pxy ^ *xd;
        }

        zio_buf_free(rm->rm_col[VDEV_RAIDZ_P].rc_data,
            rm->rm_col[VDEV_RAIDZ_P].rc_size);
        zio_buf_free(rm->rm_col[VDEV_RAIDZ_Q].rc_data,
            rm->rm_col[VDEV_RAIDZ_Q].rc_size);

        /*
         * Restore the saved parity data.
         */
        rm->rm_col[VDEV_RAIDZ_P].rc_data = pdata;
        rm->rm_col[VDEV_RAIDZ_Q].rc_data = qdata;

        return ((1 << VDEV_RAIDZ_P) | (1 << VDEV_RAIDZ_Q));
}

/* BEGIN CSTYLED */
/*
 * In the general case of reconstruction, we must solve the system of linear
 * equations defined by the coeffecients used to generate parity as well as
 * the contents of the data and parity disks. This can be expressed with
 * vectors for the original data (D) and the actual data (d) and parity (p)
 * and a matrix composed of the identity matrix (I) and a dispersal matrix (V):
 *
 *            __   __                     __     __
 *            |     |         __     __   |  p_0  |
 *            |  V  |         |  D_0  |   | p_m-1 |
 *            |     |    x    |   :   | = |  d_0  |
 *            |  I  |         | D_n-1 |   |   :   |
 *            |     |         ~~     ~~   | d_n-1 |
 *            ~~   ~~                     ~~     ~~
 *
 * I is simply a square identity matrix of size n, and V is a vandermonde
 * matrix defined by the coeffecients we chose for the various parity columns
 * (1, 2, 4). Note that these values were chosen both for simplicity, speedy
 * computation as well as linear separability.
 *
 *      __               __               __     __
 *      |   1   ..  1 1 1 |               |  p_0  |
 *      | 2^n-1 ..  4 2 1 |   __     __   |   :   |
 *      | 4^n-1 .. 16 4 1 |   |  D_0  |   | p_m-1 |
 *      |   1   ..  0 0 0 |   |  D_1  |   |  d_0  |
 *      |   0   ..  0 0 0 | x |  D_2  | = |  d_1  |
 *      |   :       : : : |   |   :   |   |  d_2  |
 *      |   0   ..  1 0 0 |   | D_n-1 |   |   :   |
 *      |   0   ..  0 1 0 |   ~~     ~~   |   :   |
 *      |   0   ..  0 0 1 |               | d_n-1 |
 *      ~~               ~~               ~~     ~~
 *
 * Note that I, V, d, and p are known. To compute D, we must invert the
 * matrix and use the known data and parity values to reconstruct the unknown
 * data values. We begin by removing the rows in V|I and d|p that correspond
 * to failed or missing columns; we then make V|I square (n x n) and d|p
 * sized n by removing rows corresponding to unused parity from the bottom up
 * to generate (V|I)' and (d|p)'. We can then generate the inverse of (V|I)'
 * using Gauss-Jordan elimination. In the example below we use m=3 parity
 * columns, n=8 data columns, with errors in d_1, d_2, and p_1:
 *           __                               __
 *           |  1   1   1   1   1   1   1   1  |
 *           | 128  64  32  16  8   4   2   1  | <-----+-+-- missing disks
 *           |  19 205 116  29  64  16  4   1  |      / /
 *           |  1   0   0   0   0   0   0   0  |     / /
 *           |  0   1   0   0   0   0   0   0  | <--' /
 *  (V|I)  = |  0   0   1   0   0   0   0   0  | <---'
 *           |  0   0   0   1   0   0   0   0  |
 *           |  0   0   0   0   1   0   0   0  |
 *           |  0   0   0   0   0   1   0   0  |
 *           |  0   0   0   0   0   0   1   0  |
 *           |  0   0   0   0   0   0   0   1  |
 *           ~~                               ~~
 *           __                               __
 *           |  1   1   1   1   1   1   1   1  |
 *           |  19 205 116  29  64  16  4   1  |
 *           |  1   0   0   0   0   0   0   0  |
 *  (V|I)' = |  0   0   0   1   0   0   0   0  |
 *           |  0   0   0   0   1   0   0   0  |
 *           |  0   0   0   0   0   1   0   0  |
 *           |  0   0   0   0   0   0   1   0  |
 *           |  0   0   0   0   0   0   0   1  |
 *           ~~                               ~~
 *
 * Here we employ Gauss-Jordan elimination to find the inverse of (V|I)'. We
 * have carefully chosen the seed values 1, 2, and 4 to ensure that this
 * matrix is not singular.
 * __                                                                 __
 * |  1   1   1   1   1   1   1   1     1   0   0   0   0   0   0   0  |
 * |  19 205 116  29  64  16  4   1     0   1   0   0   0   0   0   0  |
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 * __                                                                 __
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  1   1   1   1   1   1   1   1     1   0   0   0   0   0   0   0  |
 * |  19 205 116  29  64  16  4   1     0   1   0   0   0   0   0   0  |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 * __                                                                 __
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  0   1   1   0   0   0   0   0     1   0   1   1   1   1   1   1  |
 * |  0  205 116  0   0   0   0   0     0   1   19  29  64  16  4   1  |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 * __                                                                 __
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  0   1   1   0   0   0   0   0     1   0   1   1   1   1   1   1  |
 * |  0   0  185  0   0   0   0   0    205  1  222 208 141 221 201 204 |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 * __                                                                 __
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  0   1   1   0   0   0   0   0     1   0   1   1   1   1   1   1  |
 * |  0   0   1   0   0   0   0   0    166 100  4   40 158 168 216 209 |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 * __                                                                 __
 * |  1   0   0   0   0   0   0   0     0   0   1   0   0   0   0   0  |
 * |  0   1   0   0   0   0   0   0    167 100  5   41 159 169 217 208 |
 * |  0   0   1   0   0   0   0   0    166 100  4   40 158 168 216 209 |
 * |  0   0   0   1   0   0   0   0     0   0   0   1   0   0   0   0  |
 * |  0   0   0   0   1   0   0   0     0   0   0   0   1   0   0   0  |
 * |  0   0   0   0   0   1   0   0     0   0   0   0   0   1   0   0  |
 * |  0   0   0   0   0   0   1   0     0   0   0   0   0   0   1   0  |
 * |  0   0   0   0   0   0   0   1     0   0   0   0   0   0   0   1  |
 * ~~                                                                 ~~
 *                   __                               __
 *                   |  0   0   1   0   0   0   0   0  |
 *                   | 167 100  5   41 159 169 217 208 |
 *                   | 166 100  4   40 158 168 216 209 |
 *       (V|I)'^-1 = |  0   0   0   1   0   0   0   0  |
 *                   |  0   0   0   0   1   0   0   0  |
 *                   |  0   0   0   0   0   1   0   0  |
 *                   |  0   0   0   0   0   0   1   0  |
 *                   |  0   0   0   0   0   0   0   1  |
 *                   ~~                               ~~
 *
 * We can then simply compute D = (V|I)'^-1 x (d|p)' to discover the values
 * of the missing data.
 *
 * As is apparent from the example above, the only non-trivial rows in the
 * inverse matrix correspond to the data disks that we're trying to
 * reconstruct. Indeed, those are the only rows we need as the others would
 * only be useful for reconstructing data known or assumed to be valid. For
 * that reason, we only build the coefficients in the rows that correspond to
 * targeted columns.
 */
/* END CSTYLED */

static void
vdev_raidz_matrix_init(raidz_map_t *rm, int n, int nmap, int *map,
    uint8_t **rows)
{
        int i, j;
        int pow;

        ASSERT(n == rm->rm_cols - rm->rm_firstdatacol);

        /*
         * Fill in the missing rows of interest.
         */
        for (i = 0; i < nmap; i++) {
                ASSERT3S(0, <=, map[i]);
                ASSERT3S(map[i], <=, 2);

                pow = map[i] * n;
                if (pow > 255)
                        pow -= 255;
                ASSERT(pow <= 255);

                for (j = 0; j < n; j++) {
                        pow -= map[i];
                        if (pow < 0)
                                pow += 255;
                        rows[i][j] = vdev_raidz_pow2[pow];
                }
        }
}

static void
vdev_raidz_matrix_invert(raidz_map_t *rm, int n, int nmissing, int *missing,
    uint8_t **rows, uint8_t **invrows, const uint8_t *used)
{
        int i, j, ii, jj;
        uint8_t log;

        /*
         * Assert that the first nmissing entries from the array of used
         * columns correspond to parity columns and that subsequent entries
         * correspond to data columns.
         */
        for (i = 0; i < nmissing; i++) {
                ASSERT3S(used[i], <, rm->rm_firstdatacol);
        }
        for (; i < n; i++) {
                ASSERT3S(used[i], >=, rm->rm_firstdatacol);
        }

        /*
         * First initialize the storage where we'll compute the inverse rows.
         */
        for (i = 0; i < nmissing; i++) {
                for (j = 0; j < n; j++) {
                        invrows[i][j] = (i == j) ? 1 : 0;
                }
        }

        /*
         * Subtract all trivial rows from the rows of consequence.
         */
        for (i = 0; i < nmissing; i++) {
                for (j = nmissing; j < n; j++) {
                        ASSERT3U(used[j], >=, rm->rm_firstdatacol);
                        jj = used[j] - rm->rm_firstdatacol;
                        ASSERT3S(jj, <, n);
                        invrows[i][j] = rows[i][jj];
                        rows[i][jj] = 0;
                }
        }

        /*
         * For each of the rows of interest, we must normalize it and subtract
         * a multiple of it from the other rows.
         */
        for (i = 0; i < nmissing; i++) {
                for (j = 0; j < missing[i]; j++) {
                        ASSERT0(rows[i][j]);
                }
                ASSERT3U(rows[i][missing[i]], !=, 0);

                /*
                 * Compute the inverse of the first element and multiply each
                 * element in the row by that value.
                 */
                log = 255 - vdev_raidz_log2[rows[i][missing[i]]];

                for (j = 0; j < n; j++) {
                        rows[i][j] = vdev_raidz_exp2(rows[i][j], log);
                        invrows[i][j] = vdev_raidz_exp2(invrows[i][j], log);
                }

                for (ii = 0; ii < nmissing; ii++) {
                        if (i == ii)
                                continue;

                        ASSERT3U(rows[ii][missing[i]], !=, 0);

                        log = vdev_raidz_log2[rows[ii][missing[i]]];

                        for (j = 0; j < n; j++) {
                                rows[ii][j] ^=
                                    vdev_raidz_exp2(rows[i][j], log);
                                invrows[ii][j] ^=
                                    vdev_raidz_exp2(invrows[i][j], log);
                        }
                }
        }

        /*
         * Verify that the data that is left in the rows are properly part of
         * an identity matrix.
         */
        for (i = 0; i < nmissing; i++) {
                for (j = 0; j < n; j++) {
                        if (j == missing[i]) {
                                ASSERT3U(rows[i][j], ==, 1);
                        } else {
                                ASSERT0(rows[i][j]);
                        }
                }
        }
}

static void
vdev_raidz_matrix_reconstruct(raidz_map_t *rm, int n, int nmissing,
    int *missing, uint8_t **invrows, const uint8_t *used)
{
        int i, j, x, cc, c;
        uint8_t *src;
        uint64_t ccount;
        uint8_t *dst[VDEV_RAIDZ_MAXPARITY];
        uint64_t dcount[VDEV_RAIDZ_MAXPARITY];
        uint8_t log = 0;
        uint8_t val;
        int ll;
        uint8_t *invlog[VDEV_RAIDZ_MAXPARITY];
        uint8_t *p, *pp;
        size_t psize;

        psize = sizeof (invlog[0][0]) * n * nmissing;
        p = kmem_alloc(psize, KM_SLEEP);

        for (pp = p, i = 0; i < nmissing; i++) {
                invlog[i] = pp;
                pp += n;
        }

        for (i = 0; i < nmissing; i++) {
                for (j = 0; j < n; j++) {
                        ASSERT3U(invrows[i][j], !=, 0);
                        invlog[i][j] = vdev_raidz_log2[invrows[i][j]];
                }
        }

        for (i = 0; i < n; i++) {
                c = used[i];
                ASSERT3U(c, <, rm->rm_cols);

                src = rm->rm_col[c].rc_data;
                ccount = rm->rm_col[c].rc_size;
                for (j = 0; j < nmissing; j++) {
                        cc = missing[j] + rm->rm_firstdatacol;
                        ASSERT3U(cc, >=, rm->rm_firstdatacol);
                        ASSERT3U(cc, <, rm->rm_cols);
                        ASSERT3U(cc, !=, c);

                        dst[j] = rm->rm_col[cc].rc_data;
                        dcount[j] = rm->rm_col[cc].rc_size;
                }

                ASSERT(ccount >= rm->rm_col[missing[0]].rc_size || i > 0);

                for (x = 0; x < ccount; x++, src++) {
                        if (*src != 0)
                                log = vdev_raidz_log2[*src];

                        for (cc = 0; cc < nmissing; cc++) {
                                if (x >= dcount[cc])
                                        continue;

                                if (*src == 0) {
                                        val = 0;
                                } else {
                                        if ((ll = log + invlog[cc][i]) >= 255)
                                                ll -= 255;
                                        val = vdev_raidz_pow2[ll];
                                }

                                if (i == 0)
                                        dst[cc][x] = val;
                                else
                                        dst[cc][x] ^= val;
                        }
                }
        }

        kmem_free(p, psize);
}

static int
vdev_raidz_reconstruct_general(raidz_map_t *rm, int *tgts, int ntgts)
{
        int n, i, c, t, tt;
        int nmissing_rows;
        int missing_rows[VDEV_RAIDZ_MAXPARITY];
        int parity_map[VDEV_RAIDZ_MAXPARITY];

        uint8_t *p, *pp;
        size_t psize;

        uint8_t *rows[VDEV_RAIDZ_MAXPARITY];
        uint8_t *invrows[VDEV_RAIDZ_MAXPARITY];
        uint8_t *used;

        int code = 0;


        n = rm->rm_cols - rm->rm_firstdatacol;

        /*
         * Figure out which data columns are missing.
         */
        nmissing_rows = 0;
        for (t = 0; t < ntgts; t++) {
                if (tgts[t] >= rm->rm_firstdatacol) {
                        missing_rows[nmissing_rows++] =
                            tgts[t] - rm->rm_firstdatacol;
                }
        }

        /*
         * Figure out which parity columns to use to help generate the missing
         * data columns.
         */
        for (tt = 0, c = 0, i = 0; i < nmissing_rows; c++) {
                ASSERT(tt < ntgts);
                ASSERT(c < rm->rm_firstdatacol);

                /*
                 * Skip any targeted parity columns.
                 */
                if (c == tgts[tt]) {
                        tt++;
                        continue;
                }

                code |= 1 << c;

                parity_map[i] = c;
                i++;
        }

        ASSERT(code != 0);
        ASSERT3U(code, <, 1 << VDEV_RAIDZ_MAXPARITY);

        psize = (sizeof (rows[0][0]) + sizeof (invrows[0][0])) *
            nmissing_rows * n + sizeof (used[0]) * n;
        p = kmem_alloc(psize, KM_SLEEP);

        for (pp = p, i = 0; i < nmissing_rows; i++) {
                rows[i] = pp;
                pp += n;
                invrows[i] = pp;
                pp += n;
        }
        used = pp;

        for (i = 0; i < nmissing_rows; i++) {
                used[i] = parity_map[i];
        }

        for (tt = 0, c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                if (tt < nmissing_rows &&
                    c == missing_rows[tt] + rm->rm_firstdatacol) {
                        tt++;
                        continue;
                }

                ASSERT3S(i, <, n);
                used[i] = c;
                i++;
        }

        /*
         * Initialize the interesting rows of the matrix.
         */
        vdev_raidz_matrix_init(rm, n, nmissing_rows, parity_map, rows);

        /*
         * Invert the matrix.
         */
        vdev_raidz_matrix_invert(rm, n, nmissing_rows, missing_rows, rows,
            invrows, used);

        /*
         * Reconstruct the missing data using the generated matrix.
         */
        vdev_raidz_matrix_reconstruct(rm, n, nmissing_rows, missing_rows,
            invrows, used);

        kmem_free(p, psize);

        return (code);
}

static int
vdev_raidz_reconstruct(raidz_map_t *rm, int *t, int nt)
{
        int tgts[VDEV_RAIDZ_MAXPARITY], *dt;
        int ntgts;
        int i, c;
        int code;
        int nbadparity, nbaddata;
        int parity_valid[VDEV_RAIDZ_MAXPARITY];

        /*
         * The tgts list must already be sorted.
         */
        for (i = 1; i < nt; i++) {
                ASSERT(t[i] > t[i - 1]);
        }

        nbadparity = rm->rm_firstdatacol;
        nbaddata = rm->rm_cols - nbadparity;
        ntgts = 0;
        for (i = 0, c = 0; c < rm->rm_cols; c++) {
                if (c < rm->rm_firstdatacol)
                        parity_valid[c] = B_FALSE;

                if (i < nt && c == t[i]) {
                        tgts[ntgts++] = c;
                        i++;
                } else if (rm->rm_col[c].rc_error != 0) {
                        tgts[ntgts++] = c;
                } else if (c >= rm->rm_firstdatacol) {
                        nbaddata--;
                } else {
                        parity_valid[c] = B_TRUE;
                        nbadparity--;
                }
        }

        ASSERT(ntgts >= nt);
        ASSERT(nbaddata >= 0);
        ASSERT(nbaddata + nbadparity == ntgts);

        dt = &tgts[nbadparity];

        /*
         * See if we can use any of our optimized reconstruction routines.
         */
        if (!vdev_raidz_default_to_general) {
                switch (nbaddata) {
                case 1:
                        if (parity_valid[VDEV_RAIDZ_P])
                                return (vdev_raidz_reconstruct_p(rm, dt, 1));

                        ASSERT(rm->rm_firstdatacol > 1);

                        if (parity_valid[VDEV_RAIDZ_Q])
                                return (vdev_raidz_reconstruct_q(rm, dt, 1));

                        ASSERT(rm->rm_firstdatacol > 2);
                        break;

                case 2:
                        ASSERT(rm->rm_firstdatacol > 1);

                        if (parity_valid[VDEV_RAIDZ_P] &&
                            parity_valid[VDEV_RAIDZ_Q])
                                return (vdev_raidz_reconstruct_pq(rm, dt, 2));

                        ASSERT(rm->rm_firstdatacol > 2);

                        break;
                }
        }

        code = vdev_raidz_reconstruct_general(rm, tgts, ntgts);
        ASSERT(code < (1 << VDEV_RAIDZ_MAXPARITY));
        ASSERT(code > 0);
        return (code);
}

static int
vdev_raidz_open(vdev_t *vd, uint64_t *asize, uint64_t *max_asize,
    uint64_t *logical_ashift, uint64_t *physical_ashift)
{
        vdev_t *cvd;
        uint64_t nparity = vd->vdev_nparity;
        int c;
        int lasterror = 0;
        int numerrors = 0;

        ASSERT(nparity > 0);

        if (nparity > VDEV_RAIDZ_MAXPARITY ||
            vd->vdev_children < nparity + 1) {
                vd->vdev_stat.vs_aux = VDEV_AUX_BAD_LABEL;
                return (SET_ERROR(EINVAL));
        }

        vdev_open_children(vd);

        for (c = 0; c < vd->vdev_children; c++) {
                cvd = vd->vdev_child[c];

                if (cvd->vdev_open_error != 0) {
                        lasterror = cvd->vdev_open_error;
                        numerrors++;
                        continue;
                }

                *asize = MIN(*asize - 1, cvd->vdev_asize - 1) + 1;
                *max_asize = MIN(*max_asize - 1, cvd->vdev_max_asize - 1) + 1;
                *logical_ashift = MAX(*logical_ashift, cvd->vdev_ashift);
                *physical_ashift = MAX(*physical_ashift,
                    cvd->vdev_physical_ashift);
        }

        *asize *= vd->vdev_children;
        *max_asize *= vd->vdev_children;

        if (numerrors > nparity) {
                vd->vdev_stat.vs_aux = VDEV_AUX_NO_REPLICAS;
                return (lasterror);
        }

        return (0);
}

static void
vdev_raidz_close(vdev_t *vd)
{
        int c;

        for (c = 0; c < vd->vdev_children; c++)
                vdev_close(vd->vdev_child[c]);
}

#ifdef illumos
/*
 * Handle a read or write I/O to a RAID-Z dump device.
 *
 * The dump device is in a unique situation compared to other ZFS datasets:
 * writing to this device should be as simple and fast as possible.  In
 * addition, durability matters much less since the dump will be extracted
 * once the machine reboots.  For that reason, this function eschews parity for
 * performance and simplicity.  The dump device uses the checksum setting
 * ZIO_CHECKSUM_NOPARITY to indicate that parity is not maintained for this
 * dataset.
 *
 * Blocks of size 128 KB have been preallocated for this volume.  I/Os less than
 * 128 KB will not fill an entire block; in addition, they may not be properly
 * aligned.  In that case, this function uses the preallocated 128 KB block and
 * omits reading or writing any "empty" portions of that block, as opposed to
 * allocating a fresh appropriately-sized block.
 *
 * Looking at an example of a 32 KB I/O to a RAID-Z vdev with 5 child vdevs:
 *
 *     vdev_raidz_io_start(data, size: 32 KB, offset: 64 KB)
 *
 * If this were a standard RAID-Z dataset, a block of at least 40 KB would be
 * allocated which spans all five child vdevs.  8 KB of data would be written to
 * each of four vdevs, with the fifth containing the parity bits.
 *
 *       parity    data     data     data     data
 *     |   PP   |   XX   |   XX   |   XX   |   XX   |
 *         ^        ^        ^        ^        ^
 *         |        |        |        |        |
 *   8 KB parity    ------8 KB data blocks------
 *
 * However, when writing to the dump device, the behavior is different:
 *
 *     vdev_raidz_physio(data, size: 32 KB, offset: 64 KB)
 *
 * Unlike the normal RAID-Z case in which the block is allocated based on the
 * I/O size, reads and writes here always use a 128 KB logical I/O size.  If the
 * I/O size is less than 128 KB, only the actual portions of data are written.
 * In this example the data is written to the third data vdev since that vdev
 * contains the offset [64 KB, 96 KB).
 *
 *       parity    data     data     data     data
 *     |        |        |        |   XX   |        |
 *                                    ^
 *                                    |
 *                             32 KB data block
 *
 * As a result, an individual I/O may not span all child vdevs; moreover, a
 * small I/O may only operate on a single child vdev.
 *
 * Note that since there are no parity bits calculated or written, this format
 * remains the same no matter how many parity bits are used in a normal RAID-Z
 * stripe.  On a RAID-Z3 configuration with seven child vdevs, the example above
 * would look like:
 *
 *       parity   parity   parity    data     data     data     data
 *     |        |        |        |        |        |   XX   |        |
 *                                                      ^
 *                                                      |
 *                                               32 KB data block
 */
int
vdev_raidz_physio(vdev_t *vd, caddr_t data, size_t size,
    uint64_t offset, uint64_t origoffset, boolean_t doread, boolean_t isdump)
{
        vdev_t *tvd = vd->vdev_top;
        vdev_t *cvd;
        raidz_map_t *rm;
        raidz_col_t *rc;
        int c, err = 0;

        uint64_t start, end, colstart, colend;
        uint64_t coloffset, colsize, colskip;

        int flags = doread ? BIO_READ : BIO_WRITE;

#ifdef  _KERNEL

        /*
         * Don't write past the end of the block
         */
        VERIFY3U(offset + size, <=, origoffset + SPA_OLD_MAXBLOCKSIZE);

        start = offset;
        end = start + size;

        /*
         * Allocate a RAID-Z map for this block.  Note that this block starts
         * from the "original" offset, this is, the offset of the extent which
         * contains the requisite offset of the data being read or written.
         *
         * Even if this I/O operation doesn't span the full block size, let's
         * treat the on-disk format as if the only blocks are the complete 128
         * KB size.
         */
        rm = vdev_raidz_map_alloc(data - (offset - origoffset),
            SPA_OLD_MAXBLOCKSIZE, origoffset, B_FALSE, tvd->vdev_ashift,
            vd->vdev_children, vd->vdev_nparity);

        coloffset = origoffset;

        for (c = rm->rm_firstdatacol; c < rm->rm_cols;
            c++, coloffset += rc->rc_size) {
                rc = &rm->rm_col[c];
                cvd = vd->vdev_child[rc->rc_devidx];

                /*
                 * Find the start and end of this column in the RAID-Z map,
                 * keeping in mind that the stated size and offset of the
                 * operation may not fill the entire column for this vdev.
                 *
                 * If any portion of the data spans this column, issue the
                 * appropriate operation to the vdev.
                 */
                if (coloffset + rc->rc_size <= start)
                        continue;
                if (coloffset >= end)
                        continue;

                colstart = MAX(coloffset, start);
                colend = MIN(end, coloffset + rc->rc_size);
                colsize = colend - colstart;
                colskip = colstart - coloffset;

                VERIFY3U(colsize, <=, rc->rc_size);
                VERIFY3U(colskip, <=, rc->rc_size);

                /*
                 * Note that the child vdev will have a vdev label at the start
                 * of its range of offsets, hence the need for
                 * VDEV_LABEL_OFFSET().  See zio_vdev_child_io() for another
                 * example of why this calculation is needed.
                 */
                if ((err = vdev_disk_physio(cvd,
                    ((char *)rc->rc_data) + colskip, colsize,
                    VDEV_LABEL_OFFSET(rc->rc_offset) + colskip,
                    flags, isdump)) != 0)
                        break;
        }

        vdev_raidz_map_free(rm);
#endif  /* KERNEL */

        return (err);
}
#endif

static uint64_t
vdev_raidz_asize(vdev_t *vd, uint64_t psize)
{
        uint64_t asize;
        uint64_t ashift = vd->vdev_top->vdev_ashift;
        uint64_t cols = vd->vdev_children;
        uint64_t nparity = vd->vdev_nparity;

        asize = ((psize - 1) >> ashift) + 1;
        asize += nparity * ((asize + cols - nparity - 1) / (cols - nparity));
        asize = roundup(asize, nparity + 1) << ashift;

        return (asize);
}

static void
vdev_raidz_child_done(zio_t *zio)
{
        raidz_col_t *rc = zio->io_private;

        rc->rc_error = zio->io_error;
        rc->rc_tried = 1;
        rc->rc_skipped = 0;
}

/*
 * Start an IO operation on a RAIDZ VDev
 *
 * Outline:
 * - For write operations:
 *   1. Generate the parity data
 *   2. Create child zio write operations to each column's vdev, for both
 *      data and parity.
 *   3. If the column skips any sectors for padding, create optional dummy
 *      write zio children for those areas to improve aggregation continuity.
 * - For read operations:
 *   1. Create child zio read operations to each data column's vdev to read
 *      the range of data required for zio.
 *   2. If this is a scrub or resilver operation, or if any of the data
 *      vdevs have had errors, then create zio read operations to the parity
 *      columns' VDevs as well.
 */
static void
vdev_raidz_io_start(zio_t *zio)
{
        vdev_t *vd = zio->io_vd;
        vdev_t *tvd = vd->vdev_top;
        vdev_t *cvd;
        raidz_map_t *rm;
        raidz_col_t *rc;
        int c, i;

        rm = vdev_raidz_map_alloc(zio->io_data, zio->io_size, zio->io_offset,
            zio->io_type == ZIO_TYPE_FREE,
            tvd->vdev_ashift, vd->vdev_children,
            vd->vdev_nparity);

        zio->io_vsd = rm;
        zio->io_vsd_ops = &vdev_raidz_vsd_ops;

        ASSERT3U(rm->rm_asize, ==, vdev_psize_to_asize(vd, zio->io_size));

        if (zio->io_type == ZIO_TYPE_FREE) {
                for (c = 0; c < rm->rm_cols; c++) {
                        rc = &rm->rm_col[c];
                        cvd = vd->vdev_child[rc->rc_devidx];
                        zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                            rc->rc_offset, rc->rc_data, rc->rc_size,
                            zio->io_type, zio->io_priority, 0,
                            vdev_raidz_child_done, rc));
                }

                zio_execute(zio);
                return;
        }

        if (zio->io_type == ZIO_TYPE_WRITE) {
                vdev_raidz_generate_parity(rm);

                for (c = 0; c < rm->rm_cols; c++) {
                        rc = &rm->rm_col[c];
                        cvd = vd->vdev_child[rc->rc_devidx];
                        zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                            rc->rc_offset, rc->rc_data, rc->rc_size,
                            zio->io_type, zio->io_priority, 0,
                            vdev_raidz_child_done, rc));
                }

                /*
                 * Generate optional I/Os for any skipped sectors to improve
                 * aggregation contiguity.
                 */
                for (c = rm->rm_skipstart, i = 0; i < rm->rm_nskip; c++, i++) {
                        ASSERT(c <= rm->rm_scols);
                        if (c == rm->rm_scols)
                                c = 0;
                        rc = &rm->rm_col[c];
                        cvd = vd->vdev_child[rc->rc_devidx];
                        zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                            rc->rc_offset + rc->rc_size, NULL,
                            1 << tvd->vdev_ashift,
                            zio->io_type, zio->io_priority,
                            ZIO_FLAG_NODATA | ZIO_FLAG_OPTIONAL, NULL, NULL));
                }

                zio_execute(zio);
                return;
        }

        ASSERT(zio->io_type == ZIO_TYPE_READ);

        /*
         * Iterate over the columns in reverse order so that we hit the parity
         * last -- any errors along the way will force us to read the parity.
         */
        for (c = rm->rm_cols - 1; c >= 0; c--) {
                rc = &rm->rm_col[c];
                cvd = vd->vdev_child[rc->rc_devidx];
                if (!vdev_readable(cvd)) {
                        if (c >= rm->rm_firstdatacol)
                                rm->rm_missingdata++;
                        else
                                rm->rm_missingparity++;
                        rc->rc_error = SET_ERROR(ENXIO);
                        rc->rc_tried = 1;       /* don't even try */
                        rc->rc_skipped = 1;
                        continue;
                }
                if (vdev_dtl_contains(cvd, DTL_MISSING, zio->io_txg, 1)) {
                        if (c >= rm->rm_firstdatacol)
                                rm->rm_missingdata++;
                        else
                                rm->rm_missingparity++;
                        rc->rc_error = SET_ERROR(ESTALE);
                        rc->rc_skipped = 1;
                        continue;
                }
                if (c >= rm->rm_firstdatacol || rm->rm_missingdata > 0 ||
                    (zio->io_flags & (ZIO_FLAG_SCRUB | ZIO_FLAG_RESILVER))) {
                        zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                            rc->rc_offset, rc->rc_data, rc->rc_size,
                            zio->io_type, zio->io_priority, 0,
                            vdev_raidz_child_done, rc));
                }
        }

        zio_execute(zio);
}


/*
 * Report a checksum error for a child of a RAID-Z device.
 */
static void
raidz_checksum_error(zio_t *zio, raidz_col_t *rc, void *bad_data)
{
        vdev_t *vd = zio->io_vd->vdev_child[rc->rc_devidx];

        if (!(zio->io_flags & ZIO_FLAG_SPECULATIVE)) {
                zio_bad_cksum_t zbc;
                raidz_map_t *rm = zio->io_vsd;

                mutex_enter(&vd->vdev_stat_lock);
                vd->vdev_stat.vs_checksum_errors++;
                mutex_exit(&vd->vdev_stat_lock);

                zbc.zbc_has_cksum = 0;
                zbc.zbc_injected = rm->rm_ecksuminjected;

                zfs_ereport_post_checksum(zio->io_spa, vd, zio,
                    rc->rc_offset, rc->rc_size, rc->rc_data, bad_data,
                    &zbc);
        }
}

/*
 * We keep track of whether or not there were any injected errors, so that
 * any ereports we generate can note it.
 */
static int
raidz_checksum_verify(zio_t *zio)
{
        zio_bad_cksum_t zbc;
        raidz_map_t *rm = zio->io_vsd;

        int ret = zio_checksum_error(zio, &zbc);
        if (ret != 0 && zbc.zbc_injected != 0)
                rm->rm_ecksuminjected = 1;

        return (ret);
}

/*
 * Generate the parity from the data columns. If we tried and were able to
 * read the parity without error, verify that the generated parity matches the
 * data we read. If it doesn't, we fire off a checksum error. Return the
 * number such failures.
 */
static int
raidz_parity_verify(zio_t *zio, raidz_map_t *rm)
{
        void *orig[VDEV_RAIDZ_MAXPARITY];
        int c, ret = 0;
        raidz_col_t *rc;

        blkptr_t *bp = zio->io_bp;
        enum zio_checksum checksum = (bp == NULL ? zio->io_prop.zp_checksum :
            (BP_IS_GANG(bp) ? ZIO_CHECKSUM_GANG_HEADER : BP_GET_CHECKSUM(bp)));

        if (checksum == ZIO_CHECKSUM_NOPARITY)
                return (ret);

        for (c = 0; c < rm->rm_firstdatacol; c++) {
                rc = &rm->rm_col[c];
                if (!rc->rc_tried || rc->rc_error != 0)
                        continue;
                orig[c] = zio_buf_alloc(rc->rc_size);
                bcopy(rc->rc_data, orig[c], rc->rc_size);
        }

        vdev_raidz_generate_parity(rm);

        for (c = 0; c < rm->rm_firstdatacol; c++) {
                rc = &rm->rm_col[c];
                if (!rc->rc_tried || rc->rc_error != 0)
                        continue;
                if (bcmp(orig[c], rc->rc_data, rc->rc_size) != 0) {
                        raidz_checksum_error(zio, rc, orig[c]);
                        rc->rc_error = SET_ERROR(ECKSUM);
                        ret++;
                }
                zio_buf_free(orig[c], rc->rc_size);
        }

        return (ret);
}

/*
 * Keep statistics on all the ways that we used parity to correct data.
 */
static uint64_t raidz_corrected[1 << VDEV_RAIDZ_MAXPARITY];

static int
vdev_raidz_worst_error(raidz_map_t *rm)
{
        int error = 0;

        for (int c = 0; c < rm->rm_cols; c++)
                error = zio_worst_error(error, rm->rm_col[c].rc_error);

        return (error);
}

/*
 * Iterate over all combinations of bad data and attempt a reconstruction.
 * Note that the algorithm below is non-optimal because it doesn't take into
 * account how reconstruction is actually performed. For example, with
 * triple-parity RAID-Z the reconstruction procedure is the same if column 4
 * is targeted as invalid as if columns 1 and 4 are targeted since in both
 * cases we'd only use parity information in column 0.
 */
static int
vdev_raidz_combrec(zio_t *zio, int total_errors, int data_errors)
{
        raidz_map_t *rm = zio->io_vsd;
        raidz_col_t *rc;
        void *orig[VDEV_RAIDZ_MAXPARITY];
        int tstore[VDEV_RAIDZ_MAXPARITY + 2];
        int *tgts = &tstore[1];
        int current, next, i, c, n;
        int code, ret = 0;

        ASSERT(total_errors < rm->rm_firstdatacol);

        /*
         * This simplifies one edge condition.
         */
        tgts[-1] = -1;

        for (n = 1; n <= rm->rm_firstdatacol - total_errors; n++) {
                /*
                 * Initialize the targets array by finding the first n columns
                 * that contain no error.
                 *
                 * If there were no data errors, we need to ensure that we're
                 * always explicitly attempting to reconstruct at least one
                 * data column. To do this, we simply push the highest target
                 * up into the data columns.
                 */
                for (c = 0, i = 0; i < n; i++) {
                        if (i == n - 1 && data_errors == 0 &&
                            c < rm->rm_firstdatacol) {
                                c = rm->rm_firstdatacol;
                        }

                        while (rm->rm_col[c].rc_error != 0) {
                                c++;
                                ASSERT3S(c, <, rm->rm_cols);
                        }

                        tgts[i] = c++;
                }

                /*
                 * Setting tgts[n] simplifies the other edge condition.
                 */
                tgts[n] = rm->rm_cols;

                /*
                 * These buffers were allocated in previous iterations.
                 */
                for (i = 0; i < n - 1; i++) {
                        ASSERT(orig[i] != NULL);
                }

                orig[n - 1] = zio_buf_alloc(rm->rm_col[0].rc_size);

                current = 0;
                next = tgts[current];

                while (current != n) {
                        tgts[current] = next;
                        current = 0;

                        /*
                         * Save off the original data that we're going to
                         * attempt to reconstruct.
                         */
                        for (i = 0; i < n; i++) {
                                ASSERT(orig[i] != NULL);
                                c = tgts[i];
                                ASSERT3S(c, >=, 0);
                                ASSERT3S(c, <, rm->rm_cols);
                                rc = &rm->rm_col[c];
                                bcopy(rc->rc_data, orig[i], rc->rc_size);
                        }

                        /*
                         * Attempt a reconstruction and exit the outer loop on
                         * success.
                         */
                        code = vdev_raidz_reconstruct(rm, tgts, n);
                        if (raidz_checksum_verify(zio) == 0) {
                                atomic_inc_64(&raidz_corrected[code]);

                                for (i = 0; i < n; i++) {
                                        c = tgts[i];
                                        rc = &rm->rm_col[c];
                                        ASSERT(rc->rc_error == 0);
                                        if (rc->rc_tried)
                                                raidz_checksum_error(zio, rc,
                                                    orig[i]);
                                        rc->rc_error = SET_ERROR(ECKSUM);
                                }

                                ret = code;
                                goto done;
                        }

                        /*
                         * Restore the original data.
                         */
                        for (i = 0; i < n; i++) {
                                c = tgts[i];
                                rc = &rm->rm_col[c];
                                bcopy(orig[i], rc->rc_data, rc->rc_size);
                        }

                        do {
                                /*
                                 * Find the next valid column after the current
                                 * position..
                                 */
                                for (next = tgts[current] + 1;
                                    next < rm->rm_cols &&
                                    rm->rm_col[next].rc_error != 0; next++)
                                        continue;

                                ASSERT(next <= tgts[current + 1]);

                                /*
                                 * If that spot is available, we're done here.
                                 */
                                if (next != tgts[current + 1])
                                        break;

                                /*
                                 * Otherwise, find the next valid column after
                                 * the previous position.
                                 */
                                for (c = tgts[current - 1] + 1;
                                    rm->rm_col[c].rc_error != 0; c++)
                                        continue;

                                tgts[current] = c;
                                current++;

                        } while (current != n);
                }
        }
        n--;
done:
        for (i = 0; i < n; i++) {
                zio_buf_free(orig[i], rm->rm_col[0].rc_size);
        }

        return (ret);
}

/*
 * Complete an IO operation on a RAIDZ VDev
 *
 * Outline:
 * - For write operations:
 *   1. Check for errors on the child IOs.
 *   2. Return, setting an error code if too few child VDevs were written
 *      to reconstruct the data later.  Note that partial writes are
 *      considered successful if they can be reconstructed at all.
 * - For read operations:
 *   1. Check for errors on the child IOs.
 *   2. If data errors occurred:
 *      a. Try to reassemble the data from the parity available.
 *      b. If we haven't yet read the parity drives, read them now.
 *      c. If all parity drives have been read but the data still doesn't
 *         reassemble with a correct checksum, then try combinatorial
 *         reconstruction.
 *      d. If that doesn't work, return an error.
 *   3. If there were unexpected errors or this is a resilver operation,
 *      rewrite the vdevs that had errors.
 */
static void
vdev_raidz_io_done(zio_t *zio)
{
        vdev_t *vd = zio->io_vd;
        vdev_t *cvd;
        raidz_map_t *rm = zio->io_vsd;
        raidz_col_t *rc;
        int unexpected_errors = 0;
        int parity_errors = 0;
        int parity_untried = 0;
        int data_errors = 0;
        int total_errors = 0;
        int n, c;
        int tgts[VDEV_RAIDZ_MAXPARITY];
        int code;

        ASSERT(zio->io_bp != NULL);  /* XXX need to add code to enforce this */

        ASSERT(rm->rm_missingparity <= rm->rm_firstdatacol);
        ASSERT(rm->rm_missingdata <= rm->rm_cols - rm->rm_firstdatacol);

        for (c = 0; c < rm->rm_cols; c++) {
                rc = &rm->rm_col[c];

                if (rc->rc_error) {
                        ASSERT(rc->rc_error != ECKSUM); /* child has no bp */

                        if (c < rm->rm_firstdatacol)
                                parity_errors++;
                        else
                                data_errors++;

                        if (!rc->rc_skipped)
                                unexpected_errors++;

                        total_errors++;
                } else if (c < rm->rm_firstdatacol && !rc->rc_tried) {
                        parity_untried++;
                }
        }

        if (zio->io_type == ZIO_TYPE_WRITE) {
                /*
                 * XXX -- for now, treat partial writes as a success.
                 * (If we couldn't write enough columns to reconstruct
                 * the data, the I/O failed.  Otherwise, good enough.)
                 *
                 * Now that we support write reallocation, it would be better
                 * to treat partial failure as real failure unless there are
                 * no non-degraded top-level vdevs left, and not update DTLs
                 * if we intend to reallocate.
                 */
                /* XXPOLICY */
                if (total_errors > rm->rm_firstdatacol)
                        zio->io_error = vdev_raidz_worst_error(rm);

                return;
        } else if (zio->io_type == ZIO_TYPE_FREE) {
                return;
        }

        ASSERT(zio->io_type == ZIO_TYPE_READ);
        /*
         * There are three potential phases for a read:
         *      1. produce valid data from the columns read
         *      2. read all disks and try again
         *      3. perform combinatorial reconstruction
         *
         * Each phase is progressively both more expensive and less likely to
         * occur. If we encounter more errors than we can repair or all phases
         * fail, we have no choice but to return an error.
         */

        /*
         * If the number of errors we saw was correctable -- less than or equal
         * to the number of parity disks read -- attempt to produce data that
         * has a valid checksum. Naturally, this case applies in the absence of
         * any errors.
         */
        if (total_errors <= rm->rm_firstdatacol - parity_untried) {
                if (data_errors == 0) {
                        if (raidz_checksum_verify(zio) == 0) {
                                /*
                                 * If we read parity information (unnecessarily
                                 * as it happens since no reconstruction was
                                 * needed) regenerate and verify the parity.
                                 * We also regenerate parity when resilvering
                                 * so we can write it out to the failed device
                                 * later.
                                 */
                                if (parity_errors + parity_untried <
                                    rm->rm_firstdatacol ||
                                    (zio->io_flags & ZIO_FLAG_RESILVER)) {
                                        n = raidz_parity_verify(zio, rm);
                                        unexpected_errors += n;
                                        ASSERT(parity_errors + n <=
                                            rm->rm_firstdatacol);
                                }
                                goto done;
                        }
                } else {
                        /*
                         * We either attempt to read all the parity columns or
                         * none of them. If we didn't try to read parity, we
                         * wouldn't be here in the correctable case. There must
                         * also have been fewer parity errors than parity
                         * columns or, again, we wouldn't be in this code path.
                         */
                        ASSERT(parity_untried == 0);
                        ASSERT(parity_errors < rm->rm_firstdatacol);

                        /*
                         * Identify the data columns that reported an error.
                         */
                        n = 0;
                        for (c = rm->rm_firstdatacol; c < rm->rm_cols; c++) {
                                rc = &rm->rm_col[c];
                                if (rc->rc_error != 0) {
                                        ASSERT(n < VDEV_RAIDZ_MAXPARITY);
                                        tgts[n++] = c;
                                }
                        }

                        ASSERT(rm->rm_firstdatacol >= n);

                        code = vdev_raidz_reconstruct(rm, tgts, n);

                        if (raidz_checksum_verify(zio) == 0) {
                                atomic_inc_64(&raidz_corrected[code]);

                                /*
                                 * If we read more parity disks than were used
                                 * for reconstruction, confirm that the other
                                 * parity disks produced correct data. This
                                 * routine is suboptimal in that it regenerates
                                 * the parity that we already used in addition
                                 * to the parity that we're attempting to
                                 * verify, but this should be a relatively
                                 * uncommon case, and can be optimized if it
                                 * becomes a problem. Note that we regenerate
                                 * parity when resilvering so we can write it
                                 * out to failed devices later.
                                 */
                                if (parity_errors < rm->rm_firstdatacol - n ||
                                    (zio->io_flags & ZIO_FLAG_RESILVER)) {
                                        n = raidz_parity_verify(zio, rm);
                                        unexpected_errors += n;
                                        ASSERT(parity_errors + n <=
                                            rm->rm_firstdatacol);
                                }

                                goto done;
                        }
                }
        }

        /*
         * This isn't a typical situation -- either we got a read error or
         * a child silently returned bad data. Read every block so we can
         * try again with as much data and parity as we can track down. If
         * we've already been through once before, all children will be marked
         * as tried so we'll proceed to combinatorial reconstruction.
         */
        unexpected_errors = 1;
        rm->rm_missingdata = 0;
        rm->rm_missingparity = 0;

        for (c = 0; c < rm->rm_cols; c++) {
                if (rm->rm_col[c].rc_tried)
                        continue;

                zio_vdev_io_redone(zio);
                do {
                        rc = &rm->rm_col[c];
                        if (rc->rc_tried)
                                continue;
                        zio_nowait(zio_vdev_child_io(zio, NULL,
                            vd->vdev_child[rc->rc_devidx],
                            rc->rc_offset, rc->rc_data, rc->rc_size,
                            zio->io_type, zio->io_priority, 0,
                            vdev_raidz_child_done, rc));
                } while (++c < rm->rm_cols);

                return;
        }

        /*
         * At this point we've attempted to reconstruct the data given the
         * errors we detected, and we've attempted to read all columns. There
         * must, therefore, be one or more additional problems -- silent errors
         * resulting in invalid data rather than explicit I/O errors resulting
         * in absent data. We check if there is enough additional data to
         * possibly reconstruct the data and then perform combinatorial
         * reconstruction over all possible combinations. If that fails,
         * we're cooked.
         */
        if (total_errors > rm->rm_firstdatacol) {
                zio->io_error = vdev_raidz_worst_error(rm);

        } else if (total_errors < rm->rm_firstdatacol &&
            (code = vdev_raidz_combrec(zio, total_errors, data_errors)) != 0) {
                /*
                 * If we didn't use all the available parity for the
                 * combinatorial reconstruction, verify that the remaining
                 * parity is correct.
                 */
                if (code != (1 << rm->rm_firstdatacol) - 1)
                        (void) raidz_parity_verify(zio, rm);
        } else {
                /*
                 * We're here because either:
                 *
                 *      total_errors == rm_first_datacol, or
                 *      vdev_raidz_combrec() failed
                 *
                 * In either case, there is enough bad data to prevent
                 * reconstruction.
                 *
                 * Start checksum ereports for all children which haven't
                 * failed, and the IO wasn't speculative.
                 */
                zio->io_error = SET_ERROR(ECKSUM);

                if (!(zio->io_flags & ZIO_FLAG_SPECULATIVE)) {
                        for (c = 0; c < rm->rm_cols; c++) {
                                rc = &rm->rm_col[c];
                                if (rc->rc_error == 0) {
                                        zio_bad_cksum_t zbc;
                                        zbc.zbc_has_cksum = 0;
                                        zbc.zbc_injected =
                                            rm->rm_ecksuminjected;

                                        zfs_ereport_start_checksum(
                                            zio->io_spa,
                                            vd->vdev_child[rc->rc_devidx],
                                            zio, rc->rc_offset, rc->rc_size,
                                            (void *)(uintptr_t)c, &zbc);
                                }
                        }
                }
        }

done:
        zio_checksum_verified(zio);

        if (zio->io_error == 0 && spa_writeable(zio->io_spa) &&
            (unexpected_errors || (zio->io_flags & ZIO_FLAG_RESILVER))) {
                /*
                 * Use the good data we have in hand to repair damaged children.
                 */
                for (c = 0; c < rm->rm_cols; c++) {
                        rc = &rm->rm_col[c];
                        cvd = vd->vdev_child[rc->rc_devidx];

                        if (rc->rc_error == 0)
                                continue;

                        zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                            rc->rc_offset, rc->rc_data, rc->rc_size,
                            ZIO_TYPE_WRITE, ZIO_PRIORITY_ASYNC_WRITE,
                            ZIO_FLAG_IO_REPAIR | (unexpected_errors ?
                            ZIO_FLAG_SELF_HEAL : 0), NULL, NULL));
                }
        }
}

static void
vdev_raidz_state_change(vdev_t *vd, int faulted, int degraded)
{
        if (faulted > vd->vdev_nparity)
                vdev_set_state(vd, B_FALSE, VDEV_STATE_CANT_OPEN,
                    VDEV_AUX_NO_REPLICAS);
        else if (degraded + faulted != 0)
                vdev_set_state(vd, B_FALSE, VDEV_STATE_DEGRADED, VDEV_AUX_NONE);
        else
                vdev_set_state(vd, B_FALSE, VDEV_STATE_HEALTHY, VDEV_AUX_NONE);
}

vdev_ops_t vdev_raidz_ops = {
        vdev_raidz_open,
        vdev_raidz_close,
        vdev_raidz_asize,
        vdev_raidz_io_start,
        vdev_raidz_io_done,
        vdev_raidz_state_change,
        NULL,
        NULL,
        VDEV_TYPE_RAIDZ,        /* name of this vdev type */
        B_FALSE                 /* not a leaf vdev */
};