zfs: merge openzfs/zfs@75b4cbf62 (master) into main

[FreeBSD/FreeBSD.git] / sys / contrib / openzfs / module / 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, 2020 by Delphix. All rights reserved.
 * Copyright (c) 2016 Gvozden Nešković. All rights reserved.
 */

#include <sys/zfs_context.h>
#include <sys/spa.h>
#include <sys/vdev_impl.h>
#include <sys/zio.h>
#include <sys/zio_checksum.h>
#include <sys/abd.h>
#include <sys/fs/zfs.h>
#include <sys/fm/fs/zfs.h>
#include <sys/vdev_raidz.h>
#include <sys/vdev_raidz_impl.h>
#include <sys/vdev_draid.h>

#ifdef ZFS_DEBUG
#include <sys/vdev.h>   /* For vdev_xlate() in vdev_raidz_io_verify() */
#endif

/*
 * 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 trivial
 * 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.
 */

#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) & 0x1d1d1d1d1d1d1d1dULL); \
}

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

static void
vdev_raidz_row_free(raidz_row_t *rr)
{
        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];

                if (rc->rc_size != 0)
                        abd_free(rc->rc_abd);
                if (rc->rc_orig_data != NULL)
                        abd_free(rc->rc_orig_data);
        }

        if (rr->rr_abd_empty != NULL)
                abd_free(rr->rr_abd_empty);

        kmem_free(rr, offsetof(raidz_row_t, rr_col[rr->rr_scols]));
}

void
vdev_raidz_map_free(raidz_map_t *rm)
{
        for (int i = 0; i < rm->rm_nrows; i++)
                vdev_raidz_row_free(rm->rm_row[i]);

        kmem_free(rm, offsetof(raidz_map_t, rm_row[rm->rm_nrows]));
}

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

        vdev_raidz_map_free(rm);
}

const zio_vsd_ops_t vdev_raidz_vsd_ops = {
        .vsd_free = vdev_raidz_map_free_vsd,
};

/*
 * Divides the IO evenly across all child vdevs; usually, dcols is
 * the number of children in the target vdev.
 *
 * Avoid inlining the function to keep vdev_raidz_io_start(), which
 * is this functions only caller, as small as possible on the stack.
 */
noinline raidz_map_t *
vdev_raidz_map_alloc(zio_t *zio, uint64_t ashift, uint64_t dcols,
    uint64_t nparity)
{
        raidz_row_t *rr;
        /* The starting RAIDZ (parent) vdev sector of the block. */
        uint64_t b = zio->io_offset >> ashift;
        /* The zio's size in units of the vdev's minimum sector size. */
        uint64_t s = zio->io_size >> ashift;
        /* The first column for this stripe. */
        uint64_t f = b % dcols;
        /* The starting byte offset on each child vdev. */
        uint64_t o = (b / dcols) << ashift;
        uint64_t q, r, c, bc, col, acols, scols, coff, devidx, asize, tot;

        raidz_map_t *rm =
            kmem_zalloc(offsetof(raidz_map_t, rm_row[1]), KM_SLEEP);
        rm->rm_nrows = 1;

        /*
         * "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);

        rr = kmem_alloc(offsetof(raidz_row_t, rr_col[scols]), KM_SLEEP);
        rm->rm_row[0] = rr;

        rr->rr_cols = acols;
        rr->rr_scols = scols;
        rr->rr_bigcols = bc;
        rr->rr_missingdata = 0;
        rr->rr_missingparity = 0;
        rr->rr_firstdatacol = nparity;
        rr->rr_abd_empty = NULL;
        rr->rr_nempty = 0;
#ifdef ZFS_DEBUG
        rr->rr_offset = zio->io_offset;
        rr->rr_size = zio->io_size;
#endif

        asize = 0;

        for (c = 0; c < scols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];
                col = f + c;
                coff = o;
                if (col >= dcols) {
                        col -= dcols;
                        coff += 1ULL << ashift;
                }
                rc->rc_devidx = col;
                rc->rc_offset = coff;
                rc->rc_abd = NULL;
                rc->rc_orig_data = NULL;
                rc->rc_error = 0;
                rc->rc_tried = 0;
                rc->rc_skipped = 0;
                rc->rc_force_repair = 0;
                rc->rc_allow_repair = 1;
                rc->rc_need_orig_restore = B_FALSE;

                if (c >= acols)
                        rc->rc_size = 0;
                else if (c < bc)
                        rc->rc_size = (q + 1) << ashift;
                else
                        rc->rc_size = q << ashift;

                asize += rc->rc_size;
        }

        ASSERT3U(asize, ==, tot << ashift);
        rm->rm_nskip = roundup(tot, nparity + 1) - tot;
        rm->rm_skipstart = bc;

        for (c = 0; c < rr->rr_firstdatacol; c++)
                rr->rr_col[c].rc_abd =
                    abd_alloc_linear(rr->rr_col[c].rc_size, B_FALSE);

        for (uint64_t off = 0; c < acols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];
                rc->rc_abd = abd_get_offset_struct(&rc->rc_abdstruct,
                    zio->io_abd, off, rc->rc_size);
                off += rc->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 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(rr->rr_cols >= 2);
        ASSERT(rr->rr_col[0].rc_size == rr->rr_col[1].rc_size);

        if (rr->rr_firstdatacol == 1 && (zio->io_offset & (1ULL << 20))) {
                devidx = rr->rr_col[0].rc_devidx;
                o = rr->rr_col[0].rc_offset;
                rr->rr_col[0].rc_devidx = rr->rr_col[1].rc_devidx;
                rr->rr_col[0].rc_offset = rr->rr_col[1].rc_offset;
                rr->rr_col[1].rc_devidx = devidx;
                rr->rr_col[1].rc_offset = o;

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

        /* init RAIDZ parity ops */
        rm->rm_ops = vdev_raidz_math_get_ops();

        return (rm);
}

struct pqr_struct {
        uint64_t *p;
        uint64_t *q;
        uint64_t *r;
};

static int
vdev_raidz_p_func(void *buf, size_t size, void *private)
{
        struct pqr_struct *pqr = private;
        const uint64_t *src = buf;
        int i, cnt = size / sizeof (src[0]);

        ASSERT(pqr->p && !pqr->q && !pqr->r);

        for (i = 0; i < cnt; i++, src++, pqr->p++)
                *pqr->p ^= *src;

        return (0);
}

static int
vdev_raidz_pq_func(void *buf, size_t size, void *private)
{
        struct pqr_struct *pqr = private;
        const uint64_t *src = buf;
        uint64_t mask;
        int i, cnt = size / sizeof (src[0]);

        ASSERT(pqr->p && pqr->q && !pqr->r);

        for (i = 0; i < cnt; i++, src++, pqr->p++, pqr->q++) {
                *pqr->p ^= *src;
                VDEV_RAIDZ_64MUL_2(*pqr->q, mask);
                *pqr->q ^= *src;
        }

        return (0);
}

static int
vdev_raidz_pqr_func(void *buf, size_t size, void *private)
{
        struct pqr_struct *pqr = private;
        const uint64_t *src = buf;
        uint64_t mask;
        int i, cnt = size / sizeof (src[0]);

        ASSERT(pqr->p && pqr->q && pqr->r);

        for (i = 0; i < cnt; i++, src++, pqr->p++, pqr->q++, pqr->r++) {
                *pqr->p ^= *src;
                VDEV_RAIDZ_64MUL_2(*pqr->q, mask);
                *pqr->q ^= *src;
                VDEV_RAIDZ_64MUL_4(*pqr->r, mask);
                *pqr->r ^= *src;
        }

        return (0);
}

static void
vdev_raidz_generate_parity_p(raidz_row_t *rr)
{
        uint64_t *p = abd_to_buf(rr->rr_col[VDEV_RAIDZ_P].rc_abd);

        for (int c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                abd_t *src = rr->rr_col[c].rc_abd;

                if (c == rr->rr_firstdatacol) {
                        abd_copy_to_buf(p, src, rr->rr_col[c].rc_size);
                } else {
                        struct pqr_struct pqr = { p, NULL, NULL };
                        (void) abd_iterate_func(src, 0, rr->rr_col[c].rc_size,
                            vdev_raidz_p_func, &pqr);
                }
        }
}

static void
vdev_raidz_generate_parity_pq(raidz_row_t *rr)
{
        uint64_t *p = abd_to_buf(rr->rr_col[VDEV_RAIDZ_P].rc_abd);
        uint64_t *q = abd_to_buf(rr->rr_col[VDEV_RAIDZ_Q].rc_abd);
        uint64_t pcnt = rr->rr_col[VDEV_RAIDZ_P].rc_size / sizeof (p[0]);
        ASSERT(rr->rr_col[VDEV_RAIDZ_P].rc_size ==
            rr->rr_col[VDEV_RAIDZ_Q].rc_size);

        for (int c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                abd_t *src = rr->rr_col[c].rc_abd;

                uint64_t ccnt = rr->rr_col[c].rc_size / sizeof (p[0]);

                if (c == rr->rr_firstdatacol) {
                        ASSERT(ccnt == pcnt || ccnt == 0);
                        abd_copy_to_buf(p, src, rr->rr_col[c].rc_size);
                        (void) memcpy(q, p, rr->rr_col[c].rc_size);

                        for (uint64_t i = ccnt; i < pcnt; i++) {
                                p[i] = 0;
                                q[i] = 0;
                        }
                } else {
                        struct pqr_struct pqr = { p, q, NULL };

                        ASSERT(ccnt <= pcnt);
                        (void) abd_iterate_func(src, 0, rr->rr_col[c].rc_size,
                            vdev_raidz_pq_func, &pqr);

                        /*
                         * Treat short columns as though they are full of 0s.
                         * Note that there's therefore nothing needed for P.
                         */
                        uint64_t mask;
                        for (uint64_t i = ccnt; i < pcnt; i++) {
                                VDEV_RAIDZ_64MUL_2(q[i], mask);
                        }
                }
        }
}

static void
vdev_raidz_generate_parity_pqr(raidz_row_t *rr)
{
        uint64_t *p = abd_to_buf(rr->rr_col[VDEV_RAIDZ_P].rc_abd);
        uint64_t *q = abd_to_buf(rr->rr_col[VDEV_RAIDZ_Q].rc_abd);
        uint64_t *r = abd_to_buf(rr->rr_col[VDEV_RAIDZ_R].rc_abd);
        uint64_t pcnt = rr->rr_col[VDEV_RAIDZ_P].rc_size / sizeof (p[0]);
        ASSERT(rr->rr_col[VDEV_RAIDZ_P].rc_size ==
            rr->rr_col[VDEV_RAIDZ_Q].rc_size);
        ASSERT(rr->rr_col[VDEV_RAIDZ_P].rc_size ==
            rr->rr_col[VDEV_RAIDZ_R].rc_size);

        for (int c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                abd_t *src = rr->rr_col[c].rc_abd;

                uint64_t ccnt = rr->rr_col[c].rc_size / sizeof (p[0]);

                if (c == rr->rr_firstdatacol) {
                        ASSERT(ccnt == pcnt || ccnt == 0);
                        abd_copy_to_buf(p, src, rr->rr_col[c].rc_size);
                        (void) memcpy(q, p, rr->rr_col[c].rc_size);
                        (void) memcpy(r, p, rr->rr_col[c].rc_size);

                        for (uint64_t i = ccnt; i < pcnt; i++) {
                                p[i] = 0;
                                q[i] = 0;
                                r[i] = 0;
                        }
                } else {
                        struct pqr_struct pqr = { p, q, r };

                        ASSERT(ccnt <= pcnt);
                        (void) abd_iterate_func(src, 0, rr->rr_col[c].rc_size,
                            vdev_raidz_pqr_func, &pqr);

                        /*
                         * Treat short columns as though they are full of 0s.
                         * Note that there's therefore nothing needed for P.
                         */
                        uint64_t mask;
                        for (uint64_t i = ccnt; i < pcnt; i++) {
                                VDEV_RAIDZ_64MUL_2(q[i], mask);
                                VDEV_RAIDZ_64MUL_4(r[i], mask);
                        }
                }
        }
}

/*
 * Generate RAID parity in the first virtual columns according to the number of
 * parity columns available.
 */
void
vdev_raidz_generate_parity_row(raidz_map_t *rm, raidz_row_t *rr)
{
        ASSERT3U(rr->rr_cols, !=, 0);

        /* Generate using the new math implementation */
        if (vdev_raidz_math_generate(rm, rr) != RAIDZ_ORIGINAL_IMPL)
                return;

        switch (rr->rr_firstdatacol) {
        case 1:
                vdev_raidz_generate_parity_p(rr);
                break;
        case 2:
                vdev_raidz_generate_parity_pq(rr);
                break;
        case 3:
                vdev_raidz_generate_parity_pqr(rr);
                break;
        default:
                cmn_err(CE_PANIC, "invalid RAID-Z configuration");
        }
}

void
vdev_raidz_generate_parity(raidz_map_t *rm)
{
        for (int i = 0; i < rm->rm_nrows; i++) {
                raidz_row_t *rr = rm->rm_row[i];
                vdev_raidz_generate_parity_row(rm, rr);
        }
}

/* ARGSUSED */
static int
vdev_raidz_reconst_p_func(void *dbuf, void *sbuf, size_t size, void *private)
{
        uint64_t *dst = dbuf;
        uint64_t *src = sbuf;
        int cnt = size / sizeof (src[0]);

        for (int i = 0; i < cnt; i++) {
                dst[i] ^= src[i];
        }

        return (0);
}

/* ARGSUSED */
static int
vdev_raidz_reconst_q_pre_func(void *dbuf, void *sbuf, size_t size,
    void *private)
{
        uint64_t *dst = dbuf;
        uint64_t *src = sbuf;
        uint64_t mask;
        int cnt = size / sizeof (dst[0]);

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

        return (0);
}

/* ARGSUSED */
static int
vdev_raidz_reconst_q_pre_tail_func(void *buf, size_t size, void *private)
{
        uint64_t *dst = buf;
        uint64_t mask;
        int cnt = size / sizeof (dst[0]);

        for (int i = 0; i < cnt; i++, dst++) {
                /* same operation as vdev_raidz_reconst_q_pre_func() on dst */
                VDEV_RAIDZ_64MUL_2(*dst, mask);
        }

        return (0);
}

struct reconst_q_struct {
        uint64_t *q;
        int exp;
};

static int
vdev_raidz_reconst_q_post_func(void *buf, size_t size, void *private)
{
        struct reconst_q_struct *rq = private;
        uint64_t *dst = buf;
        int cnt = size / sizeof (dst[0]);

        for (int i = 0; i < cnt; i++, dst++, rq->q++) {
                int j;
                uint8_t *b;

                *dst ^= *rq->q;
                for (j = 0, b = (uint8_t *)dst; j < 8; j++, b++) {
                        *b = vdev_raidz_exp2(*b, rq->exp);
                }
        }

        return (0);
}

struct reconst_pq_struct {
        uint8_t *p;
        uint8_t *q;
        uint8_t *pxy;
        uint8_t *qxy;
        int aexp;
        int bexp;
};

static int
vdev_raidz_reconst_pq_func(void *xbuf, void *ybuf, size_t size, void *private)
{
        struct reconst_pq_struct *rpq = private;
        uint8_t *xd = xbuf;
        uint8_t *yd = ybuf;

        for (int i = 0; i < size;
            i++, rpq->p++, rpq->q++, rpq->pxy++, rpq->qxy++, xd++, yd++) {
                *xd = vdev_raidz_exp2(*rpq->p ^ *rpq->pxy, rpq->aexp) ^
                    vdev_raidz_exp2(*rpq->q ^ *rpq->qxy, rpq->bexp);
                *yd = *rpq->p ^ *rpq->pxy ^ *xd;
        }

        return (0);
}

static int
vdev_raidz_reconst_pq_tail_func(void *xbuf, size_t size, void *private)
{
        struct reconst_pq_struct *rpq = private;
        uint8_t *xd = xbuf;

        for (int i = 0; i < size;
            i++, rpq->p++, rpq->q++, rpq->pxy++, rpq->qxy++, xd++) {
                /* same operation as vdev_raidz_reconst_pq_func() on xd */
                *xd = vdev_raidz_exp2(*rpq->p ^ *rpq->pxy, rpq->aexp) ^
                    vdev_raidz_exp2(*rpq->q ^ *rpq->qxy, rpq->bexp);
        }

        return (0);
}

static void
vdev_raidz_reconstruct_p(raidz_row_t *rr, int *tgts, int ntgts)
{
        int x = tgts[0];
        abd_t *dst, *src;

        ASSERT3U(ntgts, ==, 1);
        ASSERT3U(x, >=, rr->rr_firstdatacol);
        ASSERT3U(x, <, rr->rr_cols);

        ASSERT3U(rr->rr_col[x].rc_size, <=, rr->rr_col[VDEV_RAIDZ_P].rc_size);

        src = rr->rr_col[VDEV_RAIDZ_P].rc_abd;
        dst = rr->rr_col[x].rc_abd;

        abd_copy_from_buf(dst, abd_to_buf(src), rr->rr_col[x].rc_size);

        for (int c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                uint64_t size = MIN(rr->rr_col[x].rc_size,
                    rr->rr_col[c].rc_size);

                src = rr->rr_col[c].rc_abd;

                if (c == x)
                        continue;

                (void) abd_iterate_func2(dst, src, 0, 0, size,
                    vdev_raidz_reconst_p_func, NULL);
        }
}

static void
vdev_raidz_reconstruct_q(raidz_row_t *rr, int *tgts, int ntgts)
{
        int x = tgts[0];
        int c, exp;
        abd_t *dst, *src;

        ASSERT(ntgts == 1);

        ASSERT(rr->rr_col[x].rc_size <= rr->rr_col[VDEV_RAIDZ_Q].rc_size);

        for (c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                uint64_t size = (c == x) ? 0 : MIN(rr->rr_col[x].rc_size,
                    rr->rr_col[c].rc_size);

                src = rr->rr_col[c].rc_abd;
                dst = rr->rr_col[x].rc_abd;

                if (c == rr->rr_firstdatacol) {
                        abd_copy(dst, src, size);
                        if (rr->rr_col[x].rc_size > size) {
                                abd_zero_off(dst, size,
                                    rr->rr_col[x].rc_size - size);
                        }
                } else {
                        ASSERT3U(size, <=, rr->rr_col[x].rc_size);
                        (void) abd_iterate_func2(dst, src, 0, 0, size,
                            vdev_raidz_reconst_q_pre_func, NULL);
                        (void) abd_iterate_func(dst,
                            size, rr->rr_col[x].rc_size - size,
                            vdev_raidz_reconst_q_pre_tail_func, NULL);
                }
        }

        src = rr->rr_col[VDEV_RAIDZ_Q].rc_abd;
        dst = rr->rr_col[x].rc_abd;
        exp = 255 - (rr->rr_cols - 1 - x);

        struct reconst_q_struct rq = { abd_to_buf(src), exp };
        (void) abd_iterate_func(dst, 0, rr->rr_col[x].rc_size,
            vdev_raidz_reconst_q_post_func, &rq);
}

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

        ASSERT(ntgts == 2);
        ASSERT(x < y);
        ASSERT(x >= rr->rr_firstdatacol);
        ASSERT(y < rr->rr_cols);

        ASSERT(rr->rr_col[x].rc_size >= rr->rr_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 = rr->rr_col[VDEV_RAIDZ_P].rc_abd;
        qdata = rr->rr_col[VDEV_RAIDZ_Q].rc_abd;
        xsize = rr->rr_col[x].rc_size;
        ysize = rr->rr_col[y].rc_size;

        rr->rr_col[VDEV_RAIDZ_P].rc_abd =
            abd_alloc_linear(rr->rr_col[VDEV_RAIDZ_P].rc_size, B_TRUE);
        rr->rr_col[VDEV_RAIDZ_Q].rc_abd =
            abd_alloc_linear(rr->rr_col[VDEV_RAIDZ_Q].rc_size, B_TRUE);
        rr->rr_col[x].rc_size = 0;
        rr->rr_col[y].rc_size = 0;

        vdev_raidz_generate_parity_pq(rr);

        rr->rr_col[x].rc_size = xsize;
        rr->rr_col[y].rc_size = ysize;

        p = abd_to_buf(pdata);
        q = abd_to_buf(qdata);
        pxy = abd_to_buf(rr->rr_col[VDEV_RAIDZ_P].rc_abd);
        qxy = abd_to_buf(rr->rr_col[VDEV_RAIDZ_Q].rc_abd);
        xd = rr->rr_col[x].rc_abd;
        yd = rr->rr_col[y].rc_abd;

        /*
         * 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 - (rr->rr_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)];

        ASSERT3U(xsize, >=, ysize);
        struct reconst_pq_struct rpq = { p, q, pxy, qxy, aexp, bexp };

        (void) abd_iterate_func2(xd, yd, 0, 0, ysize,
            vdev_raidz_reconst_pq_func, &rpq);
        (void) abd_iterate_func(xd, ysize, xsize - ysize,
            vdev_raidz_reconst_pq_tail_func, &rpq);

        abd_free(rr->rr_col[VDEV_RAIDZ_P].rc_abd);
        abd_free(rr->rr_col[VDEV_RAIDZ_Q].rc_abd);

        /*
         * Restore the saved parity data.
         */
        rr->rr_col[VDEV_RAIDZ_P].rc_abd = pdata;
        rr->rr_col[VDEV_RAIDZ_Q].rc_abd = qdata;
}

/* BEGIN CSTYLED */
/*
 * In the general case of reconstruction, we must solve the system of linear
 * equations defined by the coefficients 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 coefficients 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  |
 *           | 128  64  32  16  8   4   2   1  |
 *           |  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  |
 *           ~~                               ~~
 *
 * 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_row_t *rr, int n, int nmap, int *map,
    uint8_t **rows)
{
        int i, j;
        int pow;

        ASSERT(n == rr->rr_cols - rr->rr_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_row_t *rr, 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], <, rr->rr_firstdatacol);
        }
        for (; i < n; i++) {
                ASSERT3S(used[i], >=, rr->rr_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], >=, rr->rr_firstdatacol);
                        jj = used[j] - rr->rr_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_row_t *rr, 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] = { NULL };
        uint64_t dcount[VDEV_RAIDZ_MAXPARITY] = { 0 };
        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, <, rr->rr_cols);

                ccount = rr->rr_col[c].rc_size;
                ASSERT(ccount >= rr->rr_col[missing[0]].rc_size || i > 0);
                if (ccount == 0)
                        continue;
                src = abd_to_buf(rr->rr_col[c].rc_abd);
                for (j = 0; j < nmissing; j++) {
                        cc = missing[j] + rr->rr_firstdatacol;
                        ASSERT3U(cc, >=, rr->rr_firstdatacol);
                        ASSERT3U(cc, <, rr->rr_cols);
                        ASSERT3U(cc, !=, c);

                        dcount[j] = rr->rr_col[cc].rc_size;
                        if (dcount[j] != 0)
                                dst[j] = abd_to_buf(rr->rr_col[cc].rc_abd);
                }

                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 void
vdev_raidz_reconstruct_general(raidz_row_t *rr, 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;

        abd_t **bufs = NULL;

        /*
         * Matrix reconstruction can't use scatter ABDs yet, so we allocate
         * temporary linear ABDs if any non-linear ABDs are found.
         */
        for (i = rr->rr_firstdatacol; i < rr->rr_cols; i++) {
                if (!abd_is_linear(rr->rr_col[i].rc_abd)) {
                        bufs = kmem_alloc(rr->rr_cols * sizeof (abd_t *),
                            KM_PUSHPAGE);

                        for (c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                                raidz_col_t *col = &rr->rr_col[c];

                                bufs[c] = col->rc_abd;
                                if (bufs[c] != NULL) {
                                        col->rc_abd = abd_alloc_linear(
                                            col->rc_size, B_TRUE);
                                        abd_copy(col->rc_abd, bufs[c],
                                            col->rc_size);
                                }
                        }

                        break;
                }
        }

        n = rr->rr_cols - rr->rr_firstdatacol;

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

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

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

        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 = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                if (tt < nmissing_rows &&
                    c == missing_rows[tt] + rr->rr_firstdatacol) {
                        tt++;
                        continue;
                }

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

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

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

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

        kmem_free(p, psize);

        /*
         * copy back from temporary linear abds and free them
         */
        if (bufs) {
                for (c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                        raidz_col_t *col = &rr->rr_col[c];

                        if (bufs[c] != NULL) {
                                abd_copy(bufs[c], col->rc_abd, col->rc_size);
                                abd_free(col->rc_abd);
                        }
                        col->rc_abd = bufs[c];
                }
                kmem_free(bufs, rr->rr_cols * sizeof (abd_t *));
        }
}

static void
vdev_raidz_reconstruct_row(raidz_map_t *rm, raidz_row_t *rr,
    const int *t, int nt)
{
        int tgts[VDEV_RAIDZ_MAXPARITY], *dt;
        int ntgts;
        int i, c, ret;
        int nbadparity, nbaddata;
        int parity_valid[VDEV_RAIDZ_MAXPARITY];

        nbadparity = rr->rr_firstdatacol;
        nbaddata = rr->rr_cols - nbadparity;
        ntgts = 0;
        for (i = 0, c = 0; c < rr->rr_cols; c++) {
                if (c < rr->rr_firstdatacol)
                        parity_valid[c] = B_FALSE;

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

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

        dt = &tgts[nbadparity];

        /* Reconstruct using the new math implementation */
        ret = vdev_raidz_math_reconstruct(rm, rr, parity_valid, dt, nbaddata);
        if (ret != RAIDZ_ORIGINAL_IMPL)
                return;

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

                ASSERT(rr->rr_firstdatacol > 1);

                if (parity_valid[VDEV_RAIDZ_Q]) {
                        vdev_raidz_reconstruct_q(rr, dt, 1);
                        return;
                }

                ASSERT(rr->rr_firstdatacol > 2);
                break;

        case 2:
                ASSERT(rr->rr_firstdatacol > 1);

                if (parity_valid[VDEV_RAIDZ_P] &&
                    parity_valid[VDEV_RAIDZ_Q]) {
                        vdev_raidz_reconstruct_pq(rr, dt, 2);
                        return;
                }

                ASSERT(rr->rr_firstdatacol > 2);

                break;
        }

        vdev_raidz_reconstruct_general(rr, tgts, ntgts);
}

static int
vdev_raidz_open(vdev_t *vd, uint64_t *asize, uint64_t *max_asize,
    uint64_t *logical_ashift, uint64_t *physical_ashift)
{
        vdev_raidz_t *vdrz = vd->vdev_tsd;
        uint64_t nparity = vdrz->vd_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++) {
                vdev_t *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)
{
        for (int c = 0; c < vd->vdev_children; c++) {
                if (vd->vdev_child[c] != NULL)
                        vdev_close(vd->vdev_child[c]);
        }
}

static uint64_t
vdev_raidz_asize(vdev_t *vd, uint64_t psize)
{
        vdev_raidz_t *vdrz = vd->vdev_tsd;
        uint64_t asize;
        uint64_t ashift = vd->vdev_top->vdev_ashift;
        uint64_t cols = vdrz->vd_logical_width;
        uint64_t nparity = vdrz->vd_nparity;

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

        return (asize);
}

/*
 * The allocatable space for a raidz vdev is N * sizeof(smallest child)
 * so each child must provide at least 1/Nth of its asize.
 */
static uint64_t
vdev_raidz_min_asize(vdev_t *vd)
{
        return ((vd->vdev_min_asize + vd->vdev_children - 1) /
            vd->vdev_children);
}

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;
}

static void
vdev_raidz_io_verify(vdev_t *vd, raidz_row_t *rr, int col)
{
#ifdef ZFS_DEBUG
        vdev_t *tvd = vd->vdev_top;

        range_seg64_t logical_rs, physical_rs, remain_rs;
        logical_rs.rs_start = rr->rr_offset;
        logical_rs.rs_end = logical_rs.rs_start +
            vdev_raidz_asize(vd, rr->rr_size);

        raidz_col_t *rc = &rr->rr_col[col];
        vdev_t *cvd = vd->vdev_child[rc->rc_devidx];

        vdev_xlate(cvd, &logical_rs, &physical_rs, &remain_rs);
        ASSERT(vdev_xlate_is_empty(&remain_rs));
        ASSERT3U(rc->rc_offset, ==, physical_rs.rs_start);
        ASSERT3U(rc->rc_offset, <, physical_rs.rs_end);
        /*
         * It would be nice to assert that rs_end is equal
         * to rc_offset + rc_size but there might be an
         * optional I/O at the end that is not accounted in
         * rc_size.
         */
        if (physical_rs.rs_end > rc->rc_offset + rc->rc_size) {
                ASSERT3U(physical_rs.rs_end, ==, rc->rc_offset +
                    rc->rc_size + (1 << tvd->vdev_ashift));
        } else {
                ASSERT3U(physical_rs.rs_end, ==, rc->rc_offset + rc->rc_size);
        }
#endif
}

static void
vdev_raidz_io_start_write(zio_t *zio, raidz_row_t *rr, uint64_t ashift)
{
        vdev_t *vd = zio->io_vd;
        raidz_map_t *rm = zio->io_vsd;
        int c, i;

        vdev_raidz_generate_parity_row(rm, rr);

        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];
                if (rc->rc_size == 0)
                        continue;

                /* Verify physical to logical translation */
                vdev_raidz_io_verify(vd, rr, c);

                zio_nowait(zio_vdev_child_io(zio, NULL,
                    vd->vdev_child[rc->rc_devidx], rc->rc_offset,
                    rc->rc_abd, rc->rc_size, zio->io_type, zio->io_priority,
                    0, vdev_raidz_child_done, rc));
        }

        /*
         * Generate optional I/Os for skip sectors to improve aggregation
         * contiguity.
         */
        for (c = rm->rm_skipstart, i = 0; i < rm->rm_nskip; c++, i++) {
                ASSERT(c <= rr->rr_scols);
                if (c == rr->rr_scols)
                        c = 0;

                raidz_col_t *rc = &rr->rr_col[c];
                vdev_t *cvd = vd->vdev_child[rc->rc_devidx];

                zio_nowait(zio_vdev_child_io(zio, NULL, cvd,
                    rc->rc_offset + rc->rc_size, NULL, 1ULL << ashift,
                    zio->io_type, zio->io_priority,
                    ZIO_FLAG_NODATA | ZIO_FLAG_OPTIONAL, NULL, NULL));
        }
}

static void
vdev_raidz_io_start_read(zio_t *zio, raidz_row_t *rr)
{
        vdev_t *vd = zio->io_vd;

        /*
         * 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 (int c = rr->rr_cols - 1; c >= 0; c--) {
                raidz_col_t *rc = &rr->rr_col[c];
                if (rc->rc_size == 0)
                        continue;
                vdev_t *cvd = vd->vdev_child[rc->rc_devidx];
                if (!vdev_readable(cvd)) {
                        if (c >= rr->rr_firstdatacol)
                                rr->rr_missingdata++;
                        else
                                rr->rr_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 >= rr->rr_firstdatacol)
                                rr->rr_missingdata++;
                        else
                                rr->rr_missingparity++;
                        rc->rc_error = SET_ERROR(ESTALE);
                        rc->rc_skipped = 1;
                        continue;
                }
                if (c >= rr->rr_firstdatacol || rr->rr_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_abd, rc->rc_size,
                            zio->io_type, zio->io_priority, 0,
                            vdev_raidz_child_done, rc));
                }
        }
}

/*
 * 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_raidz_t *vdrz = vd->vdev_tsd;

        raidz_map_t *rm = vdev_raidz_map_alloc(zio, tvd->vdev_ashift,
            vdrz->vd_logical_width, vdrz->vd_nparity);
        zio->io_vsd = rm;
        zio->io_vsd_ops = &vdev_raidz_vsd_ops;

        /*
         * Until raidz expansion is implemented all maps for a raidz vdev
         * contain a single row.
         */
        ASSERT3U(rm->rm_nrows, ==, 1);
        raidz_row_t *rr = rm->rm_row[0];

        if (zio->io_type == ZIO_TYPE_WRITE) {
                vdev_raidz_io_start_write(zio, rr, tvd->vdev_ashift);
        } else {
                ASSERT(zio->io_type == ZIO_TYPE_READ);
                vdev_raidz_io_start_read(zio, rr);
        }

        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, abd_t *bad_data)
{
        vdev_t *vd = zio->io_vd->vdev_child[rc->rc_devidx];

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

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

                (void) zfs_ereport_post_checksum(zio->io_spa, vd,
                    &zio->io_bookmark, zio, rc->rc_offset, rc->rc_size,
                    rc->rc_abd, bad_data, &zbc);
                mutex_enter(&vd->vdev_stat_lock);
                vd->vdev_stat.vs_checksum_errors++;
                mutex_exit(&vd->vdev_stat_lock);
        }
}

/*
 * 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;

        bzero(&zbc, sizeof (zio_bad_cksum_t));

        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 of such failures.
 */
static int
raidz_parity_verify(zio_t *zio, raidz_row_t *rr)
{
        abd_t *orig[VDEV_RAIDZ_MAXPARITY];
        int c, ret = 0;
        raidz_map_t *rm = zio->io_vsd;
        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 < rr->rr_firstdatacol; c++) {
                rc = &rr->rr_col[c];
                if (!rc->rc_tried || rc->rc_error != 0)
                        continue;

                orig[c] = abd_alloc_sametype(rc->rc_abd, rc->rc_size);
                abd_copy(orig[c], rc->rc_abd, rc->rc_size);
        }

        /*
         * Regenerates parity even for !tried||rc_error!=0 columns.  This
         * isn't harmful but it does have the side effect of fixing stuff
         * we didn't realize was necessary (i.e. even if we return 0).
         */
        vdev_raidz_generate_parity_row(rm, rr);

        for (c = 0; c < rr->rr_firstdatacol; c++) {
                rc = &rr->rr_col[c];

                if (!rc->rc_tried || rc->rc_error != 0)
                        continue;

                if (abd_cmp(orig[c], rc->rc_abd) != 0) {
                        raidz_checksum_error(zio, rc, orig[c]);
                        rc->rc_error = SET_ERROR(ECKSUM);
                        ret++;
                }
                abd_free(orig[c]);
        }

        return (ret);
}

static int
vdev_raidz_worst_error(raidz_row_t *rr)
{
        int error = 0;

        for (int c = 0; c < rr->rr_cols; c++)
                error = zio_worst_error(error, rr->rr_col[c].rc_error);

        return (error);
}

static void
vdev_raidz_io_done_verified(zio_t *zio, raidz_row_t *rr)
{
        int unexpected_errors = 0;
        int parity_errors = 0;
        int parity_untried = 0;
        int data_errors = 0;

        ASSERT3U(zio->io_type, ==, ZIO_TYPE_READ);

        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];

                if (rc->rc_error) {
                        if (c < rr->rr_firstdatacol)
                                parity_errors++;
                        else
                                data_errors++;

                        if (!rc->rc_skipped)
                                unexpected_errors++;
                } else if (c < rr->rr_firstdatacol && !rc->rc_tried) {
                        parity_untried++;
                }
        }

        /*
         * If we read more parity disks than were used for
         * reconstruction, confirm that the other parity disks produced
         * correct data.
         *
         * Note that we also regenerate parity when resilvering so we
         * can write it out to failed devices later.
         */
        if (parity_errors + parity_untried <
            rr->rr_firstdatacol - data_errors ||
            (zio->io_flags & ZIO_FLAG_RESILVER)) {
                int n = raidz_parity_verify(zio, rr);
                unexpected_errors += n;
                ASSERT3U(parity_errors + n, <=, rr->rr_firstdatacol);
        }

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

                        if (!rc->rc_allow_repair) {
                                continue;
                        } else if (!rc->rc_force_repair &&
                            (rc->rc_error == 0 || rc->rc_size == 0)) {
                                continue;
                        }

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

static void
raidz_restore_orig_data(raidz_map_t *rm)
{
        for (int i = 0; i < rm->rm_nrows; i++) {
                raidz_row_t *rr = rm->rm_row[i];
                for (int c = 0; c < rr->rr_cols; c++) {
                        raidz_col_t *rc = &rr->rr_col[c];
                        if (rc->rc_need_orig_restore) {
                                abd_copy(rc->rc_abd,
                                    rc->rc_orig_data, rc->rc_size);
                                rc->rc_need_orig_restore = B_FALSE;
                        }
                }
        }
}

/*
 * returns EINVAL if reconstruction of the block will not be possible
 * returns ECKSUM if this specific reconstruction failed
 * returns 0 on successful reconstruction
 */
static int
raidz_reconstruct(zio_t *zio, int *ltgts, int ntgts, int nparity)
{
        raidz_map_t *rm = zio->io_vsd;

        /* Reconstruct each row */
        for (int r = 0; r < rm->rm_nrows; r++) {
                raidz_row_t *rr = rm->rm_row[r];
                int my_tgts[VDEV_RAIDZ_MAXPARITY]; /* value is child id */
                int t = 0;
                int dead = 0;
                int dead_data = 0;

                for (int c = 0; c < rr->rr_cols; c++) {
                        raidz_col_t *rc = &rr->rr_col[c];
                        ASSERT0(rc->rc_need_orig_restore);
                        if (rc->rc_error != 0) {
                                dead++;
                                if (c >= nparity)
                                        dead_data++;
                                continue;
                        }
                        if (rc->rc_size == 0)
                                continue;
                        for (int lt = 0; lt < ntgts; lt++) {
                                if (rc->rc_devidx == ltgts[lt]) {
                                        if (rc->rc_orig_data == NULL) {
                                                rc->rc_orig_data =
                                                    abd_alloc_linear(
                                                    rc->rc_size, B_TRUE);
                                                abd_copy(rc->rc_orig_data,
                                                    rc->rc_abd, rc->rc_size);
                                        }
                                        rc->rc_need_orig_restore = B_TRUE;

                                        dead++;
                                        if (c >= nparity)
                                                dead_data++;
                                        my_tgts[t++] = c;
                                        break;
                                }
                        }
                }
                if (dead > nparity) {
                        /* reconstruction not possible */
                        raidz_restore_orig_data(rm);
                        return (EINVAL);
                }
                if (dead_data > 0)
                        vdev_raidz_reconstruct_row(rm, rr, my_tgts, t);
        }

        /* Check for success */
        if (raidz_checksum_verify(zio) == 0) {

                /* Reconstruction succeeded - report errors */
                for (int i = 0; i < rm->rm_nrows; i++) {
                        raidz_row_t *rr = rm->rm_row[i];

                        for (int c = 0; c < rr->rr_cols; c++) {
                                raidz_col_t *rc = &rr->rr_col[c];
                                if (rc->rc_need_orig_restore) {
                                        /*
                                         * Note: if this is a parity column,
                                         * we don't really know if it's wrong.
                                         * We need to let
                                         * vdev_raidz_io_done_verified() check
                                         * it, and if we set rc_error, it will
                                         * think that it is a "known" error
                                         * that doesn't need to be checked
                                         * or corrected.
                                         */
                                        if (rc->rc_error == 0 &&
                                            c >= rr->rr_firstdatacol) {
                                                raidz_checksum_error(zio,
                                                    rc, rc->rc_orig_data);
                                                rc->rc_error =
                                                    SET_ERROR(ECKSUM);
                                        }
                                        rc->rc_need_orig_restore = B_FALSE;
                                }
                        }

                        vdev_raidz_io_done_verified(zio, rr);
                }

                zio_checksum_verified(zio);

                return (0);
        }

        /* Reconstruction failed - restore original data */
        raidz_restore_orig_data(rm);
        return (ECKSUM);
}

/*
 * Iterate over all combinations of N bad vdevs 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.
 *
 * The order that we find the various possible combinations of failed
 * disks is dictated by these rules:
 * - Examine each "slot" (the "i" in tgts[i])
 *   - Try to increment this slot (tgts[i] = tgts[i] + 1)
 *   - if we can't increment because it runs into the next slot,
 *     reset our slot to the minimum, and examine the next slot
 *
 *  For example, with a 6-wide RAIDZ3, and no known errors (so we have to choose
 *  3 columns to reconstruct), we will generate the following sequence:
 *
 *  STATE        ACTION
 *  0 1 2        special case: skip since these are all parity
 *  0 1   3      first slot: reset to 0; middle slot: increment to 2
 *  0   2 3      first slot: increment to 1
 *    1 2 3      first: reset to 0; middle: reset to 1; last: increment to 4
 *  0 1     4    first: reset to 0; middle: increment to 2
 *  0   2   4    first: increment to 1
 *    1 2   4    first: reset to 0; middle: increment to 3
 *  0     3 4    first: increment to 1
 *    1   3 4    first: increment to 2
 *      2 3 4    first: reset to 0; middle: reset to 1; last: increment to 5
 *  0 1       5  first: reset to 0; middle: increment to 2
 *  0   2     5  first: increment to 1
 *    1 2     5  first: reset to 0; middle: increment to 3
 *  0     3   5  first: increment to 1
 *    1   3   5  first: increment to 2
 *      2 3   5  first: reset to 0; middle: increment to 4
 *  0       4 5  first: increment to 1
 *    1     4 5  first: increment to 2
 *      2   4 5  first: increment to 3
 *        3 4 5  done
 *
 * This strategy works for dRAID but is less efficient when there are a large
 * number of child vdevs and therefore permutations to check. Furthermore,
 * since the raidz_map_t rows likely do not overlap reconstruction would be
 * possible as long as there are no more than nparity data errors per row.
 * These additional permutations are not currently checked but could be as
 * a future improvement.
 */
static int
vdev_raidz_combrec(zio_t *zio)
{
        int nparity = vdev_get_nparity(zio->io_vd);
        raidz_map_t *rm = zio->io_vsd;

        /* Check if there's enough data to attempt reconstrution. */
        for (int i = 0; i < rm->rm_nrows; i++) {
                raidz_row_t *rr = rm->rm_row[i];
                int total_errors = 0;

                for (int c = 0; c < rr->rr_cols; c++) {
                        if (rr->rr_col[c].rc_error)
                                total_errors++;
                }

                if (total_errors > nparity)
                        return (vdev_raidz_worst_error(rr));
        }

        for (int num_failures = 1; num_failures <= nparity; num_failures++) {
                int tstore[VDEV_RAIDZ_MAXPARITY + 2];
                int *ltgts = &tstore[1]; /* value is logical child ID */

                /* Determine number of logical children, n */
                int n = zio->io_vd->vdev_children;

                ASSERT3U(num_failures, <=, nparity);
                ASSERT3U(num_failures, <=, VDEV_RAIDZ_MAXPARITY);

                /* Handle corner cases in combrec logic */
                ltgts[-1] = -1;
                for (int i = 0; i < num_failures; i++) {
                        ltgts[i] = i;
                }
                ltgts[num_failures] = n;

                for (;;) {
                        int err = raidz_reconstruct(zio, ltgts, num_failures,
                            nparity);
                        if (err == EINVAL) {
                                /*
                                 * Reconstruction not possible with this #
                                 * failures; try more failures.
                                 */
                                break;
                        } else if (err == 0)
                                return (0);

                        /* Compute next targets to try */
                        for (int t = 0; ; t++) {
                                ASSERT3U(t, <, num_failures);
                                ltgts[t]++;
                                if (ltgts[t] == n) {
                                        /* try more failures */
                                        ASSERT3U(t, ==, num_failures - 1);
                                        break;
                                }

                                ASSERT3U(ltgts[t], <, n);
                                ASSERT3U(ltgts[t], <=, ltgts[t + 1]);

                                /*
                                 * If that spot is available, we're done here.
                                 * Try the next combination.
                                 */
                                if (ltgts[t] != ltgts[t + 1])
                                        break;

                                /*
                                 * Otherwise, reset this tgt to the minimum,
                                 * and move on to the next tgt.
                                 */
                                ltgts[t] = ltgts[t - 1] + 1;
                                ASSERT3U(ltgts[t], ==, t);
                        }

                        /* Increase the number of failures and keep trying. */
                        if (ltgts[num_failures - 1] == n)
                                break;
                }
        }

        return (ECKSUM);
}

void
vdev_raidz_reconstruct(raidz_map_t *rm, const int *t, int nt)
{
        for (uint64_t row = 0; row < rm->rm_nrows; row++) {
                raidz_row_t *rr = rm->rm_row[row];
                vdev_raidz_reconstruct_row(rm, rr, t, nt);
        }
}

/*
 * Complete a write IO operation on a RAIDZ VDev
 *
 * Outline:
 *   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.
 */
static void
vdev_raidz_io_done_write_impl(zio_t *zio, raidz_row_t *rr)
{
        int total_errors = 0;

        ASSERT3U(rr->rr_missingparity, <=, rr->rr_firstdatacol);
        ASSERT3U(rr->rr_missingdata, <=, rr->rr_cols - rr->rr_firstdatacol);
        ASSERT3U(zio->io_type, ==, ZIO_TYPE_WRITE);

        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];

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

                        total_errors++;
                }
        }

        /*
         * 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.
         */
        if (total_errors > rr->rr_firstdatacol) {
                zio->io_error = zio_worst_error(zio->io_error,
                    vdev_raidz_worst_error(rr));
        }
}

static void
vdev_raidz_io_done_reconstruct_known_missing(zio_t *zio, raidz_map_t *rm,
    raidz_row_t *rr)
{
        int parity_errors = 0;
        int parity_untried = 0;
        int data_errors = 0;
        int total_errors = 0;

        ASSERT3U(rr->rr_missingparity, <=, rr->rr_firstdatacol);
        ASSERT3U(rr->rr_missingdata, <=, rr->rr_cols - rr->rr_firstdatacol);
        ASSERT3U(zio->io_type, ==, ZIO_TYPE_READ);

        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];

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

                        if (c < rr->rr_firstdatacol)
                                parity_errors++;
                        else
                                data_errors++;

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

        /*
         * If there were data errors and the number of errors we saw was
         * correctable -- less than or equal to the number of parity disks read
         * -- reconstruct based on the missing data.
         */
        if (data_errors != 0 &&
            total_errors <= rr->rr_firstdatacol - parity_untried) {
                /*
                 * 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 < rr->rr_firstdatacol);

                /*
                 * Identify the data columns that reported an error.
                 */
                int n = 0;
                int tgts[VDEV_RAIDZ_MAXPARITY];
                for (int c = rr->rr_firstdatacol; c < rr->rr_cols; c++) {
                        raidz_col_t *rc = &rr->rr_col[c];
                        if (rc->rc_error != 0) {
                                ASSERT(n < VDEV_RAIDZ_MAXPARITY);
                                tgts[n++] = c;
                        }
                }

                ASSERT(rr->rr_firstdatacol >= n);

                vdev_raidz_reconstruct_row(rm, rr, tgts, n);
        }
}

/*
 * Return the number of reads issued.
 */
static int
vdev_raidz_read_all(zio_t *zio, raidz_row_t *rr)
{
        vdev_t *vd = zio->io_vd;
        int nread = 0;

        rr->rr_missingdata = 0;
        rr->rr_missingparity = 0;

        /*
         * If this rows contains empty sectors which are not required
         * for a normal read then allocate an ABD for them now so they
         * may be read, verified, and any needed repairs performed.
         */
        if (rr->rr_nempty && rr->rr_abd_empty == NULL)
                vdev_draid_map_alloc_empty(zio, rr);

        for (int c = 0; c < rr->rr_cols; c++) {
                raidz_col_t *rc = &rr->rr_col[c];
                if (rc->rc_tried || rc->rc_size == 0)
                        continue;

                zio_nowait(zio_vdev_child_io(zio, NULL,
                    vd->vdev_child[rc->rc_devidx],
                    rc->rc_offset, rc->rc_abd, rc->rc_size,
                    zio->io_type, zio->io_priority, 0,
                    vdev_raidz_child_done, rc));
                nread++;
        }
        return (nread);
}

/*
 * We're here because either there were too many errors to even attempt
 * reconstruction (total_errors == rm_first_datacol), or vdev_*_combrec()
 * failed. In either case, there is enough bad data to prevent reconstruction.
 * Start checksum ereports for all children which haven't failed.
 */
static void
vdev_raidz_io_done_unrecoverable(zio_t *zio)
{
        raidz_map_t *rm = zio->io_vsd;

        for (int i = 0; i < rm->rm_nrows; i++) {
                raidz_row_t *rr = rm->rm_row[i];

                for (int c = 0; c < rr->rr_cols; c++) {
                        raidz_col_t *rc = &rr->rr_col[c];
                        vdev_t *cvd = zio->io_vd->vdev_child[rc->rc_devidx];

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

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

                        (void) zfs_ereport_start_checksum(zio->io_spa,
                            cvd, &zio->io_bookmark, zio, rc->rc_offset,
                            rc->rc_size, &zbc);
                        mutex_enter(&cvd->vdev_stat_lock);
                        cvd->vdev_stat.vs_checksum_errors++;
                        mutex_exit(&cvd->vdev_stat_lock);
                }
        }
}

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

        if (zio->io_type == ZIO_TYPE_WRITE) {
                for (int i = 0; i < rm->rm_nrows; i++) {
                        vdev_raidz_io_done_write_impl(zio, rm->rm_row[i]);
                }
        } else {
                for (int i = 0; i < rm->rm_nrows; i++) {
                        raidz_row_t *rr = rm->rm_row[i];
                        vdev_raidz_io_done_reconstruct_known_missing(zio,
                            rm, rr);
                }

                if (raidz_checksum_verify(zio) == 0) {
                        for (int i = 0; i < rm->rm_nrows; i++) {
                                raidz_row_t *rr = rm->rm_row[i];
                                vdev_raidz_io_done_verified(zio, rr);
                        }
                        zio_checksum_verified(zio);
                } else {
                        /*
                         * A sequential resilver has no checksum which makes
                         * combinatoral reconstruction impossible. This code
                         * path is unreachable since raidz_checksum_verify()
                         * has no checksum to verify and must succeed.
                         */
                        ASSERT3U(zio->io_priority, !=, ZIO_PRIORITY_REBUILD);

                        /*
                         * 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.
                         */
                        int nread = 0;
                        for (int i = 0; i < rm->rm_nrows; i++) {
                                nread += vdev_raidz_read_all(zio,
                                    rm->rm_row[i]);
                        }
                        if (nread != 0) {
                                /*
                                 * Normally our stage is VDEV_IO_DONE, but if
                                 * we've already called redone(), it will have
                                 * changed to VDEV_IO_START, in which case we
                                 * don't want to call redone() again.
                                 */
                                if (zio->io_stage != ZIO_STAGE_VDEV_IO_START)
                                        zio_vdev_io_redone(zio);
                                return;
                        }

                        zio->io_error = vdev_raidz_combrec(zio);
                        if (zio->io_error == ECKSUM &&
                            !(zio->io_flags & ZIO_FLAG_SPECULATIVE)) {
                                vdev_raidz_io_done_unrecoverable(zio);
                        }
                }
        }
}

static void
vdev_raidz_state_change(vdev_t *vd, int faulted, int degraded)
{
        vdev_raidz_t *vdrz = vd->vdev_tsd;
        if (faulted > vdrz->vd_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);
}

/*
 * Determine if any portion of the provided block resides on a child vdev
 * with a dirty DTL and therefore needs to be resilvered.  The function
 * assumes that at least one DTL is dirty which implies that full stripe
 * width blocks must be resilvered.
 */
static boolean_t
vdev_raidz_need_resilver(vdev_t *vd, const dva_t *dva, size_t psize,
    uint64_t phys_birth)
{
        vdev_raidz_t *vdrz = vd->vdev_tsd;
        uint64_t dcols = vd->vdev_children;
        uint64_t nparity = vdrz->vd_nparity;
        uint64_t ashift = vd->vdev_top->vdev_ashift;
        /* The starting RAIDZ (parent) vdev sector of the block. */
        uint64_t b = DVA_GET_OFFSET(dva) >> ashift;
        /* The zio's size in units of the vdev's minimum sector size. */
        uint64_t s = ((psize - 1) >> ashift) + 1;
        /* The first column for this stripe. */
        uint64_t f = b % dcols;

        /* Unreachable by sequential resilver. */
        ASSERT3U(phys_birth, !=, TXG_UNKNOWN);

        if (!vdev_dtl_contains(vd, DTL_PARTIAL, phys_birth, 1))
                return (B_FALSE);

        if (s + nparity >= dcols)
                return (B_TRUE);

        for (uint64_t c = 0; c < s + nparity; c++) {
                uint64_t devidx = (f + c) % dcols;
                vdev_t *cvd = vd->vdev_child[devidx];

                /*
                 * dsl_scan_need_resilver() already checked vd with
                 * vdev_dtl_contains(). So here just check cvd with
                 * vdev_dtl_empty(), cheaper and a good approximation.
                 */
                if (!vdev_dtl_empty(cvd, DTL_PARTIAL))
                        return (B_TRUE);
        }

        return (B_FALSE);
}

static void
vdev_raidz_xlate(vdev_t *cvd, const range_seg64_t *logical_rs,
    range_seg64_t *physical_rs, range_seg64_t *remain_rs)
{
        vdev_t *raidvd = cvd->vdev_parent;
        ASSERT(raidvd->vdev_ops == &vdev_raidz_ops);

        uint64_t width = raidvd->vdev_children;
        uint64_t tgt_col = cvd->vdev_id;
        uint64_t ashift = raidvd->vdev_top->vdev_ashift;

        /* make sure the offsets are block-aligned */
        ASSERT0(logical_rs->rs_start % (1 << ashift));
        ASSERT0(logical_rs->rs_end % (1 << ashift));
        uint64_t b_start = logical_rs->rs_start >> ashift;
        uint64_t b_end = logical_rs->rs_end >> ashift;

        uint64_t start_row = 0;
        if (b_start > tgt_col) /* avoid underflow */
                start_row = ((b_start - tgt_col - 1) / width) + 1;

        uint64_t end_row = 0;
        if (b_end > tgt_col)
                end_row = ((b_end - tgt_col - 1) / width) + 1;

        physical_rs->rs_start = start_row << ashift;
        physical_rs->rs_end = end_row << ashift;

        ASSERT3U(physical_rs->rs_start, <=, logical_rs->rs_start);
        ASSERT3U(physical_rs->rs_end - physical_rs->rs_start, <=,
            logical_rs->rs_end - logical_rs->rs_start);
}

/*
 * Initialize private RAIDZ specific fields from the nvlist.
 */
static int
vdev_raidz_init(spa_t *spa, nvlist_t *nv, void **tsd)
{
        vdev_raidz_t *vdrz;
        uint64_t nparity;

        uint_t children;
        nvlist_t **child;
        int error = nvlist_lookup_nvlist_array(nv,
            ZPOOL_CONFIG_CHILDREN, &child, &children);
        if (error != 0)
                return (SET_ERROR(EINVAL));

        if (nvlist_lookup_uint64(nv, ZPOOL_CONFIG_NPARITY, &nparity) == 0) {
                if (nparity == 0 || nparity > VDEV_RAIDZ_MAXPARITY)
                        return (SET_ERROR(EINVAL));

                /*
                 * Previous versions could only support 1 or 2 parity
                 * device.
                 */
                if (nparity > 1 && spa_version(spa) < SPA_VERSION_RAIDZ2)
                        return (SET_ERROR(EINVAL));
                else if (nparity > 2 && spa_version(spa) < SPA_VERSION_RAIDZ3)
                        return (SET_ERROR(EINVAL));
        } else {
                /*
                 * We require the parity to be specified for SPAs that
                 * support multiple parity levels.
                 */
                if (spa_version(spa) >= SPA_VERSION_RAIDZ2)
                        return (SET_ERROR(EINVAL));

                /*
                 * Otherwise, we default to 1 parity device for RAID-Z.
                 */
                nparity = 1;
        }

        vdrz = kmem_zalloc(sizeof (*vdrz), KM_SLEEP);
        vdrz->vd_logical_width = children;
        vdrz->vd_nparity = nparity;

        *tsd = vdrz;

        return (0);
}

static void
vdev_raidz_fini(vdev_t *vd)
{
        kmem_free(vd->vdev_tsd, sizeof (vdev_raidz_t));
}

/*
 * Add RAIDZ specific fields to the config nvlist.
 */
static void
vdev_raidz_config_generate(vdev_t *vd, nvlist_t *nv)
{
        ASSERT3P(vd->vdev_ops, ==, &vdev_raidz_ops);
        vdev_raidz_t *vdrz = vd->vdev_tsd;

        /*
         * Make sure someone hasn't managed to sneak a fancy new vdev
         * into a crufty old storage pool.
         */
        ASSERT(vdrz->vd_nparity == 1 ||
            (vdrz->vd_nparity <= 2 &&
            spa_version(vd->vdev_spa) >= SPA_VERSION_RAIDZ2) ||
            (vdrz->vd_nparity <= 3 &&
            spa_version(vd->vdev_spa) >= SPA_VERSION_RAIDZ3));

        /*
         * Note that we'll add these even on storage pools where they
         * aren't strictly required -- older software will just ignore
         * it.
         */
        fnvlist_add_uint64(nv, ZPOOL_CONFIG_NPARITY, vdrz->vd_nparity);
}

static uint64_t
vdev_raidz_nparity(vdev_t *vd)
{
        vdev_raidz_t *vdrz = vd->vdev_tsd;
        return (vdrz->vd_nparity);
}

static uint64_t
vdev_raidz_ndisks(vdev_t *vd)
{
        return (vd->vdev_children);
}

vdev_ops_t vdev_raidz_ops = {
        .vdev_op_init = vdev_raidz_init,
        .vdev_op_fini = vdev_raidz_fini,
        .vdev_op_open = vdev_raidz_open,
        .vdev_op_close = vdev_raidz_close,
        .vdev_op_asize = vdev_raidz_asize,
        .vdev_op_min_asize = vdev_raidz_min_asize,
        .vdev_op_min_alloc = NULL,
        .vdev_op_io_start = vdev_raidz_io_start,
        .vdev_op_io_done = vdev_raidz_io_done,
        .vdev_op_state_change = vdev_raidz_state_change,
        .vdev_op_need_resilver = vdev_raidz_need_resilver,
        .vdev_op_hold = NULL,
        .vdev_op_rele = NULL,
        .vdev_op_remap = NULL,
        .vdev_op_xlate = vdev_raidz_xlate,
        .vdev_op_rebuild_asize = NULL,
        .vdev_op_metaslab_init = NULL,
        .vdev_op_config_generate = vdev_raidz_config_generate,
        .vdev_op_nparity = vdev_raidz_nparity,
        .vdev_op_ndisks = vdev_raidz_ndisks,
        .vdev_op_type = VDEV_TYPE_RAIDZ,        /* name of this vdev type */
        .vdev_op_leaf = B_FALSE                 /* not a leaf vdev */
};