| 1 | /*
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| 2 | * Copyright 1995-2022 The OpenSSL Project Authors. All Rights Reserved.
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| 3 | *
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| 4 | * Licensed under the Apache License 2.0 (the "License"). You may not use
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| 5 | * this file except in compliance with the License. You can obtain a copy
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| 6 | * in the file LICENSE in the source distribution or at
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| 7 | * https://www.openssl.org/source/license.html
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| 8 | */
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| 9 |
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| 10 | #include <assert.h>
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| 11 | #include <openssl/bn.h>
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| 12 | #include "internal/cryptlib.h"
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| 13 | #include "bn_local.h"
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| 14 |
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| 15 | /* The old slow way */
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| 16 | #if 0
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| 17 | int BN_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m, const BIGNUM *d,
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| 18 | BN_CTX *ctx)
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| 19 | {
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| 20 | int i, nm, nd;
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| 21 | int ret = 0;
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| 22 | BIGNUM *D;
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| 23 |
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| 24 | bn_check_top(m);
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| 25 | bn_check_top(d);
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| 26 | if (BN_is_zero(d)) {
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| 27 | ERR_raise(ERR_LIB_BN, BN_R_DIV_BY_ZERO);
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| 28 | return 0;
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| 29 | }
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| 30 |
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| 31 | if (BN_ucmp(m, d) < 0) {
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| 32 | if (rem != NULL) {
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| 33 | if (BN_copy(rem, m) == NULL)
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| 34 | return 0;
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| 35 | }
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| 36 | if (dv != NULL)
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| 37 | BN_zero(dv);
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| 38 | return 1;
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| 39 | }
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| 40 |
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| 41 | BN_CTX_start(ctx);
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| 42 | D = BN_CTX_get(ctx);
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| 43 | if (dv == NULL)
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| 44 | dv = BN_CTX_get(ctx);
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| 45 | if (rem == NULL)
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| 46 | rem = BN_CTX_get(ctx);
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| 47 | if (D == NULL || dv == NULL || rem == NULL)
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| 48 | goto end;
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| 49 |
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| 50 | nd = BN_num_bits(d);
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| 51 | nm = BN_num_bits(m);
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| 52 | if (BN_copy(D, d) == NULL)
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| 53 | goto end;
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| 54 | if (BN_copy(rem, m) == NULL)
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| 55 | goto end;
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| 56 |
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| 57 | /*
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| 58 | * The next 2 are needed so we can do a dv->d[0]|=1 later since
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| 59 | * BN_lshift1 will only work once there is a value :-)
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| 60 | */
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| 61 | BN_zero(dv);
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| 62 | if (bn_wexpand(dv, 1) == NULL)
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| 63 | goto end;
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| 64 | dv->top = 1;
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| 65 |
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| 66 | if (!BN_lshift(D, D, nm - nd))
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| 67 | goto end;
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| 68 | for (i = nm - nd; i >= 0; i--) {
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| 69 | if (!BN_lshift1(dv, dv))
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| 70 | goto end;
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| 71 | if (BN_ucmp(rem, D) >= 0) {
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| 72 | dv->d[0] |= 1;
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| 73 | if (!BN_usub(rem, rem, D))
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| 74 | goto end;
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| 75 | }
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| 76 | /* CAN IMPROVE (and have now :=) */
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| 77 | if (!BN_rshift1(D, D))
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| 78 | goto end;
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| 79 | }
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| 80 | rem->neg = BN_is_zero(rem) ? 0 : m->neg;
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| 81 | dv->neg = m->neg ^ d->neg;
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| 82 | ret = 1;
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| 83 | end:
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| 84 | BN_CTX_end(ctx);
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| 85 | return ret;
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| 86 | }
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| 87 |
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| 88 | #else
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| 89 |
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| 90 | # if defined(BN_DIV3W)
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| 91 | BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0);
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| 92 | # elif 0
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| 93 | /*
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| 94 | * This is #if-ed away, because it's a reference for assembly implementations,
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| 95 | * where it can and should be made constant-time. But if you want to test it,
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| 96 | * just replace 0 with 1.
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| 97 | */
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| 98 | # if BN_BITS2 == 64 && defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
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| 99 | # undef BN_ULLONG
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| 100 | # define BN_ULLONG uint128_t
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| 101 | # define BN_LLONG
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| 102 | # endif
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| 103 |
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| 104 | # ifdef BN_LLONG
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| 105 | # define BN_DIV3W
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| 106 | /*
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| 107 | * Interface is somewhat quirky, |m| is pointer to most significant limb,
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| 108 | * and less significant limb is referred at |m[-1]|. This means that caller
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| 109 | * is responsible for ensuring that |m[-1]| is valid. Second condition that
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| 110 | * has to be met is that |d0|'s most significant bit has to be set. Or in
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| 111 | * other words divisor has to be "bit-aligned to the left." bn_div_fixed_top
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| 112 | * does all this. The subroutine considers four limbs, two of which are
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| 113 | * "overlapping," hence the name...
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| 114 | */
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| 115 | static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0)
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| 116 | {
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| 117 | BN_ULLONG R = ((BN_ULLONG)m[0] << BN_BITS2) | m[-1];
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| 118 | BN_ULLONG D = ((BN_ULLONG)d0 << BN_BITS2) | d1;
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| 119 | BN_ULONG Q = 0, mask;
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| 120 | int i;
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| 121 |
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| 122 | for (i = 0; i < BN_BITS2; i++) {
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| 123 | Q <<= 1;
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| 124 | if (R >= D) {
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| 125 | Q |= 1;
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| 126 | R -= D;
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| 127 | }
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| 128 | D >>= 1;
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| 129 | }
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| 130 |
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| 131 | mask = 0 - (Q >> (BN_BITS2 - 1)); /* does it overflow? */
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| 132 |
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| 133 | Q <<= 1;
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| 134 | Q |= (R >= D);
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| 135 |
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| 136 | return (Q | mask) & BN_MASK2;
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| 137 | }
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| 138 | # endif
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| 139 | # endif
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| 140 |
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| 141 | static int bn_left_align(BIGNUM *num)
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| 142 | {
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| 143 | BN_ULONG *d = num->d, n, m, rmask;
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| 144 | int top = num->top;
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| 145 | int rshift = BN_num_bits_word(d[top - 1]), lshift, i;
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| 146 |
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| 147 | lshift = BN_BITS2 - rshift;
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| 148 | rshift %= BN_BITS2; /* say no to undefined behaviour */
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| 149 | rmask = (BN_ULONG)0 - rshift; /* rmask = 0 - (rshift != 0) */
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| 150 | rmask |= rmask >> 8;
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| 151 |
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| 152 | for (i = 0, m = 0; i < top; i++) {
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| 153 | n = d[i];
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| 154 | d[i] = ((n << lshift) | m) & BN_MASK2;
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| 155 | m = (n >> rshift) & rmask;
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| 156 | }
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| 157 |
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| 158 | return lshift;
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| 159 | }
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| 160 |
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| 161 | # if !defined(OPENSSL_NO_ASM) && !defined(OPENSSL_NO_INLINE_ASM) \
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| 162 | && !defined(PEDANTIC) && !defined(BN_DIV3W)
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| 163 | # if defined(__GNUC__) && __GNUC__>=2
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| 164 | # if defined(__i386) || defined (__i386__)
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| 165 | /*-
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| 166 | * There were two reasons for implementing this template:
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| 167 | * - GNU C generates a call to a function (__udivdi3 to be exact)
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| 168 | * in reply to ((((BN_ULLONG)n0)<<BN_BITS2)|n1)/d0 (I fail to
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| 169 | * understand why...);
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| 170 | * - divl doesn't only calculate quotient, but also leaves
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| 171 | * remainder in %edx which we can definitely use here:-)
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| 172 | */
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| 173 | # undef bn_div_words
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| 174 | # define bn_div_words(n0,n1,d0) \
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| 175 | ({ asm volatile ( \
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| 176 | "divl %4" \
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| 177 | : "=a"(q), "=d"(rem) \
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| 178 | : "a"(n1), "d"(n0), "r"(d0) \
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| 179 | : "cc"); \
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| 180 | q; \
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| 181 | })
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| 182 | # define REMAINDER_IS_ALREADY_CALCULATED
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| 183 | # elif defined(__x86_64) && defined(SIXTY_FOUR_BIT_LONG)
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| 184 | /*
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| 185 | * Same story here, but it's 128-bit by 64-bit division. Wow!
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| 186 | */
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| 187 | # undef bn_div_words
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| 188 | # define bn_div_words(n0,n1,d0) \
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| 189 | ({ asm volatile ( \
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| 190 | "divq %4" \
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| 191 | : "=a"(q), "=d"(rem) \
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| 192 | : "a"(n1), "d"(n0), "r"(d0) \
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| 193 | : "cc"); \
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| 194 | q; \
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| 195 | })
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| 196 | # define REMAINDER_IS_ALREADY_CALCULATED
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| 197 | # endif /* __<cpu> */
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| 198 | # endif /* __GNUC__ */
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| 199 | # endif /* OPENSSL_NO_ASM */
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| 200 |
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| 201 | /*-
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| 202 | * BN_div computes dv := num / divisor, rounding towards
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| 203 | * zero, and sets up rm such that dv*divisor + rm = num holds.
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| 204 | * Thus:
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| 205 | * dv->neg == num->neg ^ divisor->neg (unless the result is zero)
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| 206 | * rm->neg == num->neg (unless the remainder is zero)
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| 207 | * If 'dv' or 'rm' is NULL, the respective value is not returned.
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| 208 | */
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| 209 | int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
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| 210 | BN_CTX *ctx)
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| 211 | {
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| 212 | int ret;
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| 213 |
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| 214 | if (BN_is_zero(divisor)) {
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| 215 | ERR_raise(ERR_LIB_BN, BN_R_DIV_BY_ZERO);
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| 216 | return 0;
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| 217 | }
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| 218 |
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| 219 | /*
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| 220 | * Invalid zero-padding would have particularly bad consequences so don't
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| 221 | * just rely on bn_check_top() here (bn_check_top() works only for
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| 222 | * BN_DEBUG builds)
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| 223 | */
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| 224 | if (divisor->d[divisor->top - 1] == 0) {
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| 225 | ERR_raise(ERR_LIB_BN, BN_R_NOT_INITIALIZED);
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| 226 | return 0;
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| 227 | }
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| 228 |
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| 229 | ret = bn_div_fixed_top(dv, rm, num, divisor, ctx);
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| 230 |
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| 231 | if (ret) {
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| 232 | if (dv != NULL)
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| 233 | bn_correct_top(dv);
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| 234 | if (rm != NULL)
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| 235 | bn_correct_top(rm);
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| 236 | }
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| 237 |
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| 238 | return ret;
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| 239 | }
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| 240 |
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| 241 | /*
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| 242 | * It's argued that *length* of *significant* part of divisor is public.
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| 243 | * Even if it's private modulus that is. Again, *length* is assumed
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| 244 | * public, but not *value*. Former is likely to be pre-defined by
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| 245 | * algorithm with bit granularity, though below subroutine is invariant
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| 246 | * of limb length. Thanks to this assumption we can require that |divisor|
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| 247 | * may not be zero-padded, yet claim this subroutine "constant-time"(*).
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| 248 | * This is because zero-padded dividend, |num|, is tolerated, so that
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| 249 | * caller can pass dividend of public length(*), but with smaller amount
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| 250 | * of significant limbs. This naturally means that quotient, |dv|, would
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| 251 | * contain correspongly less significant limbs as well, and will be zero-
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| 252 | * padded accordingly. Returned remainder, |rm|, will have same bit length
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| 253 | * as divisor, also zero-padded if needed. These actually leave sign bits
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| 254 | * in ambiguous state. In sense that we try to avoid negative zeros, while
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| 255 | * zero-padded zeros would retain sign.
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| 256 | *
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| 257 | * (*) "Constant-time-ness" has two pre-conditions:
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| 258 | *
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| 259 | * - availability of constant-time bn_div_3_words;
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| 260 | * - dividend is at least as "wide" as divisor, limb-wise, zero-padded
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| 261 | * if so required, which shouldn't be a privacy problem, because
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| 262 | * divisor's length is considered public;
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| 263 | */
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| 264 | int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num,
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| 265 | const BIGNUM *divisor, BN_CTX *ctx)
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| 266 | {
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| 267 | int norm_shift, i, j, loop;
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| 268 | BIGNUM *tmp, *snum, *sdiv, *res;
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| 269 | BN_ULONG *resp, *wnum, *wnumtop;
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| 270 | BN_ULONG d0, d1;
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| 271 | int num_n, div_n, num_neg;
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| 272 |
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| 273 | assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0);
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| 274 |
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| 275 | bn_check_top(num);
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| 276 | bn_check_top(divisor);
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| 277 | bn_check_top(dv);
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| 278 | bn_check_top(rm);
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| 279 |
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| 280 | BN_CTX_start(ctx);
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| 281 | res = (dv == NULL) ? BN_CTX_get(ctx) : dv;
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| 282 | tmp = BN_CTX_get(ctx);
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| 283 | snum = BN_CTX_get(ctx);
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| 284 | sdiv = BN_CTX_get(ctx);
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| 285 | if (sdiv == NULL)
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| 286 | goto err;
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| 287 |
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| 288 | /* First we normalise the numbers */
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| 289 | if (!BN_copy(sdiv, divisor))
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| 290 | goto err;
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| 291 | norm_shift = bn_left_align(sdiv);
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| 292 | sdiv->neg = 0;
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| 293 | /*
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| 294 | * Note that bn_lshift_fixed_top's output is always one limb longer
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| 295 | * than input, even when norm_shift is zero. This means that amount of
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| 296 | * inner loop iterations is invariant of dividend value, and that one
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| 297 | * doesn't need to compare dividend and divisor if they were originally
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| 298 | * of the same bit length.
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| 299 | */
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| 300 | if (!(bn_lshift_fixed_top(snum, num, norm_shift)))
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| 301 | goto err;
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| 302 |
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| 303 | div_n = sdiv->top;
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| 304 | num_n = snum->top;
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| 305 |
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| 306 | if (num_n <= div_n) {
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| 307 | /* caller didn't pad dividend -> no constant-time guarantee... */
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| 308 | if (bn_wexpand(snum, div_n + 1) == NULL)
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| 309 | goto err;
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| 310 | memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG));
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| 311 | snum->top = num_n = div_n + 1;
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| 312 | }
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| 313 |
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| 314 | loop = num_n - div_n;
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| 315 | /*
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| 316 | * Lets setup a 'window' into snum This is the part that corresponds to
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| 317 | * the current 'area' being divided
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| 318 | */
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| 319 | wnum = &(snum->d[loop]);
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| 320 | wnumtop = &(snum->d[num_n - 1]);
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| 321 |
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| 322 | /* Get the top 2 words of sdiv */
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| 323 | d0 = sdiv->d[div_n - 1];
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| 324 | d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2];
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| 325 |
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| 326 | /* Setup quotient */
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| 327 | if (!bn_wexpand(res, loop))
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| 328 | goto err;
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| 329 | num_neg = num->neg;
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| 330 | res->neg = (num_neg ^ divisor->neg);
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| 331 | res->top = loop;
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| 332 | res->flags |= BN_FLG_FIXED_TOP;
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| 333 | resp = &(res->d[loop]);
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| 334 |
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| 335 | /* space for temp */
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| 336 | if (!bn_wexpand(tmp, (div_n + 1)))
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| 337 | goto err;
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| 338 |
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| 339 | for (i = 0; i < loop; i++, wnumtop--) {
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| 340 | BN_ULONG q, l0;
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| 341 | /*
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| 342 | * the first part of the loop uses the top two words of snum and sdiv
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| 343 | * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
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| 344 | */
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| 345 | # if defined(BN_DIV3W)
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| 346 | q = bn_div_3_words(wnumtop, d1, d0);
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| 347 | # else
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| 348 | BN_ULONG n0, n1, rem = 0;
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| 349 |
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| 350 | n0 = wnumtop[0];
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| 351 | n1 = wnumtop[-1];
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| 352 | if (n0 == d0)
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| 353 | q = BN_MASK2;
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| 354 | else { /* n0 < d0 */
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| 355 | BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
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| 356 | # ifdef BN_LLONG
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| 357 | BN_ULLONG t2;
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| 358 |
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| 359 | # if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words)
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| 360 | q = (BN_ULONG)(((((BN_ULLONG) n0) << BN_BITS2) | n1) / d0);
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| 361 | # else
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| 362 | q = bn_div_words(n0, n1, d0);
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| 363 | # endif
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| 364 |
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| 365 | # ifndef REMAINDER_IS_ALREADY_CALCULATED
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| 366 | /*
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| 367 | * rem doesn't have to be BN_ULLONG. The least we
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| 368 | * know it's less that d0, isn't it?
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| 369 | */
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| 370 | rem = (n1 - q * d0) & BN_MASK2;
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| 371 | # endif
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| 372 | t2 = (BN_ULLONG) d1 *q;
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| 373 |
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| 374 | for (;;) {
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| 375 | if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2))
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| 376 | break;
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| 377 | q--;
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| 378 | rem += d0;
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| 379 | if (rem < d0)
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| 380 | break; /* don't let rem overflow */
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| 381 | t2 -= d1;
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| 382 | }
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| 383 | # else /* !BN_LLONG */
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| 384 | BN_ULONG t2l, t2h;
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| 385 |
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| 386 | q = bn_div_words(n0, n1, d0);
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| 387 | # ifndef REMAINDER_IS_ALREADY_CALCULATED
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| 388 | rem = (n1 - q * d0) & BN_MASK2;
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| 389 | # endif
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| 390 |
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| 391 | # if defined(BN_UMULT_LOHI)
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| 392 | BN_UMULT_LOHI(t2l, t2h, d1, q);
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| 393 | # elif defined(BN_UMULT_HIGH)
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| 394 | t2l = d1 * q;
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| 395 | t2h = BN_UMULT_HIGH(d1, q);
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| 396 | # else
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| 397 | {
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| 398 | BN_ULONG ql, qh;
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| 399 | t2l = LBITS(d1);
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| 400 | t2h = HBITS(d1);
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| 401 | ql = LBITS(q);
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| 402 | qh = HBITS(q);
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| 403 | mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */
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| 404 | }
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| 405 | # endif
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| 406 |
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| 407 | for (;;) {
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| 408 | if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
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| 409 | break;
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| 410 | q--;
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| 411 | rem += d0;
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| 412 | if (rem < d0)
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| 413 | break; /* don't let rem overflow */
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| 414 | if (t2l < d1)
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| 415 | t2h--;
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| 416 | t2l -= d1;
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| 417 | }
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| 418 | # endif /* !BN_LLONG */
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| 419 | }
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| 420 | # endif /* !BN_DIV3W */
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| 421 |
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| 422 | l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
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| 423 | tmp->d[div_n] = l0;
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| 424 | wnum--;
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| 425 | /*
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| 426 | * ignore top values of the bignums just sub the two BN_ULONG arrays
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| 427 | * with bn_sub_words
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| 428 | */
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| 429 | l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
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| 430 | q -= l0;
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|---|
| 431 | /*
|
|---|
| 432 | * Note: As we have considered only the leading two BN_ULONGs in
|
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| 433 | * the calculation of q, sdiv * q might be greater than wnum (but
|
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| 434 | * then (q-1) * sdiv is less or equal than wnum)
|
|---|
| 435 | */
|
|---|
| 436 | for (l0 = 0 - l0, j = 0; j < div_n; j++)
|
|---|
| 437 | tmp->d[j] = sdiv->d[j] & l0;
|
|---|
| 438 | l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
|
|---|
| 439 | (*wnumtop) += l0;
|
|---|
| 440 | assert((*wnumtop) == 0);
|
|---|
| 441 |
|
|---|
| 442 | /* store part of the result */
|
|---|
| 443 | *--resp = q;
|
|---|
| 444 | }
|
|---|
| 445 | /* snum holds remainder, it's as wide as divisor */
|
|---|
| 446 | snum->neg = num_neg;
|
|---|
| 447 | snum->top = div_n;
|
|---|
| 448 | snum->flags |= BN_FLG_FIXED_TOP;
|
|---|
| 449 |
|
|---|
| 450 | if (rm != NULL && bn_rshift_fixed_top(rm, snum, norm_shift) == 0)
|
|---|
| 451 | goto err;
|
|---|
| 452 |
|
|---|
| 453 | BN_CTX_end(ctx);
|
|---|
| 454 | return 1;
|
|---|
| 455 | err:
|
|---|
| 456 | bn_check_top(rm);
|
|---|
| 457 | BN_CTX_end(ctx);
|
|---|
| 458 | return 0;
|
|---|
| 459 | }
|
|---|
| 460 | #endif
|
|---|
