bn/bn_{div|shift}.c: introduce fixed-top interfaces.

Fixed-top interfaces tolerate zero-padded inputs and facilitate
constant-time-ness. bn_div_fixed_top tolerates zero-padded dividend,
but not divisor. It's argued that divisor's length is public even
when value is secret.

[extended tests]
Reviewed-by: NPaul Dale <paul.dale@oracle.com>
Reviewed-by: NMatt Caswell <matt@openssl.org>
(Merged from https://github.com/openssl/openssl/pull/7589)

(cherry picked from commit 3a4a88f436ed1dd1165e0b59c1ca4a25e9e1d690)

crypto/bn/bn_div.c

@@ -7,6 +7,7 @@
  * https://www.openssl.org/source/license.html
  */
 
+#include <assert.h>
 #include <openssl/bn.h>
 #include "internal/cryptlib.h"
 #include "bn_lcl.h"
@@ -137,6 +138,26 @@ static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0)
 #  endif
 # endif
 
+static int bn_left_align(BIGNUM *num)
+{
+    BN_ULONG *d = num->d, n, m, rmask;
+    int top = num->top;
+    int rshift = BN_num_bits_word(d[top - 1]), lshift, i;
+
+    lshift = BN_BITS2 - rshift;
+    rshift %= BN_BITS2;            /* say no to undefined behaviour */
+    rmask = (BN_ULONG)0 - rshift;  /* rmask = 0 - (rshift != 0) */
+    rmask |= rmask >> 8;
+
+    for (i = 0, m = 0; i < top; i++) {
+        n = d[i];
+        d[i] = ((n << lshift) | m) & BN_MASK2;
+        m = (n >> rshift) & rmask;
+    }
+
+    return lshift;
+}
+
 # if !defined(OPENSSL_NO_ASM) && !defined(OPENSSL_NO_INLINE_ASM) \
     && !defined(PEDANTIC) && !defined(BN_DIV3W)
 #  if defined(__GNUC__) && __GNUC__>=2
@@ -188,55 +209,73 @@ static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0)
 int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
            BN_CTX *ctx)
 {
-    int norm_shift, i, j, loop;
-    BIGNUM *tmp, wnum, *snum, *sdiv, *res;
-    BN_ULONG *resp, *wnump;
-    BN_ULONG d0, d1;
-    int num_n, div_n;
-    int no_branch = 0;
+    int ret;
+
+    if (BN_is_zero(divisor)) {
+        BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
+        return 0;
+    }
 
     /*
      * Invalid zero-padding would have particularly bad consequences so don't
      * just rely on bn_check_top() here (bn_check_top() works only for
      * BN_DEBUG builds)
      */
-    if ((num->top > 0 && num->d[num->top - 1] == 0) ||
-        (divisor->top > 0 && divisor->d[divisor->top - 1] == 0)) {
+    if (divisor->d[divisor->top - 1] == 0) {
         BNerr(BN_F_BN_DIV, BN_R_NOT_INITIALIZED);
         return 0;
     }
 
-    bn_check_top(num);
-    bn_check_top(divisor);
+    ret = bn_div_fixed_top(dv, rm, num, divisor, ctx);
 
-    if ((BN_get_flags(num, BN_FLG_CONSTTIME) != 0)
-        || (BN_get_flags(divisor, BN_FLG_CONSTTIME) != 0)) {
-        no_branch = 1;
+    if (ret) {
+        if (dv != NULL)
+            bn_correct_top(dv);
+        if (rm != NULL)
+            bn_correct_top(rm);
     }
 
-    bn_check_top(dv);
-    bn_check_top(rm);
-    /*- bn_check_top(num); *//*
-     * 'num' has been checked already
-     */
-    /*- bn_check_top(divisor); *//*
-     * 'divisor' has been checked already
-     */
+    return ret;
+}
 
-    if (BN_is_zero(divisor)) {
-        BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
-        return 0;
-    }
+/*
+ * It's argued that *length* of *significant* part of divisor is public.
+ * Even if it's private modulus that is. Again, *length* is assumed
+ * public, but not *value*. Former is likely to be pre-defined by
+ * algorithm with bit granularity, though below subroutine is invariant
+ * of limb length. Thanks to this assumption we can require that |divisor|
+ * may not be zero-padded, yet claim this subroutine "constant-time"(*).
+ * This is because zero-padded dividend, |num|, is tolerated, so that
+ * caller can pass dividend of public length(*), but with smaller amount
+ * of significant limbs. This naturally means that quotient, |dv|, would
+ * contain correspongly less significant limbs as well, and will be zero-
+ * padded accordingly. Returned remainder, |rm|, will have same bit length
+ * as divisor, also zero-padded if needed. These actually leave sign bits
+ * in ambiguous state. In sense that we try to avoid negative zeros, while
+ * zero-padded zeros would retain sign.
+ *
+ * (*) "Constant-time-ness" has two pre-conditions:
+ *
+ *     - availability of constant-time bn_div_3_words;
+ *     - dividend is at least as "wide" as divisor, limb-wise, zero-padded
+ *       if so requied, which shouldn't be a privacy problem, because
+ *       divisor's length is considered public;
+ */
+int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num,
+                     const BIGNUM *divisor, BN_CTX *ctx)
+{
+    int norm_shift, i, j, loop;
+    BIGNUM *tmp, *snum, *sdiv, *res;
+    BN_ULONG *resp, *wnum, *wnumtop;
+    BN_ULONG d0, d1;
+    int num_n, div_n;
 
-    if (!no_branch && BN_ucmp(num, divisor) < 0) {
-        if (rm != NULL) {
-            if (BN_copy(rm, num) == NULL)
-                return 0;
-        }
-        if (dv != NULL)
-            BN_zero(dv);
-        return 1;
-    }
+    assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0);
+
+    bn_check_top(num);
+    bn_check_top(divisor);
+    bn_check_top(dv);
+    bn_check_top(rm);
 
     BN_CTX_start(ctx);
     res = (dv == NULL) ? BN_CTX_get(ctx) : dv;
@@ -247,112 +286,72 @@ int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
         goto err;
 
     /* First we normalise the numbers */
-    norm_shift = BN_BITS2 - ((BN_num_bits(divisor)) % BN_BITS2);
-    if (!(BN_lshift(sdiv, divisor, norm_shift)))
+    if (!BN_copy(sdiv, divisor))
         goto err;
+    norm_shift = bn_left_align(sdiv);
     sdiv->neg = 0;
-    norm_shift += BN_BITS2;
-    if (!(BN_lshift(snum, num, norm_shift)))
+    /*
+     * Note that bn_lshift_fixed_top's output is always one limb longer
+     * than input, even when norm_shift is zero. This means that amount of
+     * inner loop iterations is invariant of dividend value, and that one
+     * doesn't need to compare dividend and divisor if they were originally
+     * of the same bit length.
+     */
+    if (!(bn_lshift_fixed_top(snum, num, norm_shift)))
         goto err;
-    snum->neg = 0;
-
-    if (no_branch) {
-        /*
-         * Since we don't know whether snum is larger than sdiv, we pad snum
-         * with enough zeroes without changing its value.
-         */
-        if (snum->top <= sdiv->top + 1) {
-            if (bn_wexpand(snum, sdiv->top + 2) == NULL)
-                goto err;
-            for (i = snum->top; i < sdiv->top + 2; i++)
-                snum->d[i] = 0;
-            snum->top = sdiv->top + 2;
-        } else {
-            if (bn_wexpand(snum, snum->top + 1) == NULL)
-                goto err;
-            snum->d[snum->top] = 0;
-            snum->top++;
-        }
-    }
 
     div_n = sdiv->top;
     num_n = snum->top;
+
+    if (num_n <= div_n) {
+        /* caller didn't pad dividend -> no constant-time guarantee... */
+        if (bn_wexpand(snum, div_n + 1) == NULL)
+            goto err;
+        memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG));
+        snum->top = num_n = div_n + 1;
+    }
+
     loop = num_n - div_n;
     /*
      * Lets setup a 'window' into snum This is the part that corresponds to
      * the current 'area' being divided
      */
-    wnum.neg = 0;
-    wnum.d = &(snum->d[loop]);
-    wnum.top = div_n;
-    wnum.flags = BN_FLG_STATIC_DATA;
-    /*
-     * only needed when BN_ucmp messes up the values between top and max
-     */
-    wnum.dmax = snum->dmax - loop; /* so we don't step out of bounds */
+    wnum = &(snum->d[loop]);
+    wnumtop = &(snum->d[num_n - 1]);
 
     /* Get the top 2 words of sdiv */
-    /* div_n=sdiv->top; */
     d0 = sdiv->d[div_n - 1];
     d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2];
 
-    /* pointer to the 'top' of snum */
-    wnump = &(snum->d[num_n - 1]);
-
-    /* Setup to 'res' */
-    if (!bn_wexpand(res, (loop + 1)))
+    /* Setup quotient */
+    if (!bn_wexpand(res, loop))
         goto err;
     res->neg = (num->neg ^ divisor->neg);
-    res->top = loop - no_branch;
-    resp = &(res->d[loop - 1]);
+    res->top = loop;
+    res->flags |= BN_FLG_FIXED_TOP;
+    resp = &(res->d[loop]);
 
     /* space for temp */
     if (!bn_wexpand(tmp, (div_n + 1)))
         goto err;
 
-    if (!no_branch) {
-        if (BN_ucmp(&wnum, sdiv) >= 0) {
-            /*
-             * If BN_DEBUG_RAND is defined BN_ucmp changes (via bn_pollute)
-             * the const bignum arguments => clean the values between top and
-             * max again
-             */
-            bn_clear_top2max(&wnum);
-            bn_sub_words(wnum.d, wnum.d, sdiv->d, div_n);
-            *resp = 1;
-        } else
-            res->top--;
-    }
-
-    /* Increase the resp pointer so that we never create an invalid pointer. */
-    resp++;
-
-    /*
-     * if res->top == 0 then clear the neg value otherwise decrease the resp
-     * pointer
-     */
-    if (res->top == 0)
-        res->neg = 0;
-    else
-        resp--;
-
-    for (i = 0; i < loop - 1; i++, wnump--) {
+    for (i = 0; i < loop; i++, wnumtop--) {
         BN_ULONG q, l0;
         /*
          * the first part of the loop uses the top two words of snum and sdiv
          * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
          */
 # if defined(BN_DIV3W)
-        q = bn_div_3_words(wnump, d1, d0);
+        q = bn_div_3_words(wnumtop, d1, d0);
 # else
         BN_ULONG n0, n1, rem = 0;
 
-        n0 = wnump[0];
-        n1 = wnump[-1];
+        n0 = wnumtop[0];
+        n1 = wnumtop[-1];
         if (n0 == d0)
             q = BN_MASK2;
         else {                  /* n0 < d0 */
-
+            BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
 #  ifdef BN_LLONG
             BN_ULLONG t2;
 
@@ -372,7 +371,7 @@ int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
             t2 = (BN_ULLONG) d1 *q;
 
             for (;;) {
-                if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | wnump[-2]))
+                if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2))
                     break;
                 q--;
                 rem += d0;
@@ -405,7 +404,7 @@ int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
 #   endif
 
             for (;;) {
-                if ((t2h < rem) || ((t2h == rem) && (t2l <= wnump[-2])))
+                if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
                     break;
                 q--;
                 rem += d0;
@@ -421,12 +420,12 @@ int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
 
         l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
         tmp->d[div_n] = l0;
-        wnum.d--;
+        wnum--;
         /*
-         * ingore top values of the bignums just sub the two BN_ULONG arrays
+         * ignore top values of the bignums just sub the two BN_ULONG arrays
          * with bn_sub_words
          */
-        l0 = bn_sub_words(wnum.d, wnum.d, tmp->d, div_n + 1);
+        l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
         q -= l0;
         /*
          * Note: As we have considered only the leading two BN_ULONGs in
@@ -435,31 +434,19 @@ int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
          */
         for (l0 = 0 - l0, j = 0; j < div_n; j++)
             tmp->d[j] = sdiv->d[j] & l0;
-        l0 = bn_add_words(wnum.d, wnum.d, tmp->d, div_n);
-        /*
-         * we can't have an overflow here (assuming that q != 0, but
-         * if q == 0 then tmp is zero anyway)
-         */
-        (*wnump) += l0;
+        l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
+        (*wnumtop) += l0;
+        assert((*wnumtop) == 0);
 
         /* store part of the result */
-        resp--;
-        *resp = q;
-    }
-    bn_correct_top(snum);
-    if (rm != NULL) {
-        /*
-         * Keep a copy of the neg flag in num because if rm==num BN_rshift()
-         * will overwrite it.
-         */
-        int neg = num->neg;
-        BN_rshift(rm, snum, norm_shift);
-        if (!BN_is_zero(rm))
-            rm->neg = neg;
-        bn_check_top(rm);
+        *--resp = q;
     }
-    if (no_branch)
-        bn_correct_top(res);
+    /* snum holds remainder, it's as wide as divisor */
+    snum->neg = num->neg;
+    snum->top = div_n;
+    snum->flags |= BN_FLG_FIXED_TOP;
+    if (rm != NULL)
+        bn_rshift_fixed_top(rm, snum, norm_shift);
     BN_CTX_end(ctx);
     return 1;
  err:

crypto/bn/bn_shift.c

 /*
- * Copyright 1995-2016 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
  *
  * Licensed under the OpenSSL license (the "License").  You may not use
  * this file except in compliance with the License.  You can obtain a copy
@@ -7,6 +7,7 @@
  * https://www.openssl.org/source/license.html
  */
 
+#include <assert.h>
 #include "internal/cryptlib.h"
 #include "bn_lcl.h"
 
@@ -82,40 +83,70 @@ int BN_rshift1(BIGNUM *r, const BIGNUM *a)
 
 int BN_lshift(BIGNUM *r, const BIGNUM *a, int n)
 {
-    int i, nw, lb, rb;
-    BN_ULONG *t, *f;
-    BN_ULONG l;
-
-    bn_check_top(r);
-    bn_check_top(a);
+    int ret;
 
     if (n < 0) {
         BNerr(BN_F_BN_LSHIFT, BN_R_INVALID_SHIFT);
         return 0;
     }
 
+    ret = bn_lshift_fixed_top(r, a, n);
+
+    bn_correct_top(r);
+    bn_check_top(r);
+
+    return ret;
+}
+
+/*
+ * In respect to shift factor the execution time is invariant of
+ * |n % BN_BITS2|, but not |n / BN_BITS2|. Or in other words pre-condition
+ * for constant-time-ness is |n < BN_BITS2| or |n / BN_BITS2| being
+ * non-secret.
+ */
+int bn_lshift_fixed_top(BIGNUM *r, const BIGNUM *a, int n)
+{
+    int i, nw;
+    unsigned int lb, rb;
+    BN_ULONG *t, *f;
+    BN_ULONG l, m, rmask = 0;
+
+    assert(n >= 0);
+
+    bn_check_top(r);
+    bn_check_top(a);
+
     nw = n / BN_BITS2;
     if (bn_wexpand(r, a->top + nw + 1) == NULL)
         return 0;
-    r->neg = a->neg;
-    lb = n % BN_BITS2;
-    rb = BN_BITS2 - lb;
-    f = a->d;
-    t = r->d;
-    t[a->top + nw] = 0;
-    if (lb == 0)
-        for (i = a->top - 1; i >= 0; i--)
-            t[nw + i] = f[i];
-    else
-        for (i = a->top - 1; i >= 0; i--) {
-            l = f[i];
-            t[nw + i + 1] |= (l >> rb) & BN_MASK2;
-            t[nw + i] = (l << lb) & BN_MASK2;
+
+    if (a->top != 0) {
+        lb = (unsigned int)n % BN_BITS2;
+        rb = BN_BITS2 - lb;
+        rb %= BN_BITS2;            /* say no to undefined behaviour */
+        rmask = (BN_ULONG)0 - rb;  /* rmask = 0 - (rb != 0) */
+        rmask |= rmask >> 8;
+        f = &(a->d[0]);
+        t = &(r->d[nw]);
+        l = f[a->top - 1];
+        t[a->top] = (l >> rb) & rmask;
+        for (i = a->top - 1; i > 0; i--) {
+            m = l << lb;
+            l = f[i - 1];
+            t[i] = (m | ((l >> rb) & rmask)) & BN_MASK2;
         }
-    memset(t, 0, sizeof(*t) * nw);
+        t[0] = (l << lb) & BN_MASK2;
+    } else {
+        /* shouldn't happen, but formally required */
+        r->d[nw] = 0;
+    }
+    if (nw != 0)
+        memset(r->d, 0, sizeof(*t) * nw);
+
+    r->neg = a->neg;
     r->top = a->top + nw + 1;
-    bn_correct_top(r);
-    bn_check_top(r);
+    r->flags |= BN_FLG_FIXED_TOP;
+
     return 1;
 }
 
@@ -173,3 +204,54 @@ int BN_rshift(BIGNUM *r, const BIGNUM *a, int n)
     bn_check_top(r);
     return 1;
 }
+
+/*
+ * In respect to shift factor the execution time is invariant of
+ * |n % BN_BITS2|, but not |n / BN_BITS2|. Or in other words pre-condition
+ * for constant-time-ness for sufficiently[!] zero-padded inputs is
+ * |n < BN_BITS2| or |n / BN_BITS2| being non-secret.
+ */
+int bn_rshift_fixed_top(BIGNUM *r, const BIGNUM *a, int n)
+{
+    int i, top, nw;
+    unsigned int lb, rb;
+    BN_ULONG *t, *f;
+    BN_ULONG l, m, mask;
+
+    bn_check_top(r);
+    bn_check_top(a);
+
+    assert(n >= 0);
+
+    nw = n / BN_BITS2;
+    if (nw >= a->top) {
+        /* shouldn't happen, but formally required */
+        BN_zero(r);
+        return 1;
+    }
+
+    rb = (unsigned int)n % BN_BITS2;
+    lb = BN_BITS2 - rb;
+    lb %= BN_BITS2;            /* say no to undefined behaviour */
+    mask = (BN_ULONG)0 - lb;   /* mask = 0 - (lb != 0) */
+    mask |= mask >> 8;
+    top = a->top - nw;
+    if (r != a && bn_wexpand(r, top) == NULL)
+        return 0;
+
+    t = &(r->d[0]);
+    f = &(a->d[nw]);
+    l = f[0];
+    for (i = 0; i < top - 1; i++) {
+        m = f[i + 1];
+        t[i] = (l >> rb) | ((m << lb) & mask);
+        l = m;
+    }
+    t[i] = l >> rb;
+
+    r->neg = a->neg;
+    r->top = top;
+    r->flags |= BN_FLG_FIXED_TOP;
+
+    return 1;
+}

crypto/include/internal/bn_int.h

@@ -65,7 +65,10 @@ int bn_set_words(BIGNUM *a, const BN_ULONG *words, int num_words);
  * is customarily arranged by bn_correct_top. Output from below functions
  * is not processed with bn_correct_top, and for this reason it may not be
  * returned out of public API. It may only be passed internally into other
- * functions known to support non-minimal or zero-padded BIGNUMs.
+ * functions known to support non-minimal or zero-padded BIGNUMs. Even
+ * though the goal is to facilitate constant-time-ness, not each subroutine
+ * is constant-time by itself. They all have pre-conditions, consult source
+ * code...
  */
 int bn_mul_mont_fixed_top(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
                           BN_MONT_CTX *mont, BN_CTX *ctx);
@@ -79,5 +82,9 @@ int bn_mod_sub_fixed_top(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
                          const BIGNUM *m);
 int bn_mul_fixed_top(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx);
 int bn_sqr_fixed_top(BIGNUM *r, const BIGNUM *a, BN_CTX *ctx);
+int bn_lshift_fixed_top(BIGNUM *r, const BIGNUM *a, int n);
+int bn_rshift_fixed_top(BIGNUM *r, const BIGNUM *a, int n);
+int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m,
+                     const BIGNUM *d, BN_CTX *ctx);
 
 #endif