crypto/openssl/crypto/ec/ecp_nistputil.c

   1 /* crypto/ec/ecp_nistputil.c */
   2 /*
   3  * Written by Bodo Moeller for the OpenSSL project.
   4  */
   5 /* Copyright 2011 Google Inc.
   6  *
   7  * Licensed under the Apache License, Version 2.0 (the "License");
   8  *
   9  * you may not use this file except in compliance with the License.
  10  * You may obtain a copy of the License at
  11  *
  12  *     http://www.apache.org/licenses/LICENSE-2.0
  13  *
  14  *  Unless required by applicable law or agreed to in writing, software
  15  *  distributed under the License is distributed on an "AS IS" BASIS,
  16  *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  17  *  See the License for the specific language governing permissions and
  18  *  limitations under the License.
  19  */
  20
  21 #include <openssl/opensslconf.h>
  22 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
  23
  24 /*
  25  * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
  26  */
  27
  28 # include <stddef.h>
  29 # include "ec_lcl.h"
  30
  31 /*
  32  * Convert an array of points into affine coordinates. (If the point at
  33  * infinity is found (Z = 0), it remains unchanged.) This function is
  34  * essentially an equivalent to EC_POINTs_make_affine(), but works with the
  35  * internal representation of points as used by ecp_nistp###.c rather than
  36  * with (BIGNUM-based) EC_POINT data structures. point_array is the
  37  * input/output buffer ('num' points in projective form, i.e. three
  38  * coordinates each), based on an internal representation of field elements
  39  * of size 'felem_size'. tmp_felems needs to point to a temporary array of
  40  * 'num'+1 field elements for storage of intermediate values.
  41  */
  42 void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
  43                                               size_t felem_size,
  44                                               void *tmp_felems,
  45                                               void (*felem_one) (void *out),
  46                                               int (*felem_is_zero) (const void
  47                                                                     *in),
  48                                               void (*felem_assign) (void *out,
  49                                                                     const void
  50                                                                     *in),
  51                                               void (*felem_square) (void *out,
  52                                                                     const void
  53                                                                     *in),
  54                                               void (*felem_mul) (void *out,
  55                                                                  const void
  56                                                                  *in1,
  57                                                                  const void
  58                                                                  *in2),
  59                                               void (*felem_inv) (void *out,
  60                                                                  const void
  61                                                                  *in),
  62                                               void (*felem_contract) (void
  63                                                                       *out,
  64                                                                       const
  65                                                                       void
  66                                                                       *in))
  67 {
  68     int i = 0;
  69
  70 # define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
  71 # define X(I) (&((char *)point_array)[3*(I) * felem_size])
  72 # define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
  73 # define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
  74
  75     if (!felem_is_zero(Z(0)))
  76         felem_assign(tmp_felem(0), Z(0));
  77     else
  78         felem_one(tmp_felem(0));
  79     for (i = 1; i < (int)num; i++) {
  80         if (!felem_is_zero(Z(i)))
  81             felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
  82         else
  83             felem_assign(tmp_felem(i), tmp_felem(i - 1));
  84     }
  85     /*
  86      * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
  87      * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
  88      */
  89
  90     felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
  91     for (i = num - 1; i >= 0; i--) {
  92         if (i > 0)
  93             /*
  94              * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
  95              * is the inverse of the product of Z(0) .. Z(i)
  96              */
  97             /* 1/Z(i) */
  98             felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
  99         else
 100             felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
 101
 102         if (!felem_is_zero(Z(i))) {
 103             if (i > 0)
 104                 /*
 105                  * For next iteration, replace tmp_felem(i-1) by its inverse
 106                  */
 107                 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
 108
 109             /*
 110              * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
 111              */
 112             felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
 113             felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
 114             felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
 115             felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
 116             felem_contract(X(i), X(i));
 117             felem_contract(Y(i), Y(i));
 118             felem_one(Z(i));
 119         } else {
 120             if (i > 0)
 121                 /*
 122                  * For next iteration, replace tmp_felem(i-1) by its inverse
 123                  */
 124                 felem_assign(tmp_felem(i - 1), tmp_felem(i));
 125         }
 126     }
 127 }
 128
 129 /*-
 130  * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
 131  * significant bit), and recodes them into a signed digit for use in fast point
 132  * multiplication: the use of signed rather than unsigned digits means that
 133  * fewer points need to be precomputed, given that point inversion is easy
 134  * (a precomputed point dP makes -dP available as well).
 135  *
 136  * BACKGROUND:
 137  *
 138  * Signed digits for multiplication were introduced by Booth ("A signed binary
 139  * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
 140  * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
 141  * Booth's original encoding did not generally improve the density of nonzero
 142  * digits over the binary representation, and was merely meant to simplify the
 143  * handling of signed factors given in two's complement; but it has since been
 144  * shown to be the basis of various signed-digit representations that do have
 145  * further advantages, including the wNAF, using the following general approach:
 146  *
 147  * (1) Given a binary representation
 148  *
 149  *       b_k  ...  b_2  b_1  b_0,
 150  *
 151  *     of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
 152  *     by using bit-wise subtraction as follows:
 153  *
 154  *        b_k b_(k-1)  ...  b_2  b_1  b_0
 155  *      -     b_k      ...  b_3  b_2  b_1  b_0
 156  *       -------------------------------------
 157  *        s_k b_(k-1)  ...  s_3  s_2  s_1  s_0
 158  *
 159  *     A left-shift followed by subtraction of the original value yields a new
 160  *     representation of the same value, using signed bits s_i = b_(i+1) - b_i.
 161  *     This representation from Booth's paper has since appeared in the
 162  *     literature under a variety of different names including "reversed binary
 163  *     form", "alternating greedy expansion", "mutual opposite form", and
 164  *     "sign-alternating {+-1}-representation".
 165  *
 166  *     An interesting property is that among the nonzero bits, values 1 and -1
 167  *     strictly alternate.
 168  *
 169  * (2) Various window schemes can be applied to the Booth representation of
 170  *     integers: for example, right-to-left sliding windows yield the wNAF
 171  *     (a signed-digit encoding independently discovered by various researchers
 172  *     in the 1990s), and left-to-right sliding windows yield a left-to-right
 173  *     equivalent of the wNAF (independently discovered by various researchers
 174  *     around 2004).
 175  *
 176  * To prevent leaking information through side channels in point multiplication,
 177  * we need to recode the given integer into a regular pattern: sliding windows
 178  * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
 179  * decades older: we'll be using the so-called "modified Booth encoding" due to
 180  * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
 181  * (1961), pp. 67-91), in a radix-2^5 setting.  That is, we always combine five
 182  * signed bits into a signed digit:
 183  *
 184  *       s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
 185  *
 186  * The sign-alternating property implies that the resulting digit values are
 187  * integers from -16 to 16.
 188  *
 189  * Of course, we don't actually need to compute the signed digits s_i as an
 190  * intermediate step (that's just a nice way to see how this scheme relates
 191  * to the wNAF): a direct computation obtains the recoded digit from the
 192  * six bits b_(4j + 4) ... b_(4j - 1).
 193  *
 194  * This function takes those five bits as an integer (0 .. 63), writing the
 195  * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
 196  * value, in the range 0 .. 8).  Note that this integer essentially provides the
 197  * input bits "shifted to the left" by one position: for example, the input to
 198  * compute the least significant recoded digit, given that there's no bit b_-1,
 199  * has to be b_4 b_3 b_2 b_1 b_0 0.
 200  *
 201  */
 202 void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
 203                                      unsigned char *digit, unsigned char in)
 204 {
 205     unsigned char s, d;
 206
 207     s = ~((in >> 5) - 1);       /* sets all bits to MSB(in), 'in' seen as
 208                                  * 6-bit value */
 209     d = (1 << 6) - in - 1;
 210     d = (d & s) | (in & ~s);
 211     d = (d >> 1) + (d & 1);
 212
 213     *sign = s & 1;
 214     *digit = d;
 215 }
 216 #else
 217 static void *dummy = &dummy;
 218 #endif