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 /* Convert an array of points into affine coordinates.
  32  * (If the point at infinity is found (Z = 0), it remains unchanged.)
  33  * This function is essentially an equivalent to EC_POINTs_make_affine(), but
  34  * works with the internal representation of points as used by ecp_nistp###.c
  35  * rather than with (BIGNUM-based) EC_POINT data structures.
  36  *
  37  * point_array is the input/output buffer ('num' points in projective form,
  38  * i.e. three coordinates each), based on an internal representation of
  39  * field elements of size 'felem_size'.
  40  *
  41  * tmp_felems needs to point to a temporary array of 'num'+1 field elements
  42  * for storage of intermediate values.
  43  */
  44 void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
  45         size_t felem_size, void *tmp_felems,
  46         void (*felem_one)(void *out),
  47         int (*felem_is_zero)(const void *in),
  48         void (*felem_assign)(void *out, const void *in),
  49         void (*felem_square)(void *out, const void *in),
  50         void (*felem_mul)(void *out, const void *in1, const void *in2),
  51         void (*felem_inv)(void *out, const void *in),
  52         void (*felem_contract)(void *out, const void *in))
  53         {
  54         int i = 0;
  55
  56 #define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
  57 #define X(I) (&((char *)point_array)[3*(I) * felem_size])
  58 #define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
  59 #define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
  60
  61         if (!felem_is_zero(Z(0)))
  62                 felem_assign(tmp_felem(0), Z(0));
  63         else
  64                 felem_one(tmp_felem(0));
  65         for (i = 1; i < (int)num; i++)
  66                 {
  67                 if (!felem_is_zero(Z(i)))
  68                         felem_mul(tmp_felem(i), tmp_felem(i-1), Z(i));
  69                 else
  70                         felem_assign(tmp_felem(i), tmp_felem(i-1));
  71                 }
  72         /* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any zero-valued factors:
  73          * if Z(i) = 0, we essentially pretend that Z(i) = 1 */
  74
  75         felem_inv(tmp_felem(num-1), tmp_felem(num-1));
  76         for (i = num - 1; i >= 0; i--)
  77                 {
  78                 if (i > 0)
  79                         /* tmp_felem(i-1) is the product of Z(0) .. Z(i-1),
  80                          * tmp_felem(i) is the inverse of the product of Z(0) .. Z(i)
  81                          */
  82                         felem_mul(tmp_felem(num), tmp_felem(i-1), tmp_felem(i)); /* 1/Z(i) */
  83                 else
  84                         felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
  85
  86                 if (!felem_is_zero(Z(i)))
  87                         {
  88                         if (i > 0)
  89                                 /* For next iteration, replace tmp_felem(i-1) by its inverse */
  90                                 felem_mul(tmp_felem(i-1), tmp_felem(i), Z(i));
  91
  92                         /* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1) */
  93                         felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
  94                         felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
  95                         felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
  96                         felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
  97                         felem_contract(X(i), X(i));
  98                         felem_contract(Y(i), Y(i));
  99                         felem_one(Z(i));
 100                         }
 101                 else
 102                         {
 103                         if (i > 0)
 104                                 /* For next iteration, replace tmp_felem(i-1) by its inverse */
 105                                 felem_assign(tmp_felem(i-1), tmp_felem(i));
 106                         }
 107                 }
 108         }
 109
 110 /*
 111  * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
 112  * significant bit), and recodes them into a signed digit for use in fast point
 113  * multiplication: the use of signed rather than unsigned digits means that
 114  * fewer points need to be precomputed, given that point inversion is easy
 115  * (a precomputed point dP makes -dP available as well).
 116  *
 117  * BACKGROUND:
 118  *
 119  * Signed digits for multiplication were introduced by Booth ("A signed binary
 120  * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
 121  * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
 122  * Booth's original encoding did not generally improve the density of nonzero
 123  * digits over the binary representation, and was merely meant to simplify the
 124  * handling of signed factors given in two's complement; but it has since been
 125  * shown to be the basis of various signed-digit representations that do have
 126  * further advantages, including the wNAF, using the following general approach:
 127  *
 128  * (1) Given a binary representation
 129  *
 130  *       b_k  ...  b_2  b_1  b_0,
 131  *
 132  *     of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
 133  *     by using bit-wise subtraction as follows:
 134  *
 135  *        b_k b_(k-1)  ...  b_2  b_1  b_0
 136  *      -     b_k      ...  b_3  b_2  b_1  b_0
 137  *       -------------------------------------
 138  *        s_k b_(k-1)  ...  s_3  s_2  s_1  s_0
 139  *
 140  *     A left-shift followed by subtraction of the original value yields a new
 141  *     representation of the same value, using signed bits s_i = b_(i+1) - b_i.
 142  *     This representation from Booth's paper has since appeared in the
 143  *     literature under a variety of different names including "reversed binary
 144  *     form", "alternating greedy expansion", "mutual opposite form", and
 145  *     "sign-alternating {+-1}-representation".
 146  *
 147  *     An interesting property is that among the nonzero bits, values 1 and -1
 148  *     strictly alternate.
 149  *
 150  * (2) Various window schemes can be applied to the Booth representation of
 151  *     integers: for example, right-to-left sliding windows yield the wNAF
 152  *     (a signed-digit encoding independently discovered by various researchers
 153  *     in the 1990s), and left-to-right sliding windows yield a left-to-right
 154  *     equivalent of the wNAF (independently discovered by various researchers
 155  *     around 2004).
 156  *
 157  * To prevent leaking information through side channels in point multiplication,
 158  * we need to recode the given integer into a regular pattern: sliding windows
 159  * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
 160  * decades older: we'll be using the so-called "modified Booth encoding" due to
 161  * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
 162  * (1961), pp. 67-91), in a radix-2^5 setting.  That is, we always combine five
 163  * signed bits into a signed digit:
 164  *
 165  *       s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
 166  *
 167  * The sign-alternating property implies that the resulting digit values are
 168  * integers from -16 to 16.
 169  *
 170  * Of course, we don't actually need to compute the signed digits s_i as an
 171  * intermediate step (that's just a nice way to see how this scheme relates
 172  * to the wNAF): a direct computation obtains the recoded digit from the
 173  * six bits b_(4j + 4) ... b_(4j - 1).
 174  *
 175  * This function takes those five bits as an integer (0 .. 63), writing the
 176  * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
 177  * value, in the range 0 .. 8).  Note that this integer essentially provides the
 178  * input bits "shifted to the left" by one position: for example, the input to
 179  * compute the least significant recoded digit, given that there's no bit b_-1,
 180  * has to be b_4 b_3 b_2 b_1 b_0 0.
 181  *
 182  */
 183 void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, unsigned char *digit, unsigned char in)
 184         {
 185         unsigned char s, d;
 186
 187         s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 6-bit value */
 188         d = (1 << 6) - in - 1;
 189         d = (d & s) | (in & ~s);
 190         d = (d >> 1) + (d & 1);
 191
 192         *sign = s & 1;
 193         *digit = d;
 194         }
 195 #else
 196 static void *dummy=&dummy;
 197 #endif