lib/libcrypto/ec/ecp_nistputil.c

   1 /* $OpenBSD: ecp_nistputil.c,v 1.6 2014/07/10 22:45:57 jsing Exp $ */
   2 /*
   3  * Written by Bodo Moeller for the OpenSSL project.
   4  */
   5 /*
   6  * Copyright (c) 2011 Google Inc.
   7  *
   8  * Permission to use, copy, modify, and distribute this software for any
   9  * purpose with or without fee is hereby granted, provided that the above
  10  * copyright notice and this permission notice appear in all copies.
  11  *
  12  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  13  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  14  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  15  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  16  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  17  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  18  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  19  */
  20
  21 #include <stddef.h>
  22
  23 #include <openssl/opensslconf.h>
  24
  25 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
  26
  27 /*
  28  * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
  29  */
  30
  31 #include "ec_lcl.h"
  32
  33 /* Convert an array of points into affine coordinates.
  34  * (If the point at infinity is found (Z = 0), it remains unchanged.)
  35  * This function is essentially an equivalent to EC_POINTs_make_affine(), but
  36  * works with the internal representation of points as used by ecp_nistp###.c
  37  * rather than with (BIGNUM-based) EC_POINT data structures.
  38  *
  39  * point_array is the input/output buffer ('num' points in projective form,
  40  * i.e. three coordinates each), based on an internal representation of
  41  * field elements of size 'felem_size'.
  42  *
  43  * tmp_felems needs to point to a temporary array of 'num'+1 field elements
  44  * for storage of intermediate values.
  45  */
  46 void
  47 ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
  48     size_t felem_size, void *tmp_felems,
  49     void (*felem_one) (void *out),
  50     int (*felem_is_zero) (const void *in),
  51     void (*felem_assign) (void *out, const void *in),
  52     void (*felem_square) (void *out, const void *in),
  53     void (*felem_mul) (void *out, const void *in1, const void *in2),
  54     void (*felem_inv) (void *out, const void *in),
  55     void (*felem_contract) (void *out, const void *in))
  56 {
  57         int i = 0;
  58
  59 #define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
  60 #define X(I) (&((char *)point_array)[3*(I) * felem_size])
  61 #define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
  62 #define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
  63
  64         if (!felem_is_zero(Z(0)))
  65                 felem_assign(tmp_felem(0), Z(0));
  66         else
  67                 felem_one(tmp_felem(0));
  68         for (i = 1; i < (int) num; i++) {
  69                 if (!felem_is_zero(Z(i)))
  70                         felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
  71                 else
  72                         felem_assign(tmp_felem(i), tmp_felem(i - 1));
  73         }
  74         /*
  75          * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
  76          * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i)
  77          * = 1
  78          */
  79
  80         felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
  81         for (i = num - 1; i >= 0; i--) {
  82                 if (i > 0)
  83                         /*
  84                          * tmp_felem(i-1) is the product of Z(0) .. Z(i-1),
  85                          * tmp_felem(i) is the inverse of the product of Z(0)
  86                          * .. Z(i)
  87                          */
  88                         felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));      /* 1/Z(i) */
  89                 else
  90                         felem_assign(tmp_felem(num), tmp_felem(0));     /* 1/Z(0) */
  91
  92                 if (!felem_is_zero(Z(i))) {
  93                         if (i > 0)
  94                                 /*
  95                                  * For next iteration, replace tmp_felem(i-1)
  96                                  * by its inverse
  97                                  */
  98                                 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
  99
 100                         /*
 101                          * Convert point (X, Y, Z) into affine form (X/(Z^2),
 102                          * Y/(Z^3), 1)
 103                          */
 104                         felem_square(Z(i), tmp_felem(num));     /* 1/(Z^2) */
 105                         felem_mul(X(i), X(i), Z(i));    /* X/(Z^2) */
 106                         felem_mul(Z(i), Z(i), tmp_felem(num));  /* 1/(Z^3) */
 107                         felem_mul(Y(i), Y(i), Z(i));    /* Y/(Z^3) */
 108                         felem_contract(X(i), X(i));
 109                         felem_contract(Y(i), Y(i));
 110                         felem_one(Z(i));
 111                 } else {
 112                         if (i > 0)
 113                                 /*
 114                                  * For next iteration, replace tmp_felem(i-1)
 115                                  * by its inverse
 116                                  */
 117                                 felem_assign(tmp_felem(i - 1), tmp_felem(i));
 118                 }
 119         }
 120 }
 121
 122 /*
 123  * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
 124  * significant bit), and recodes them into a signed digit for use in fast point
 125  * multiplication: the use of signed rather than unsigned digits means that
 126  * fewer points need to be precomputed, given that point inversion is easy
 127  * (a precomputed point dP makes -dP available as well).
 128  *
 129  * BACKGROUND:
 130  *
 131  * Signed digits for multiplication were introduced by Booth ("A signed binary
 132  * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
 133  * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
 134  * Booth's original encoding did not generally improve the density of nonzero
 135  * digits over the binary representation, and was merely meant to simplify the
 136  * handling of signed factors given in two's complement; but it has since been
 137  * shown to be the basis of various signed-digit representations that do have
 138  * further advantages, including the wNAF, using the following general approach:
 139  *
 140  * (1) Given a binary representation
 141  *
 142  *       b_k  ...  b_2  b_1  b_0,
 143  *
 144  *     of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
 145  *     by using bit-wise subtraction as follows:
 146  *
 147  *        b_k b_(k-1)  ...  b_2  b_1  b_0
 148  *      -     b_k      ...  b_3  b_2  b_1  b_0
 149  *       -------------------------------------
 150  *        s_k b_(k-1)  ...  s_3  s_2  s_1  s_0
 151  *
 152  *     A left-shift followed by subtraction of the original value yields a new
 153  *     representation of the same value, using signed bits s_i = b_(i+1) - b_i.
 154  *     This representation from Booth's paper has since appeared in the
 155  *     literature under a variety of different names including "reversed binary
 156  *     form", "alternating greedy expansion", "mutual opposite form", and
 157  *     "sign-alternating {+-1}-representation".
 158  *
 159  *     An interesting property is that among the nonzero bits, values 1 and -1
 160  *     strictly alternate.
 161  *
 162  * (2) Various window schemes can be applied to the Booth representation of
 163  *     integers: for example, right-to-left sliding windows yield the wNAF
 164  *     (a signed-digit encoding independently discovered by various researchers
 165  *     in the 1990s), and left-to-right sliding windows yield a left-to-right
 166  *     equivalent of the wNAF (independently discovered by various researchers
 167  *     around 2004).
 168  *
 169  * To prevent leaking information through side channels in point multiplication,
 170  * we need to recode the given integer into a regular pattern: sliding windows
 171  * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
 172  * decades older: we'll be using the so-called "modified Booth encoding" due to
 173  * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
 174  * (1961), pp. 67-91), in a radix-2^5 setting.  That is, we always combine five
 175  * signed bits into a signed digit:
 176  *
 177  *       s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
 178  *
 179  * The sign-alternating property implies that the resulting digit values are
 180  * integers from -16 to 16.
 181  *
 182  * Of course, we don't actually need to compute the signed digits s_i as an
 183  * intermediate step (that's just a nice way to see how this scheme relates
 184  * to the wNAF): a direct computation obtains the recoded digit from the
 185  * six bits b_(4j + 4) ... b_(4j - 1).
 186  *
 187  * This function takes those five bits as an integer (0 .. 63), writing the
 188  * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
 189  * value, in the range 0 .. 8).  Note that this integer essentially provides the
 190  * input bits "shifted to the left" by one position: for example, the input to
 191  * compute the least significant recoded digit, given that there's no bit b_-1,
 192  * has to be b_4 b_3 b_2 b_1 b_0 0.
 193  *
 194  */
 195 void
 196 ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, unsigned char *digit, unsigned char in)
 197 {
 198         unsigned char s, d;
 199
 200         s = ~((in >> 5) - 1);   /* sets all bits to MSB(in), 'in' seen as
 201                                  * 6-bit value */
 202         d = (1 << 6) - in - 1;
 203         d = (d & s) | (in & ~s);
 204         d = (d >> 1) + (d & 1);
 205
 206         *sign = s & 1;
 207         *digit = d;
 208 }
 209 #endif