1 /*
2  * Copyright 2011-2019 The OpenSSL Project Authors. All Rights Reserved.
3  *
4  * Licensed under the OpenSSL license (the "License").  You may not use
5  * this file except in compliance with the License.  You can obtain a copy
6  * in the file LICENSE in the source distribution or at
7  * https://www.openssl.org/source/license.html
8  */
9 
10 /* Copyright 2011 Google Inc.
11  *
12  * Licensed under the Apache License, Version 2.0 (the "License");
13  *
14  * you may not use this file except in compliance with the License.
15  * You may obtain a copy of the License at
16  *
17  *     http://www.apache.org/licenses/LICENSE-2.0
18  *
19  *  Unless required by applicable law or agreed to in writing, software
20  *  distributed under the License is distributed on an "AS IS" BASIS,
21  *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22  *  See the License for the specific language governing permissions and
23  *  limitations under the License.
24  */
25 
26 #include <openssl/opensslconf.h>
27 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
28 NON_EMPTY_TRANSLATION_UNIT
29 #else
30 
31 /*
32  * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
33  */
34 
35 # include <stddef.h>
36 # include "ec_local.h"
37 
38 /*
39  * Convert an array of points into affine coordinates. (If the point at
40  * infinity is found (Z = 0), it remains unchanged.) This function is
41  * essentially an equivalent to EC_POINTs_make_affine(), but works with the
42  * internal representation of points as used by ecp_nistp###.c rather than
43  * with (BIGNUM-based) EC_POINT data structures. point_array is the
44  * input/output buffer ('num' points in projective form, i.e. three
45  * coordinates each), based on an internal representation of field elements
46  * of size 'felem_size'. tmp_felems needs to point to a temporary array of
47  * 'num'+1 field elements for storage of intermediate values.
48  */
49 void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
50                                               size_t felem_size,
51                                               void *tmp_felems,
52                                               void (*felem_one) (void *out),
53                                               int (*felem_is_zero) (const void
54                                                                     *in),
55                                               void (*felem_assign) (void *out,
56                                                                     const void
57                                                                     *in),
58                                               void (*felem_square) (void *out,
59                                                                     const void
60                                                                     *in),
61                                               void (*felem_mul) (void *out,
62                                                                  const void
63                                                                  *in1,
64                                                                  const void
65                                                                  *in2),
66                                               void (*felem_inv) (void *out,
67                                                                  const void
68                                                                  *in),
69                                               void (*felem_contract) (void
70                                                                       *out,
71                                                                       const
72                                                                       void
73                                                                       *in))
74 {
75     int i = 0;
76 
77 # define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
78 # define X(I) (&((char *)point_array)[3*(I) * felem_size])
79 # define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
80 # define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
81 
82     if (!felem_is_zero(Z(0)))
83         felem_assign(tmp_felem(0), Z(0));
84     else
85         felem_one(tmp_felem(0));
86     for (i = 1; i < (int)num; i++) {
87         if (!felem_is_zero(Z(i)))
88             felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
89         else
90             felem_assign(tmp_felem(i), tmp_felem(i - 1));
91     }
92     /*
93      * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
94      * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
95      */
96 
97     felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
98     for (i = num - 1; i >= 0; i--) {
99         if (i > 0)
100             /*
101              * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
102              * is the inverse of the product of Z(0) .. Z(i)
103              */
104             /* 1/Z(i) */
105             felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
106         else
107             felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
108 
109         if (!felem_is_zero(Z(i))) {
110             if (i > 0)
111                 /*
112                  * For next iteration, replace tmp_felem(i-1) by its inverse
113                  */
114                 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
115 
116             /*
117              * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
118              */
119             felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
120             felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
121             felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
122             felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
123             felem_contract(X(i), X(i));
124             felem_contract(Y(i), Y(i));
125             felem_one(Z(i));
126         } else {
127             if (i > 0)
128                 /*
129                  * For next iteration, replace tmp_felem(i-1) by its inverse
130                  */
131                 felem_assign(tmp_felem(i - 1), tmp_felem(i));
132         }
133     }
134 }
135 
136 /*-
137  * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
138  * significant bit), and recodes them into a signed digit for use in fast point
139  * multiplication: the use of signed rather than unsigned digits means that
140  * fewer points need to be precomputed, given that point inversion is easy
141  * (a precomputed point dP makes -dP available as well).
142  *
143  * BACKGROUND:
144  *
145  * Signed digits for multiplication were introduced by Booth ("A signed binary
146  * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
147  * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
148  * Booth's original encoding did not generally improve the density of nonzero
149  * digits over the binary representation, and was merely meant to simplify the
150  * handling of signed factors given in two's complement; but it has since been
151  * shown to be the basis of various signed-digit representations that do have
152  * further advantages, including the wNAF, using the following general approach:
153  *
154  * (1) Given a binary representation
155  *
156  *       b_k  ...  b_2  b_1  b_0,
157  *
158  *     of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
159  *     by using bit-wise subtraction as follows:
160  *
161  *        b_k     b_(k-1)  ...  b_2  b_1  b_0
162  *      -         b_k      ...  b_3  b_2  b_1  b_0
163  *       -----------------------------------------
164  *        s_(k+1) s_k      ...  s_3  s_2  s_1  s_0
165  *
166  *     A left-shift followed by subtraction of the original value yields a new
167  *     representation of the same value, using signed bits s_i = b_(i-1) - b_i.
168  *     This representation from Booth's paper has since appeared in the
169  *     literature under a variety of different names including "reversed binary
170  *     form", "alternating greedy expansion", "mutual opposite form", and
171  *     "sign-alternating {+-1}-representation".
172  *
173  *     An interesting property is that among the nonzero bits, values 1 and -1
174  *     strictly alternate.
175  *
176  * (2) Various window schemes can be applied to the Booth representation of
177  *     integers: for example, right-to-left sliding windows yield the wNAF
178  *     (a signed-digit encoding independently discovered by various researchers
179  *     in the 1990s), and left-to-right sliding windows yield a left-to-right
180  *     equivalent of the wNAF (independently discovered by various researchers
181  *     around 2004).
182  *
183  * To prevent leaking information through side channels in point multiplication,
184  * we need to recode the given integer into a regular pattern: sliding windows
185  * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
186  * decades older: we'll be using the so-called "modified Booth encoding" due to
187  * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
188  * (1961), pp. 67-91), in a radix-2^5 setting.  That is, we always combine five
189  * signed bits into a signed digit:
190  *
191  *       s_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j)
192  *
193  * The sign-alternating property implies that the resulting digit values are
194  * integers from -16 to 16.
195  *
196  * Of course, we don't actually need to compute the signed digits s_i as an
197  * intermediate step (that's just a nice way to see how this scheme relates
198  * to the wNAF): a direct computation obtains the recoded digit from the
199  * six bits b_(5j + 4) ... b_(5j - 1).
200  *
201  * This function takes those six bits as an integer (0 .. 63), writing the
202  * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
203  * value, in the range 0 .. 16).  Note that this integer essentially provides
204  * the input bits "shifted to the left" by one position: for example, the input
205  * to compute the least significant recoded digit, given that there's no bit
206  * b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
207  *
208  */
209 void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
210                                      unsigned char *digit, unsigned char in)
211 {
212     unsigned char s, d;
213 
214     s = ~((in >> 5) - 1);       /* sets all bits to MSB(in), 'in' seen as
215                                  * 6-bit value */
216     d = (1 << 6) - in - 1;
217     d = (d & s) | (in & ~s);
218     d = (d >> 1) + (d & 1);
219 
220     *sign = s & 1;
221     *digit = d;
222 }
223 #endif
224