1 /* crypto/ec/ecp_nistputil.c */
3 * Written by Bodo Moeller for the OpenSSL project.
5 /* Copyright 2011 Google Inc.
7 * Licensed under the Apache License, Version 2.0 (the "License");
9 * you may not use this file except in compliance with the License.
10 * You may obtain a copy of the License at
12 * http://www.apache.org/licenses/LICENSE-2.0
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.
21 #ifdef EC_NISTP_64_GCC_128
24 * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
30 /* Convert an array of points into affine coordinates.
31 * (If the point at infinity is found (Z = 0), it remains unchanged.)
32 * This function is essentially an equivalent to EC_POINTs_make_affine(), but
33 * works with the internal representation of points as used by ecp_nistp###.c
34 * rather than with (BIGNUM-based) EC_POINT data structures.
36 * point_array is the input/output buffer ('num' points in projective form,
37 * i.e. three coordinates each), based on an internal representation of
38 * field elements of size 'felem_size'.
40 * tmp_felems needs to point to a temporary array of 'num'+1 field elements
41 * for storage of intermediate values.
43 void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
44 size_t felem_size, void *tmp_felems,
45 void (*felem_one)(void *out),
46 int (*felem_is_zero)(const void *in),
47 void (*felem_assign)(void *out, const void *in),
48 void (*felem_square)(void *out, const void *in),
49 void (*felem_mul)(void *out, const void *in1, const void *in2),
50 void (*felem_inv)(void *out, const void *in),
51 void (*felem_contract)(void *out, const void *in))
55 #define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
56 #define X(I) (&((char *)point_array)[3*(I) * felem_size])
57 #define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
58 #define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
60 if (!felem_is_zero(Z(0)))
61 felem_assign(tmp_felem(0), Z(0));
63 felem_one(tmp_felem(0));
64 for (i = 1; i < (int)num; i++)
66 if (!felem_is_zero(Z(i)))
67 felem_mul(tmp_felem(i), tmp_felem(i-1), Z(i));
69 felem_assign(tmp_felem(i), tmp_felem(i-1));
71 /* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any zero-valued factors:
72 * if Z(i) = 0, we essentially pretend that Z(i) = 1 */
74 felem_inv(tmp_felem(num-1), tmp_felem(num-1));
75 for (i = num - 1; i >= 0; i--)
78 /* tmp_felem(i-1) is the product of Z(0) .. Z(i-1),
79 * tmp_felem(i) is the inverse of the product of Z(0) .. Z(i)
81 felem_mul(tmp_felem(num), tmp_felem(i-1), tmp_felem(i)); /* 1/Z(i) */
83 felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
85 if (!felem_is_zero(Z(i)))
88 /* For next iteration, replace tmp_felem(i-1) by its inverse */
89 felem_mul(tmp_felem(i-1), tmp_felem(i), Z(i));
91 /* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1) */
92 felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
93 felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
94 felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
95 felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
96 felem_contract(X(i), X(i));
97 felem_contract(Y(i), Y(i));
103 /* For next iteration, replace tmp_felem(i-1) by its inverse */
104 felem_assign(tmp_felem(i-1), tmp_felem(i));
110 * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
111 * significant bit), and recodes them into a signed digit for use in fast point
112 * multiplication: the use of signed rather than unsigned digits means that
113 * fewer points need to be precomputed, given that point inversion is easy
114 * (a precomputed point dP makes -dP available as well).
118 * Signed digits for multiplication were introduced by Booth ("A signed binary
119 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
120 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
121 * Booth's original encoding did not generally improve the density of nonzero
122 * digits over the binary representation, and was merely meant to simplify the
123 * handling of signed factors given in two's complement; but it has since been
124 * shown to be the basis of various signed-digit representations that do have
125 * further advantages, including the wNAF, using the following general approach:
127 * (1) Given a binary representation
129 * b_k ... b_2 b_1 b_0,
131 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
132 * by using bit-wise subtraction as follows:
134 * b_k b_(k-1) ... b_2 b_1 b_0
135 * - b_k ... b_3 b_2 b_1 b_0
136 * -------------------------------------
137 * s_k b_(k-1) ... s_3 s_2 s_1 s_0
139 * A left-shift followed by subtraction of the original value yields a new
140 * representation of the same value, using signed bits s_i = b_(i+1) - b_i.
141 * This representation from Booth's paper has since appeared in the
142 * literature under a variety of different names including "reversed binary
143 * form", "alternating greedy expansion", "mutual opposite form", and
144 * "sign-alternating {+-1}-representation".
146 * An interesting property is that among the nonzero bits, values 1 and -1
147 * strictly alternate.
149 * (2) Various window schemes can be applied to the Booth representation of
150 * integers: for example, right-to-left sliding windows yield the wNAF
151 * (a signed-digit encoding independently discovered by various researchers
152 * in the 1990s), and left-to-right sliding windows yield a left-to-right
153 * equivalent of the wNAF (independently discovered by various researchers
156 * To prevent leaking information through side channels in point multiplication,
157 * we need to recode the given integer into a regular pattern: sliding windows
158 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
159 * decades older: we'll be using the so-called "modified Booth encoding" due to
160 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
161 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
162 * signed bits into a signed digit:
164 * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
166 * The sign-alternating property implies that the resulting digit values are
167 * integers from -16 to 16.
169 * Of course, we don't actually need to compute the signed digits s_i as an
170 * intermediate step (that's just a nice way to see how this scheme relates
171 * to the wNAF): a direct computation obtains the recoded digit from the
172 * six bits b_(4j + 4) ... b_(4j - 1).
174 * This function takes those five bits as an integer (0 .. 63), writing the
175 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
176 * value, in the range 0 .. 8). Note that this integer essentially provides the
177 * input bits "shifted to the left" by one position: for example, the input to
178 * compute the least significant recoded digit, given that there's no bit b_-1,
179 * has to be b_4 b_3 b_2 b_1 b_0 0.
182 void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, unsigned char *digit, unsigned char in)
186 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 6-bit value */
187 d = (1 << 6) - in - 1;
188 d = (d & s) | (in & ~s);
189 d = (d >> 1) + (d & 1);
195 static void *dummy=&dummy;