/* Copyright (c) 2015, Google Inc. | |

* | |

* Permission to use, copy, modify, and/or distribute this software for any | |

* purpose with or without fee is hereby granted, provided that the above | |

* copyright notice and this permission notice appear in all copies. | |

* | |

* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES | |

* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF | |

* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY | |

* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES | |

* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION | |

* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN | |

* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ | |

#include <openssl/base.h> | |

#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) | |

#include <openssl/ec.h> | |

#include "internal.h" | |

/* Convert an array of points into affine coordinates. (If the point at | |

* infinity is found (Z = 0), it remains unchanged.) This function is | |

* essentially an equivalent to EC_POINTs_make_affine(), but works with the | |

* internal representation of points as used by ecp_nistp###.c rather than | |

* with (BIGNUM-based) EC_POINT data structures. point_array is the | |

* input/output buffer ('num' points in projective form, i.e. three | |

* coordinates each), based on an internal representation of field elements | |

* of size 'felem_size'. tmp_felems needs to point to a temporary array of | |

* 'num'+1 field elements for storage of intermediate values. */ | |

void ec_GFp_nistp_points_make_affine_internal( | |

size_t num, void *point_array, size_t felem_size, void *tmp_felems, | |

void (*felem_one)(void *out), int (*felem_is_zero)(const void *in), | |

void (*felem_assign)(void *out, const void *in), | |

void (*felem_square)(void *out, const void *in), | |

void (*felem_mul)(void *out, const void *in1, const void *in2), | |

void (*felem_inv)(void *out, const void *in), | |

void (*felem_contract)(void *out, const void *in)) { | |

int i = 0; | |

#define tmp_felem(I) (&((char *)tmp_felems)[(I)*felem_size]) | |

#define X(I) (&((char *)point_array)[3 * (I)*felem_size]) | |

#define Y(I) (&((char *)point_array)[(3 * (I) + 1) * felem_size]) | |

#define Z(I) (&((char *)point_array)[(3 * (I) + 2) * felem_size]) | |

if (!felem_is_zero(Z(0))) { | |

felem_assign(tmp_felem(0), Z(0)); | |

} else { | |

felem_one(tmp_felem(0)); | |

} | |

for (i = 1; i < (int)num; i++) { | |

if (!felem_is_zero(Z(i))) { | |

felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i)); | |

} else { | |

felem_assign(tmp_felem(i), tmp_felem(i - 1)); | |

} | |

} | |

/* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any | |

* zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1. */ | |

felem_inv(tmp_felem(num - 1), tmp_felem(num - 1)); | |

for (i = num - 1; i >= 0; i--) { | |

if (i > 0) { | |

/* tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i) | |

* is the inverse of the product of Z(0) .. Z(i). */ | |

/* 1/Z(i) */ | |

felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i)); | |

} else { | |

felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */ | |

} | |

if (!felem_is_zero(Z(i))) { | |

if (i > 0) { | |

/* For next iteration, replace tmp_felem(i-1) by its inverse. */ | |

felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i)); | |

} | |

/* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1). */ | |

felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */ | |

felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */ | |

felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */ | |

felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */ | |

felem_contract(X(i), X(i)); | |

felem_contract(Y(i), Y(i)); | |

felem_one(Z(i)); | |

} else { | |

if (i > 0) { | |

/* For next iteration, replace tmp_felem(i-1) by its inverse. */ | |

felem_assign(tmp_felem(i - 1), tmp_felem(i)); | |

} | |

} | |

} | |

} | |

/* This function looks at 5+1 scalar bits (5 current, 1 adjacent less | |

* significant bit), and recodes them into a signed digit for use in fast point | |

* multiplication: the use of signed rather than unsigned digits means that | |

* fewer points need to be precomputed, given that point inversion is easy (a | |

* precomputed point dP makes -dP available as well). | |

* | |

* BACKGROUND: | |

* | |

* Signed digits for multiplication were introduced by Booth ("A signed binary | |

* multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, | |

* pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. | |

* Booth's original encoding did not generally improve the density of nonzero | |

* digits over the binary representation, and was merely meant to simplify the | |

* handling of signed factors given in two's complement; but it has since been | |

* shown to be the basis of various signed-digit representations that do have | |

* further advantages, including the wNAF, using the following general | |

* approach: | |

* | |

* (1) Given a binary representation | |

* | |

* b_k ... b_2 b_1 b_0, | |

* | |

* of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 | |

* by using bit-wise subtraction as follows: | |

* | |

* b_k b_(k-1) ... b_2 b_1 b_0 | |

* - b_k ... b_3 b_2 b_1 b_0 | |

* ------------------------------------- | |

* s_k b_(k-1) ... s_3 s_2 s_1 s_0 | |

* | |

* A left-shift followed by subtraction of the original value yields a new | |

* representation of the same value, using signed bits s_i = b_(i+1) - b_i. | |

* This representation from Booth's paper has since appeared in the | |

* literature under a variety of different names including "reversed binary | |

* form", "alternating greedy expansion", "mutual opposite form", and | |

* "sign-alternating {+-1}-representation". | |

* | |

* An interesting property is that among the nonzero bits, values 1 and -1 | |

* strictly alternate. | |

* | |

* (2) Various window schemes can be applied to the Booth representation of | |

* integers: for example, right-to-left sliding windows yield the wNAF | |

* (a signed-digit encoding independently discovered by various researchers | |

* in the 1990s), and left-to-right sliding windows yield a left-to-right | |

* equivalent of the wNAF (independently discovered by various researchers | |

* around 2004). | |

* | |

* To prevent leaking information through side channels in point multiplication, | |

* we need to recode the given integer into a regular pattern: sliding windows | |

* as in wNAFs won't do, we need their fixed-window equivalent -- which is a few | |

* decades older: we'll be using the so-called "modified Booth encoding" due to | |

* MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 | |

* (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five | |

* signed bits into a signed digit: | |

* | |

* s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) | |

* | |

* The sign-alternating property implies that the resulting digit values are | |

* integers from -16 to 16. | |

* | |

* Of course, we don't actually need to compute the signed digits s_i as an | |

* intermediate step (that's just a nice way to see how this scheme relates | |

* to the wNAF): a direct computation obtains the recoded digit from the | |

* six bits b_(4j + 4) ... b_(4j - 1). | |

* | |

* This function takes those five bits as an integer (0 .. 63), writing the | |

* recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute | |

* value, in the range 0 .. 8). Note that this integer essentially provides the | |

* input bits "shifted to the left" by one position: for example, the input to | |

* compute the least significant recoded digit, given that there's no bit b_-1, | |

* has to be b_4 b_3 b_2 b_1 b_0 0. */ | |

void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, | |

uint8_t in) { | |

uint8_t s, d; | |

s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as | |

* 6-bit value */ | |

d = (1 << 6) - in - 1; | |

d = (d & s) | (in & ~s); | |

d = (d >> 1) + (d & 1); | |

*sign = s & 1; | |

*digit = d; | |

} | |

#endif /* 64_BIT && !WINDOWS */ |