fuchsia / third_party / rust-crates / 66b1ec50fe5a290df0f1bc099b2c2db48282805f / . / rustc_deps / vendor / ring / crypto / fipsmodule / ec / ecp_nistz.h

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

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#ifndef OPENSSL_HEADER_EC_ECP_NISTZ_H | |

#define OPENSSL_HEADER_EC_ECP_NISTZ_H | |

#include <GFp/base.h> | |

#include "../../limbs/limbs.h" | |

// This function looks at `w + 1` scalar bits (`w` 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**w setting. That is, we always combine `w` | |

// signed bits into a signed digit, e.g. (for `w == 5`): | |

// | |

// 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 `-2**(w-1)` to `2**(w-1)`, e.g. -16 to 16 for `w == 5`. | |

// | |

// 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 `w` bits as an integer, writing the recoded digit | |

// to |*is_negative| (a mask for `constant_time_select_s`) and |*digit| | |

// (absolute value, in the range 0 .. 2**(w-1). 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. | |

static inline void booth_recode(Limb *is_negative, unsigned *digit, | |

unsigned in, unsigned w) { | |

assert(w >= 2); | |

assert(w <= 7); | |

// Set all bits of `s` to MSB(in), similar to |constant_time_msb_s|, | |

// but 'in' seen as (`w+1`)-bit value. | |

Limb s = ~((in >> w) - 1); | |

unsigned d; | |

d = (1 << (w + 1)) - in - 1; | |

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

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

*is_negative = constant_time_is_nonzero_w(s & 1); | |

*digit = d; | |

} | |

void gfp_little_endian_bytes_from_scalar(uint8_t str[], size_t str_len, | |

const Limb scalar[], | |

size_t num_limbs); | |

#endif // OPENSSL_HEADER_EC_ECP_NISTZ_H |