| /* 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 |
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| * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ |
| |
| #include <openssl/base.h> |
| |
| #include <openssl/ec.h> |
| |
| #include "internal.h" |
| |
| |
| // 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+1) s_k ... 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_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j) |
| // |
| // 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_(5j + 4) ... b_(5j - 1). |
| // |
| // This function takes those six 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 .. 16). 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. |
| // |
| // DOUBLING CASE: |
| // |
| // Point addition formulas for short Weierstrass curves are often incomplete. |
| // Edge cases such as P + P or P + ∞ must be handled separately. This |
| // complicates constant-time requirements. P + ∞ cannot be avoided (any window |
| // may be zero) and is handled with constant-time selects. P + P (where P is not |
| // ∞) usually is not. Instead, windowing strategies are chosen to avoid this |
| // case. Whether this happens depends on the group order. |
| // |
| // Let w be the window width (in this function, w = 5). The non-trivial doubling |
| // case in single-point scalar multiplication may occur if and only if the |
| // 2^(w-1) bit of the group order is zero. |
| // |
| // Note the above only holds if the scalar is fully reduced and the group order |
| // is a prime that is much larger than 2^w. It also only holds when windows |
| // are applied from most significant to least significant, doubling between each |
| // window. It does not apply to more complex table strategies such as |
| // |EC_GFp_nistz256_method|. |
| // |
| // PROOF: |
| // |
| // Let n be the group order. Let l be the number of bits needed to represent n. |
| // Assume there exists some 0 <= k < n such that signed w-bit windowed |
| // multiplication hits the doubling case. |
| // |
| // Windowed multiplication consists of iterating over groups of s_i (defined |
| // above based on k's binary representation) from most to least significant. At |
| // iteration i (for i = ..., 3w, 2w, w, 0, starting from the most significant |
| // window), we: |
| // |
| // 1. Double the accumulator A, w times. Let A_i be the value of A at this |
| // point. |
| // |
| // 2. Set A to T_i + A_i, where T_i is a precomputed multiple of P |
| // corresponding to the window s_(i+w-1) ... s_i. |
| // |
| // Let j be the index such that A_j = T_j ≠ ∞. Looking at A_i and T_i as |
| // multiples of P, define a_i and t_i to be scalar coefficients of A_i and T_i. |
| // Thus a_j = t_j ≠ 0 (mod n). Note a_i and t_i may not be reduced mod n. t_i is |
| // the value of the w signed bits s_(i+w-1) ... s_i. a_i is computed as a_i = |
| // 2^w * (a_(i+w) + t_(i+w)). |
| // |
| // t_i is bounded by -2^(w-1) <= t_i <= 2^(w-1). Additionally, we may write it |
| // in terms of unsigned bits b_i. t_i consists of signed bits s_(i+w-1) ... s_i. |
| // This is computed as: |
| // |
| // b_(i+w-2) b_(i+w-3) ... b_i b_(i-1) |
| // - b_(i+w-1) b_(i+w-2) ... b_(i+1) b_i |
| // -------------------------------------------- |
| // t_i = s_(i+w-1) s_(i+w-2) ... s_(i+1) s_i |
| // |
| // Observe that b_(i+w-2) through b_i occur in both terms. Let x be the integer |
| // represented by that bit string, i.e. 2^(w-2)*b_(i+w-2) + ... + b_i. |
| // |
| // t_i = (2*x + b_(i-1)) - (2^(w-1)*b_(i+w-1) + x) |
| // = x - 2^(w-1)*b_(i+w-1) + b_(i-1) |
| // |
| // Or, using C notation for bit operations: |
| // |
| // t_i = (k>>i) & ((1<<(w-1)) - 1) - (k>>i) & (1<<(w-1)) + (k>>(i-1)) & 1 |
| // |
| // Note b_(i-1) is added in left-shifted by one (or doubled) from its place. |
| // This is compensated by t_(i-w)'s subtraction term. Thus, a_i may be computed |
| // by adding b_l b_(l-1) ... b_(i+1) b_i and an extra copy of b_(i-1). In C |
| // notation, this is: |
| // |
| // a_i = (k>>(i+w)) << w + ((k>>(i+w-1)) & 1) << w |
| // |
| // Observe that, while t_i may be positive or negative, a_i is bounded by |
| // 0 <= a_i < n + 2^w. Additionally, a_i can only be zero if b_(i+w-1) and up |
| // are all zero. (Note this implies a non-trivial P + (-P) is unreachable for |
| // all groups. That would imply the subsequent a_i is zero, which means all |
| // terms thus far were zero.) |
| // |
| // Returning to our doubling position, we have a_j = t_j (mod n). We now |
| // determine the value of a_j - t_j, which must be divisible by n. Our bounds on |
| // a_j and t_j imply a_j - t_j is 0 or n. If it is 0, a_j = t_j. However, 2^w |
| // divides a_j and -2^(w-1) <= t_j <= 2^(w-1), so this can only happen if |
| // a_j = t_j = 0, which is a trivial doubling. Therefore, a_j - t_j = n. |
| // |
| // Now we determine j. Suppose j > 0. w divides j, so j >= w. Then, |
| // |
| // n = a_j - t_j = (k>>(j+w)) << w + ((k>>(j+w-1)) & 1) << w - t_j |
| // <= k/2^j + 2^w - t_j |
| // < n/2^w + 2^w + 2^(w-1) |
| // |
| // n is much larger than 2^w, so this is impossible. Thus, j = 0: only the final |
| // addition may hit the doubling case. |
| // |
| // Finally, we consider bit patterns for n and k. Divide k into k_H + k_M + k_L |
| // such that k_H is the contribution from b_(l-1) .. b_w, k_M is the |
| // contribution from b_(w-1), and k_L is the contribution from b_(w-2) ... b_0. |
| // That is: |
| // |
| // - 2^w divides k_H |
| // - k_M is 0 or 2^(w-1) |
| // - 0 <= k_L < 2^(w-1) |
| // |
| // Divide n into n_H + n_M + n_L similarly. We thus have: |
| // |
| // t_0 = (k>>0) & ((1<<(w-1)) - 1) - (k>>0) & (1<<(w-1)) + (k>>(0-1)) & 1 |
| // = k & ((1<<(w-1)) - 1) - k & (1<<(w-1)) |
| // = k_L - k_M |
| // |
| // a_0 = (k>>(0+w)) << w + ((k>>(0+w-1)) & 1) << w |
| // = (k>>w) << w + ((k>>(w-1)) & 1) << w |
| // = k_H + 2*k_M |
| // |
| // n = a_0 - t_0 |
| // n_H + n_M + n_L = (k_H + 2*k_M) - (k_L - k_M) |
| // = k_H + 3*k_M - k_L |
| // |
| // k_H - k_L < k and k < n, so k_H - k_L ≠ n. Therefore k_M is not 0 and must be |
| // 2^(w-1). Now we consider k_H and n_H. We know k_H <= n_H. Suppose k_H = n_H. |
| // Then, |
| // |
| // n_M + n_L = 3*(2^(w-1)) - k_L |
| // > 3*(2^(w-1)) - 2^(w-1) |
| // = 2^w |
| // |
| // Contradiction (n_M + n_L is the bottom w bits of n). Thus k_H < n_H. Suppose |
| // k_H < n_H - 2*2^w. Then, |
| // |
| // n_H + n_M + n_L = k_H + 3*(2^(w-1)) - k_L |
| // < n_H - 2*2^w + 3*(2^(w-1)) - k_L |
| // n_M + n_L < -2^(w-1) - k_L |
| // |
| // Contradiction. Thus, k_H = n_H - 2^w. (Note 2^w divides n_H and k_H.) Thus, |
| // |
| // n_H + n_M + n_L = k_H + 3*(2^(w-1)) - k_L |
| // = n_H - 2^w + 3*(2^(w-1)) - k_L |
| // n_M + n_L = 2^(w-1) - k_L |
| // <= 2^(w-1) |
| // |
| // Equality would mean 2^(w-1) divides n, which is impossible if n is prime. |
| // Thus n_M + n_L < 2^(w-1), so n_M is zero, proving our condition. |
| // |
| // This proof constructs k, so, to show the converse, let k_H = n_H - 2^w, |
| // k_M = 2^(w-1), k_L = 2^(w-1) - n_L. This will result in a non-trivial point |
| // doubling in the final addition and is the only such scalar. |
| // |
| // COMMON CURVES: |
| // |
| // The group orders for common curves end in the following bit patterns: |
| // |
| // P-521: ...00001001; w = 4 is okay |
| // P-384: ...01110011; w = 2, 5, 6, 7 are okay |
| // P-256: ...01010001; w = 5, 7 are okay |
| // P-224: ...00111101; w = 3, 4, 5, 6 are okay |
| void ec_GFp_nistp_recode_scalar_bits(crypto_word_t *sign, crypto_word_t *digit, |
| crypto_word_t in) { |
| crypto_word_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; |
| } |
