| /* Copyright lowRISC contributors. */ |
| /* Licensed under the Apache License, Version 2.0, see LICENSE for details. */ |
| /* SPDX-License-Identifier: Apache-2.0 */ |
| /* |
| * Constant time 384 bit modular multiplication based on Barrett reduction |
| * algorithm. |
| * |
| * Provides a modular multiplication kernel for 384 bit wide numbers |
| * for the modulus and barret constant of NIST curve P384 only. |
| */ |
| |
| .section .text |
| |
| /** |
| * 384-bit modular multiplication based on Barrett reduction algorithm |
| * optimized for the special modulus of the NIST P-384 curve. |
| * |
| * Returns c = a x b % p. |
| * |
| * Expects: two operands, modulus p and pre-calculated parameter u for barrett |
| * reduction (usually greek mu in literature). u is expected without the |
| * leading 1 at bit 384. u has to be pre-calculated as u = floor(2^768/p). |
| * |
| * This implementation mostly follows the description in the |
| * "Handbook of Applied Cryptography" in Algorithm 14.42. |
| * Differences: |
| * - This implementation incorporates a multiplication before the reduction. |
| * Therefore it expects two operands (a, b) instead of a wider integer x. |
| * - The computation of q2 ignores the MSbs of q1 and u to allow using |
| * a 384x384 bit multiplication. This is compensated later by |
| * individual (conditional) additions. |
| * - The truncations in step 2 of HAC 14.42 in the form of (... mod b^(k+1) ) |
| * are not implemented here and the full register width is used. This |
| * allows to omit computation of r1 (since r1=x) and step 3 of HAC 14.42 |
| * |
| * Flags: Flags when leaving this subroutine depend on a potentially discarded |
| * value and therefore are not usable after return. |
| * |
| * @param[in] [w11, w10]: a, first operand, max. length 384 bit, a < m. |
| * @param[in] [w17, w16]: b, second operand, max. length 384 bit, b < m. |
| * @param[in] [w13, w12]: p, modulus of P384 i.e.: |
| m = 2^384 - 2^128 - 2^96 + 2^32 - 1. |
| * @param[in] [w15, w14]: u, pre-computed Barrett constant (without u[384]/MSb |
| * of u which is always 1 for the allowed range but |
| * has to be set to 0 here). |
| * @param[in] w31: all-zero. |
| * @param[out] [w17, w16]: c, result, max. length 384 bit. |
| * |
| * Clobbered registers: w16, w17, w18, w19, w20, w21, w22, w23, w24 |
| * Clobbered flag groups: FG0 |
| */ |
| .globl barrett384_p384 |
| barrett384_p384: |
| /* Compute the integer product of the operands x = a * b |
| x = [w18, w22, w21] = a * b = [w11, w10] * [w17, w16] |
| => max. length x: 768 bit */ |
| bn.mulqacc.z w10.0, w16.0, 0 |
| bn.mulqacc w10.0, w16.1, 64 |
| bn.mulqacc.so w21.L, w10.1, w16.0, 64 |
| bn.mulqacc w10.0, w16.2, 0 |
| bn.mulqacc w10.1, w16.1, 0 |
| bn.mulqacc w10.2, w16.0, 0 |
| bn.mulqacc w10.0, w16.3, 64 |
| bn.mulqacc w10.1, w16.2, 64 |
| bn.mulqacc w10.2, w16.1, 64 |
| bn.mulqacc.so w21.U, w10.3, w16.0, 64 |
| bn.mulqacc w10.0, w17.0, 0 |
| bn.mulqacc w10.1, w16.3, 0 |
| bn.mulqacc w10.2, w16.2, 0 |
| bn.mulqacc w10.3, w16.1, 0 |
| bn.mulqacc w11.0, w16.0, 0 |
| bn.mulqacc w10.0, w17.1, 64 |
| bn.mulqacc w10.1, w17.0, 64 |
| bn.mulqacc w10.2, w16.3, 64 |
| bn.mulqacc w10.3, w16.2, 64 |
| bn.mulqacc w11.0, w16.1, 64 |
| bn.mulqacc.so w22.L, w11.1, w16.0, 64 |
| bn.mulqacc w10.1, w17.1, 0 |
| bn.mulqacc w10.2, w17.0, 0 |
| bn.mulqacc w10.3, w16.3, 0 |
| bn.mulqacc w11.0, w16.2, 0 |
| bn.mulqacc w11.1, w16.1, 0 |
| bn.mulqacc w10.2, w17.1, 64 |
| bn.mulqacc w10.3, w17.0, 64 |
| bn.mulqacc w11.0, w16.3, 64 |
| bn.mulqacc.so w22.U, w11.1, w16.2, 64 |
| bn.mulqacc w10.3, w17.1, 0 |
| bn.mulqacc w11.0, w17.0, 0 |
| bn.mulqacc w11.1, w16.3, 0 |
| bn.mulqacc w11.0, w17.1, 64 |
| bn.mulqacc.so w18.L, w11.1, w17.0, 64 |
| bn.mulqacc.so w18.U, w11.1, w17.1, 0 |
| |
| /* Store correction factor to compensate for later neglected MSb of x. |
| x is 768 bit wide and therefore the 383 bit right shifted version q1 |
| (below) contains 385 bit. Bit 384 of q1 is neglected to allow using a |
| 384x384 multiplier. For the MSb of x being set we temporary store u |
| (or zero) here to be used in a later constant time correction of a |
| multiplication with u. Note that this requires the MSb flag being carried |
| over from the multiplication routine. */ |
| bn.sel w23, w14, w31, M |
| bn.sel w24, w15, w31, M |
| |
| /* Compute q1 = x >> 383 |
| q1 = [w11, w10] = [w18, w22, w21] >> 383 = [w18, w21] >> 127 |
| => max length q1: 385 bits */ |
| bn.rshi w11, w31, w18 >> 127 |
| bn.rshi w10, w18, w22 >> 127 |
| |
| /* Compute q2 = q1*u |
| Instead of full q2 (which would be up to 770 bits) we ignore the MSb of u |
| and the MSb of q1 and correct this later. This allows using a 384x384 |
| multiplier. We use the property that u for the modulus of P384 is zero in |
| the bits 383 downto 129 and use a 384x192 multiplication routine. |
| => max. length q2': 513 bit |
| q2' = q1[383:0]*u[128:0] = [w18, w17, w16] = [w11, w10] * [w15, w14] */ |
| |
| /* 576 = 384*192 bit multiplication kernel */ |
| bn.mulqacc.z w10.0, w14.0, 0 |
| bn.mulqacc w10.0, w14.1, 64 |
| bn.mulqacc.so w16.L, w10.1, w14.0, 64 |
| bn.mulqacc w10.0, w14.2, 0 |
| bn.mulqacc w10.1, w14.1, 0 |
| bn.mulqacc w10.2, w14.0, 0 |
| bn.mulqacc w10.1, w14.2, 64 |
| bn.mulqacc w10.2, w14.1, 64 |
| bn.mulqacc.so w16.U, w10.3, w14.0, 64 |
| bn.mulqacc w10.2, w14.2, 0 |
| bn.mulqacc w10.3, w14.1, 0 |
| bn.mulqacc w11.0, w14.0, 0 |
| bn.mulqacc w10.3, w14.2, 64 |
| bn.mulqacc w11.0, w14.1, 64 |
| bn.mulqacc.so w17.L, w11.1, w14.0, 64 |
| bn.mulqacc w11.0, w14.2, 0 |
| bn.mulqacc w11.1, w14.1, 0 |
| bn.mulqacc.so w17.U, w11.1, w14.2, 64 |
| |
| /* w14.3 is always zero here due to structure of Barrett constant */ |
| bn.mulqacc.wo w18, w11.1, w14.3, 64 |
| |
| /* q3 = q2 >> 385 |
| In this step, the compensation for the neglected MSbs of q1 and u is |
| carried out underway. To add them in the q2 domain, they would have to be |
| left shifted by 384 bit first. To directly add them we first shift q2' by |
| 384 bit to the right, perform the additions, and shift the result another |
| bit to the right. The additions cannot overflow due to leading zeros |
| after shift. |
| q2'' = q2' >> 384 = [w20, w19] = [w18, w17, w16] >> 384 |
| = [w18, w17] >> 128 */ |
| bn.rshi w20, w31, w18 >> 128 |
| bn.rshi w19, w18, w17 >> 128 |
| /* Add q1. This is unconditional since MSb of u is always 1. |
| This cannot overflow due to leading zeros. |
| q2''' = q2'' + q1 = [w20, w19] = [w20, w19] + [w10, w11] */ |
| bn.add w19, w19, w10 |
| bn.addc w20, w20, w11 |
| /* Conditionally add u (without leading 1) in case of MSb of x being set. |
| This is the "real" q2 but shifted by 384 bits to the right. This cannot |
| overflow due to leading zeros |
| q2'''' = x[767]?q2'''+u[383:0]:q2''' |
| = [w20, w19] + [w24, w23] = q2 >> 384 */ |
| bn.add w19, w19, w23 |
| bn.addc w20, w20, w24 |
| /* finally this gives q3 by shifting the remaining bit to the right |
| q3 = q2 >> 385 = q2'''' >> 1 = [w11, w10] = [w20, w19] >> 1 */ |
| bn.rshi w11, w31, w20 >> 1 |
| bn.rshi w10, w20, w19 >> 1 |
| |
| /* r2 = q3*m[511:0] = [w17, w16] = ([w11, w10] * [w13, w12])[511:0] |
| A 384x384 bit multiplication kernel is used here, hence both q3 or p |
| must not be wider than 384 bit. This is always the case for p. For q3 it |
| is the case if a<p and b<p. |
| The 256 highest bits of the multiplication result are not needed, |
| so we do not compute them. */ |
| bn.mulqacc.z w10.0, w12.0, 0 |
| bn.mulqacc w10.0, w12.1, 64 |
| bn.mulqacc.so w16.L, w10.1, w12.0, 64 |
| bn.mulqacc w10.0, w12.2, 0 |
| bn.mulqacc w10.1, w12.1, 0 |
| bn.mulqacc w10.2, w12.0, 0 |
| bn.mulqacc w10.0, w12.3, 64 |
| bn.mulqacc w10.1, w12.2, 64 |
| bn.mulqacc w10.2, w12.1, 64 |
| bn.mulqacc.so w16.U, w10.3, w12.0, 64 |
| bn.mulqacc w10.0, w13.0, 0 |
| bn.mulqacc w10.1, w12.3, 0 |
| bn.mulqacc w10.2, w12.2, 0 |
| bn.mulqacc w10.3, w12.1, 0 |
| bn.mulqacc w11.0, w12.0, 0 |
| bn.mulqacc w10.0, w13.1, 64 |
| bn.mulqacc w10.1, w13.0, 64 |
| bn.mulqacc w10.2, w12.3, 64 |
| bn.mulqacc w10.3, w12.2, 64 |
| bn.mulqacc w11.0, w12.1, 64 |
| bn.mulqacc.so w17.L, w11.1, w12.0, 64 |
| bn.mulqacc w10.1, w13.1, 0 |
| bn.mulqacc w10.2, w13.0, 0 |
| bn.mulqacc w10.3, w12.3, 0 |
| bn.mulqacc w11.0, w12.2, 0 |
| bn.mulqacc w11.1, w12.1, 0 |
| bn.mulqacc w10.2, w13.1, 64 |
| bn.mulqacc w10.3, w13.0, 64 |
| bn.mulqacc w11.0, w12.3, 64 |
| bn.mulqacc.so w17.U, w11.1, w12.2, 64 |
| |
| /* Compute r = x-r2 = x-q3*p |
| since 0 <= r < 3*p, we only need to consider the lower limbs of x and r2 |
| r[511:0] = [w22, w21] - [w17, w16] */ |
| bn.sub w21, w21, w16 |
| bn.subb w22, w22, w17 |
| |
| /* Barrett algorithm requires subtraction of the modulus at most two times if |
| result is too large. However in the special case of P-384 we need to |
| subtract only once */ |
| bn.sub w16, w21, w12 |
| bn.subb w17, w22, w13 |
| bn.sel w16, w21, w16, C |
| bn.sel w17, w22, w17, C |
| |
| /* return result: c =[w17, w16] = a * b % p. */ |
| ret |
