| /* Copyright lowRISC Contributors. |
| * Copyright 2016 The Chromium OS Authors. All rights reserved. |
| * Use of this source code is governed by a BSD-style license that can be |
| * found in the LICENSE.dcrypto file. |
| * |
| * Derived from code in |
| * https://chromium.googlesource.com/chromiumos/platform/ec/+/refs/heads/cr50_stab/chip/g/dcrypto/dcrypto_bn.c |
| */ |
| |
| |
| .text |
| .globl modexp_65537 |
| .globl modexp |
| .globl modload |
| |
| /** |
| * Precomputation of a constant m0' for Montgomery modular arithmetic |
| * |
| * Word-wise Montgomery modular arithmetic requires a quantity m0' to be |
| * precomputed once per modulus M. m0' is the negative of the |
| * modular multiplicative inverse of the lowest limb m0 of the modulus M, in |
| * the field GF(2^w), where w is the number of bits per limb. w is set to 256 |
| * in this subroutine. |
| * |
| * Returns: m0' = -m0^(-1) mod 2^256 |
| * with m0 being the lowest limb of the modulus M |
| * |
| * This subroutine implements the Dusse-Kaliski method for computing the |
| * multiplicative modular inverse when the modulus is of the form 2^k. |
| * [Dus] DOI https://doi.org/10.1007/3-540-46877-3_21 section 3.2 |
| * (Algorithm "Modular Inverse" on p. 235) |
| * |
| * Flags: When leaving this subroutine, flags of FG0 depend on a |
| * the final subtraction and can be used if needed. |
| * FG0.M, FG0.L, FG0.Z depend directly on the value of the result m0'. |
| * FG0.C is always set. |
| * FG1 is not modified in this subroutine. |
| * |
| * @param[in] w28: m0, the lowest 256 bit limb of the modulus M |
| * @param[in] w31: all-zero. |
| * @param[out] w29: m0', negative of inverse of m0 in GF(2^256) |
| * |
| * clobbered registers: w0, w1, w29 |
| * clobbered flag groups: FG0 |
| */ |
| m0inv: |
| /* w0 keeps track of loop iterations in one-hot encoding, i.e. |
| w0 = 2^i in the loop body below and initialized here with w0 = 1 |
| It is used for both the comparison in step 4 of [Dus] and the |
| addition in step 6 of [Dus] */ |
| bn.xor w0, w0, w0 |
| bn.addi w0, w0, 1 |
| |
| /* according to [Dus] the result variable y is initialized with 1 */ |
| /* w29 = y_0 = 1 */ |
| bn.mov w29, w0 |
| |
| /* iterate over all 256 bits of m0. |
| i refers to the loop cycle 0..255 in the loop body below. */ |
| loopi 256, 13 |
| |
| /* y_i <= m*y_{i-1] */ |
| bn.mulqacc.z w28.0, w29.0, 0 |
| bn.mulqacc w28.1, w29.0, 64 |
| bn.mulqacc.so w1.L, w28.0, w29.1, 64 |
| bn.mulqacc w28.2, w29.0, 0 |
| bn.mulqacc w28.1, w29.1, 0 |
| bn.mulqacc w28.0, w29.2, 0 |
| bn.mulqacc w28.3, w29.0, 64 |
| bn.mulqacc w28.2, w29.1, 64 |
| bn.mulqacc w28.1, w29.2, 64 |
| bn.mulqacc.so w1.U, w28.0, w29.3, 64 |
| |
| /* This checks if w1 = y_i = m0*y_(i-1) < 2^(i-1) mod 2^i |
| Due to the mathematical properties it can be shown that y_i at this point, |
| is either 1 or (10..0..01)_(i). Therefore, just probing the i_th bit is |
| the same as the full compare. */ |
| bn.and w1, w1, w0 |
| |
| /* Compute |
| y_i=w29 <= w1=m0*y_(i-1) < 2^(i-1) mod 2^i y_i ? : y_{i-1}+2^i : y_{i-1} |
| there cannot be overlaps => or'ing is as good as adding */ |
| bn.or w29, w29, w1 |
| |
| /* double w0 (w0 <= w0 << 1) i.e. w0=2^i */ |
| bn.add w0, w0, w0 |
| |
| /* finally, compute m0' (negative of inverse) |
| w29 = m0' = -(m0^-1) mod 2^256 = -y_255 = 0 - y_255 = w31 - w29 */ |
| bn.sub w29, w31, w29 |
| |
| ret |
| |
| /** |
| * Constant time conditional subtraction of modulus from a bigint |
| * |
| * Returns C <= C-s*M |
| * with C being a bigint of length 256..4096 bit |
| * M being the modulus of length 256..4096 bit |
| * s being a boolean value [0,1] |
| * |
| * Conditionally subtracts the modulus located in dmem from the bigint |
| * located in a buffer in the wide regfile (starting at w5). The subtracted |
| * value is selected when FG1.C equals 1, otherwise the unmodified value is |
| * selected. |
| * |
| * Note that the interpretation of the subtrahend as a modulus is only |
| * contextual. In theory, it can be any bigint. However, the subtrahend is |
| * expected in dmem at a location that is reserved for the modulus according |
| * to the calling conventions within this library. |
| * |
| * Flags: When leaving this subroutine, flags of FG0 depend on a |
| * potentially discarded value and therefore are not usable after |
| * return. |
| * FG1 is not modified in this subroutine. |
| * |
| * @param[in] x16: dptr_m, pointer to 1st limb of modulus M |
| * @param[in] x30: N, number of 256 bit limbs in modulus and bigint |
| * @param[in] w31: all-zero |
| * @param[in] FG1.C: s, selection flag |
| * @param[out] [w[5+N-1]:w5]: new bigint value |
| * @param[in] FG0.C: needs to be set to 0 |
| * |
| * clobbered registers: x8, x10, x11, x16, w2, w3, w4, w5 to w[5+N-1] |
| * clobbered flag groups: FG0 |
| */ |
| cond_sub_mod: |
| |
| /* setup pointers */ |
| li x8, 5 |
| li x10, 3 |
| li x11, 2 |
| |
| /* reset flags for FG0 */ |
| bn.add w31, w31, w31 |
| |
| /* iterate over all limbs for limb-wise subtraction + conditional selection*/ |
| loop x30, 5 |
| |
| /* load a limb of modulus from dmem to w3 */ |
| bn.lid x10, 0(x16++) |
| |
| /* load the limb of bigint buffer to w2 */ |
| bn.movr x11, x8 |
| |
| /* subtract the current limb of the modulus from current limb of bigint */ |
| bn.subb w4, w2, w3 |
| |
| /* conditionally select subtraction result or unmodified limb */ |
| bn.sel w3, w4, w2, FG1.C |
| |
| /* move back result from w3 to bigint buffer */ |
| bn.movr x8++, x10 |
| |
| ret |
| |
| |
| /** |
| * Compute square of Montgomery modulus |
| * |
| * Returns RR = R^2 mod M |
| * with M being the Modulus of length 256..4096 bit |
| * R = 2^(256*N), N is the number of limbs per bigint |
| * |
| * The squared Montgomery modulus (RR) is needed to transform bigints to and |
| * from the Montgomery domain. |
| * |
| * RR is computed in this subroutine by iteratively doubling and reduction. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * @param[in] x16: dptr_M, pointer to first limb of modulus in dmem |
| * @param[in] x18: dptr_RR: dmem pointer to first limb of output buffer for RR |
| * @param[in] x30: N, number of limbs |
| * @param[in] w31: all-zero |
| * @param[out] dmem[x18+N*32:x18]: computed RR |
| * |
| * clobbered registers: x3, x8, x10, x11, x16, x18 |
| * w0, w2, w3, w4, w5 to w20 depending on N |
| * clobbered flag groups: FG0, FG1 |
| */ |
| compute_rr: |
| /* save pointer to modulus */ |
| addi x22, x16, 0 |
| |
| /* zeroize w3 */ |
| bn.xor w3, w3, w3 |
| |
| /* compute full length of current bigint size in bits |
| N*w = x24 = N*256 = N*2^8 = x30 << 8 */ |
| slli x24, x30, 8 |
| |
| /* reg pointers */ |
| li x8, 5 |
| li x10, 3 |
| |
| /* zeroize w3 */ |
| bn.xor w3, w3, w3 |
| |
| /* zeroize all limbs of bigint in regfile */ |
| loop x30, 1 |
| bn.movr x8++, x10 |
| |
| /* compute R-M |
| since R = 2^(N*w), this can be computed as R-M = unsigned(0-M) */ |
| bn.addi w0, w31, 1 |
| bn.sub w3, w31, w0, FG1 |
| addi x16, x22, 0 |
| jal x1, cond_sub_mod |
| |
| /* Compute R^2 mod M = R*2^(N*w) mod M. |
| => R^2 mod M can be computed by performing N*w duplications of R. |
| We directly perform a modulo reduction in each step such that the |
| final result will already be reduced. */ |
| loop x24, 18 |
| /* reset pointer */ |
| li x8, 5 |
| |
| /* zeroize w3 reset flags of FG1 */ |
| bn.sub w3, w3, w3, FG1 |
| |
| /* Duplicate the intermediate bigint result. This can overflow such that |
| bit 2^(N*w) (represented by the carry bit after the final loop cycle) |
| is set. */ |
| loop x30, 3 |
| /* copy current limb of bigint to w2 */ |
| bn.movr x11, x8 |
| |
| /* perform the doubling */ |
| bn.addc w2, w2, w2, FG1 |
| |
| /* copy result back to bigint in regfile */ |
| bn.movr x8++, x11 |
| |
| /* Conditionally subtract the modulus from the current bigint Y if there |
| was an overflow. Again, just considering the lowest N*w bits is |
| sufficient, since (in case of an overflow) we can write |
| 2*Y as 2^(N*w) + X with M > X >= 0. |
| Then, 2*Y - M = 2^(N*w) + X - M = X + unsigned(0-M) */ |
| addi x16, x22, 0 |
| jal x1, cond_sub_mod |
| |
| /* reset pointer to 1st limb of bigint in regfile */ |
| li x8, 5 |
| |
| /* reset pointer to modulus in dmem */ |
| addi x16, x22, 0 |
| |
| /* reset flags of FG1 */ |
| bn.sub w3, w3, w3, FG1 |
| |
| /* compare intermediate bigint y with modulus |
| subtract modulus if Y > M */ |
| loop x30, 3 |
| bn.lid x10, 0(x16++) |
| bn.movr x11, x8++ |
| bn.cmpb w3, w2, FG1 |
| addi x16, x22, 0 |
| jal x1, cond_sub_mod |
| |
| li x0, 0 |
| |
| /* reset pointer to 1st limb of bigint in regfile */ |
| li x8, 5 |
| |
| /* store computed RR in dmem */ |
| loop x30, 2 |
| bn.sid x8, 0(x18++) |
| addi x8, x8, 1 |
| |
| ret |
| |
| |
| /** |
| * Unrolled 512=256x256 bit multiplication. |
| * |
| * Returns c = a x b. |
| * |
| * Flags: No flags are set in this subroutine |
| * |
| * @param[in] w30: a, first operand |
| * @param[in] w25: b, second operand |
| * @param[out] [w26, w27]: c, result |
| * |
| * clobbered registers: w26, w27 |
| * clobbered flag groups: none |
| */ |
| mul256_w30xw25: |
| bn.mulqacc.z w30.0, w25.0, 0 |
| bn.mulqacc w30.1, w25.0, 64 |
| bn.mulqacc.so w27.L, w30.0, w25.1, 64 |
| bn.mulqacc w30.2, w25.0, 0 |
| bn.mulqacc w30.1, w25.1, 0 |
| bn.mulqacc w30.0, w25.2, 0 |
| bn.mulqacc w30.3, w25.0, 64 |
| bn.mulqacc w30.2, w25.1, 64 |
| bn.mulqacc w30.1, w25.2, 64 |
| bn.mulqacc.so w27.U, w30.0, w25.3, 64 |
| bn.mulqacc w30.3, w25.1, 0 |
| bn.mulqacc w30.2, w25.2, 0 |
| bn.mulqacc w30.1, w25.3, 0 |
| bn.mulqacc w30.3, w25.2, 64 |
| bn.mulqacc.so w26.L, w30.2, w25.3, 64 |
| bn.mulqacc.so w26.U, w30.3, w25.3, 0 |
| |
| ret |
| |
| |
| /** |
| * Unrolled 512=256x256 bit multiplication. |
| * |
| * Returns c = a x b. |
| * |
| * Flags: No flags are set in this subroutine |
| * |
| * @param[in] w30: a, first operand |
| * @param[in] w2: b, second operand |
| * @param[out] [w26, w27]: c, result |
| * |
| * clobbered registers: w26, w27 |
| * clobbered flag groups: none |
| */ |
| mul256_w30xw2: |
| bn.mulqacc.z w30.0, w2.0, 0 |
| bn.mulqacc w30.1, w2.0, 64 |
| bn.mulqacc.so w27.L, w30.0, w2.1, 64 |
| bn.mulqacc w30.2, w2.0, 0 |
| bn.mulqacc w30.1, w2.1, 0 |
| bn.mulqacc w30.0, w2.2, 0 |
| bn.mulqacc w30.3, w2.0, 64 |
| bn.mulqacc w30.2, w2.1, 64 |
| bn.mulqacc w30.1, w2.2, 64 |
| bn.mulqacc.so w27.U, w30.0, w2.3, 64 |
| bn.mulqacc w30.3, w2.1, 0 |
| bn.mulqacc w30.2, w2.2, 0 |
| bn.mulqacc w30.1, w2.3, 0 |
| bn.mulqacc w30.3, w2.2, 64 |
| bn.mulqacc.so w26.L, w30.2, w2.3, 64 |
| bn.mulqacc.so w26.U, w30.3, w2.3, 0 |
| |
| ret |
| |
| |
| /** |
| * Constant time conditional bigint subtraction |
| * |
| * Returns C <= C-s*B |
| * with B being a bigint of length 256..4096 bit |
| * C being a bigint of length 256..4096 bit |
| * s being a boolean value [0,1] |
| * |
| * Depending on state of FG1.C subtracts a bigint B located in dmem from |
| * another bigint C, located in the wide reg file and stores result at same |
| * position in wide reg file. |
| * |
| * Flags: When leaving this subroutine, flags of FG0 depend on a |
| * potentially discarded value and therefore are not usable after |
| * return. FG1 is not modified in this subroutine. |
| * |
| * @param[in] x16: dmem pointer to first limb of subtrahend (B) |
| * @param[in] x8: regfile pointer to first limb of minuend and result (C) |
| * @param[in] FG.C: s, subtraction flag, subtract if 1 |
| * @param[in] x30: number of limbs |
| * @param[in] FG0.C: needs to be set to 0 |
| * |
| * clobbered registers: x8, x16, w24, w29, w30, w[x8] to w[x8+N-1] |
| * clobbered Flag Groups: FG0 |
| */ |
| cond_sub_to_reg: |
| |
| /* load pointers to temp regs */ |
| li x12, 30 |
| li x13, 24 |
| |
| /* iterate over all limbs for conditional limb-wise subtraction */ |
| loop x30, 6 |
| /* load limb of subtrahend (input B) to w24 */ |
| bn.lid x13, 0(x16++) |
| |
| /* load limb of minuend (input C) to w30 */ |
| bn.movr x12, x8 |
| |
| /* perform subtraction for a limb */ |
| bn.subb w29, w30, w24 |
| |
| bn.movr x8, x13 |
| |
| /* conditionally select subtraction result or unmodified limb */ |
| bn.sel w24, w29, w30, FG1.C |
| |
| /* store selection result in reg file */ |
| bn.movr x8++, x13 |
| ret |
| |
| |
| /** |
| * Main loop body for constant-time Montgomery Modular Multiplication |
| * |
| * Returns: C <= (C + A*b_i + M*m0'*(C[0] + A[0]*b_i))/(2^WLEN) mod R |
| * |
| * This implements the main loop body for the Montgomery Modular Multiplication |
| * as well as the conditional subtraction. See e.g. Handbook of Applied |
| * Cryptography (HAC) 14.36 (steps 2.1 and 2.2) and step 3. |
| * This subroutine has to be called for every iteration of the loop in step 2 |
| * of HAC 14.36, i.e. once per limb of operand B (x in HAC notation). The limb |
| * is supplied via w2. In the explanations below, the index i refers to the |
| * i_th call to this subroutine within one full Montgomery Multiplication run. |
| * Step 3 of HAC 14.36 is replaced by the approach to perform the conditional |
| * subtraction when the intermediate result is larger than R instead of m. See |
| * e.g. https://eprint.iacr.org/2017/1057 section 2.4.2 for a justification. |
| * This does not omit the conditional subtraction, but simplifies the |
| * comparison. The subtraction is carried out in constant time manner. |
| * Variable names of HAC are mapped as follows to the ones used in the |
| * this library: x=B, y=A, A=C, b=2^WLEN, m=M, R=R, m' = m0', n=N. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * @param[in] x16: dmem pointer to first limb of modulus M |
| * @param[in] x19: dmem pointer to first limb operand A |
| * @param[in] x31: N-1, number of limbs minus one |
| * @param[in] w2: current limb of operand B, b_i |
| * @param[in] w3: Montgomery constant m0' |
| * @param[in] w31: all-zero |
| * @param[in] [w[4+N-1]:w4] intermediate result A |
| * @param[out] [w[4+N-1]:w4] intermediate result A |
| * |
| * clobbered registers: x8, x10, x12, x13, x16, x19 |
| * w24, w25, w26, w27, w28, w29, w30, w4 to w[4+N-1] |
| * clobbered Flag Groups: FG0, FG1 |
| */ |
| mont_loop: |
| /* save pointer to modulus */ |
| addi x22, x16, 0 |
| |
| /* pointers to temp. wregs */ |
| li x12, 30 |
| li x13, 24 |
| |
| /* buffer read pointer */ |
| li x8, 4 |
| |
| /* buffer write pointer */ |
| li x10, 4 |
| |
| /* load 1st limb of input y (operand a): w30 = y[0] */ |
| bn.lid x12, 0(x19++) |
| |
| /* This is x_i*y_0 in step 2.1 of HAC 14.36 */ |
| /* [w26, w27] = w30*w2 = y[0]*x_i */ |
| jal x1, mul256_w30xw2 |
| |
| /* w24 = w4 = A[0] */ |
| bn.movr x13, x8++ |
| |
| /* add A[0]: [w29, w30] = [w26, w27] + w24 = y[0]*x_i + A[0] */ |
| bn.add w30, w27, w24 |
| |
| /* this serves as c_xy in the first cycle of the loop below */ |
| bn.addc w29, w26, w31 |
| |
| /* w25 = w3 = m0' */ |
| bn.mov w25, w3 |
| |
| /* multiply by m0', this concludes Step 2.1 of HAC 14.36 */ |
| /* [_, u_i] = [w26, w27] = w30*w25 = (y[0]*x_i + A[0])*m0'*/ |
| jal x1, mul256_w30xw25 |
| |
| |
| /* With the computation of u_i, the computations in a cycle 0 of the loop |
| below are already partly done. The following instructions (until the |
| start of the loop) implement the remainder, such that cycle 0 can be |
| omitted in the loop */ |
| |
| /* [_, u_i] = [w28, w25] = [w26, w27] */ |
| bn.mov w25, w27 |
| bn.mov w28, w26 |
| |
| /* w24 = w30 = y[0]*x_i + A[0] mod b */ |
| bn.mov w24, w30 |
| |
| /* load first limb of modulus: w30 = m[0] */ |
| bn.lid x12, 0(x16++) |
| |
| /* [w26, w27] = w30*w25 = m[0]*u_i*/ |
| jal x1, mul256_w30xw25 |
| |
| /* [w28, w27] = [w26, w27] + w24 = m[0]*u_i + (y[0]*x_i + A[0] mod b) */ |
| bn.add w27, w27, w24 |
| /* this serves as c_m in the first cycle of the loop below */ |
| bn.addc w28, w26, w31 |
| |
| |
| /* This loop implements step 2.2 of HAC 14.36 with a word-by-word approach. |
| The loop body is subdivided into two steps. Each step performs one |
| multiplication and subsequently adds two WLEN sized words to the |
| 2WLEN-sized result, such that there are no overflows at the end of each |
| step- |
| Two carry words are required between the cycles. Those are c_xy and c_m. |
| Assume that the variable j runs from 1 to N-1 in the explanations below. |
| A cycle 0 is omitted, since the results from the computations above are |
| re-used */ |
| loop x31, 14 |
| /* Step 1: First multiplication takes a limb of each of the operands and |
| computes the product. The carry word from the previous cycle c_xy and |
| the j_th limb of the buffer A, A[j] are added to the multiplication |
| result. |
| |
| /* load limb of y (operand a) and mult. with x_i: [w26, w27] <= y[j]*x_i */ |
| bn.lid x12, 0(x19++) |
| jal x1, mul256_w30xw2 |
| /* add limb of buffer: [w26, w27] <= [w26,w27] + w24 = y[j]*x_i + A[j] */ |
| bn.movr x13, x8++ |
| bn.add w27, w27, w24 |
| bn.addc w26, w26, w31 |
| /* add carry word from previous cycle: |
| [c_xy, a_tmp] = [w29, w24] <= [w26,w27] + w29 = y[j]*x_i + A[j] + c_xy*/ |
| bn.add w24, w27, w29 |
| bn.addc w29, w26, w31 |
| |
| |
| /* Step 2: Second multiplication computes the product of a limb m[j] of |
| the modulus with u_i. The 2nd carry word from the previous loop cycle |
| c_m and the lower word a_tmp of the result of Step 1 are added. */ |
| |
| /* load limb m[j] of modulus and multiply with u_i: |
| [w26, w27] = w30*w25 = m[j+1]*u_i */ |
| bn.lid x12, 0(x16++) |
| jal x1, mul256_w30xw25 |
| /* add result from first step |
| [w26, w27] <= [w26,w27] + w24 = m[j+1]*u_i + a_tmp */ |
| bn.add w27, w27, w24 |
| bn.addc w26, w26, w31 |
| /* [c_m, A[j]] = [w28, w24] = m[j+1]*u_i + a_tmp + c_m */ |
| bn.add w24, w27, w28, FG1 |
| /* store at w[4+j] = A[j-1] |
| This includes the reduction by 2^WLEN = 2^b in step 2.2 of HAC 14.36 */ |
| bn.movr x10++, x13 |
| bn.addc w28, w26, w31, FG1 |
| |
| |
| /* Most significant limb of A is sum of the carry words of last loop cycle |
| A[N-1] = w24 <= w29 + w28 = c_xy + c_m */ |
| bn.addc w24, w29, w28, FG1 |
| bn.movr x10++, x13 |
| |
| /* restore pointers */ |
| addi x16, x22, 0 |
| li x8, 4 |
| li x10, 4 |
| |
| /* This replaces Step 3 of HAC 14.36 and performs conditional constant-time |
| subtraction of the modulus from the output buffer. */ |
| jal x1, cond_sub_to_reg |
| |
| /* restore pointer */ |
| li x8, 4 |
| |
| ret |
| |
| |
| /** |
| * Constant time conditional bigint subtraction |
| * |
| * Returns C = A-x*B |
| * with A being a bigint of length 256..4096 bit |
| * B being a bigint of length 256..4096 bit |
| * C being a bigint of length 256..4096 bit |
| * x being a boolean value [0,1] |
| * |
| * Depending on state of FG1.C subtracts a bigint B located in dmem from |
| * another bigint A, located in the wide reg file and stores result C in dmem. |
| * |
| * Flags: When leaving this subroutine, flags of FG0 depend on a |
| * potentially discarded value and therefore are not usable after |
| * return. FG1 is not modified in this subroutine. |
| * |
| * @param[in] x16: dmem pointer to first limb of subtrahend (B) |
| * @param[in] x8: regfile pointer to first limb of minuend (input A) |
| * @param[in] x21: dmem pointer to first limb of result (C) |
| * @param[in] x30: N, number of limbs |
| * @param[in] FG1.C: subtraction condition, subtract if 1 (x) |
| * @param[in] x9: pointer to temp reg, must be set to 3 |
| * @param[in] x11: pointer to temp reg, must be set to 2 |
| * @param[in] FG0.C: needs to be set to 0 |
| * |
| * clobbered registers: x8, x16, x21, w2, w3 |
| * clobbered Flag Groups: FG0 |
| */ |
| cond_sub_to_dmem: |
| /* iterate over all limbs for conditional limb-wise subtraction */ |
| loop x30, 5 |
| /* load limb of subtrahend (input B): w3 = dmem[x16+i] */ |
| bn.lid x9, 0(x16++) |
| |
| /* move limb from bignum bufer to w2 */ |
| bn.movr x11, x8++ |
| |
| /* perform subtraction for a limb w3 = w2-1 */ |
| bn.subb w3, w2, w3 |
| |
| /* conditionally select subtraction result or unmodified limb */ |
| bn.sel w2, w3, w2, FG1.C |
| |
| /* store selection result in dmem */ |
| bn.sid x11, 0(x21++) |
| |
| ret |
| |
| |
| /** |
| * Constant-time Montgomery modular multiply by one |
| * |
| * Returns: C = montmul(1,A) = A*R^(-1) mod M |
| * |
| * Routine for back-conversion from Montgomery domain. |
| * This implements the limb-by-limb interleaved Montgomery Modular |
| * Multiplication Algorithm, with one operand fixed to 1. This is only a |
| * wrapper around the main loop body. For algorithmic implementation details |
| * see the mont_loop subroutine. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * @param[in] x16: dmem pointer to first limb of modulus M |
| * @param[in] x17: dptr_m0d: dmem pointer to Montgomery Constant m0' |
| * @param[in] x19: dmem pointer to first limb of operand A |
| * @param[in] x21: dmem pointer to first limb of result C |
| * @param[in] x30: N, number of limbs |
| * @param[in] x31: N-1, number of limbs minus one |
| * @param[in] x8: pointer to temp reg, must be set to 4 |
| * @param[in] x9: pointer to temp reg, must be set to 3 |
| * @param[in] x10: pointer to temp reg, must be set to 4 |
| * @param[in] x11: pointer to temp reg, must be set to 2 |
| * @param[in] w31: all-zero |
| * |
| * clobbered registers: x6, x7, x8, x9, x10, x12, x13, x16, x19, x21 |
| * w2. w3. w24, w25, w26, w27, w28, w29, w30 |
| * w4 to w[4+N-1] |
| * clobbered Flag Groups: FG0, FG1 |
| */ |
| montmul_mul1: |
| /* load Montgomery constant: w3 = dmem[x17] = dmem[dptr_m0d] = m0' */ |
| bn.lid x9, 0(x17) |
| |
| /* init regfile bigint buffer with zeros */ |
| bn.mov w2, w31 |
| loop x30, 1 |
| bn.movr x10++, x11 |
| |
| /* w2=1 this is operand B */ |
| bn.xor w2, w2, w2 |
| bn.addi w2, w2, 1 |
| |
| /* save dmem pointers for operand A and modulus */ |
| addi x6, x16, 0 |
| addi x7, x19, 0 |
| |
| /* iterate over limbs of operand B */ |
| loop x30, 4 |
| |
| /* restore dmem pointers for operand A and modulus */ |
| addi x16, x6, 0 |
| addi x19, x7, 0 |
| |
| /* Main loop body of Montgomery Multiplication algorithm */ |
| /* 1[i]*A */ |
| jal x1, mont_loop |
| |
| /* all subsequent limbs of operand B are zero since B=1 */ |
| bn.mov w2, w31 |
| |
| /* restore dmem pointers for operand A and modulus */ |
| addi x16, x6, 0 |
| addi x19, x7, 0 |
| |
| /* zeroize w2 and clear flags */ |
| bn.sub w2, w2, w2, FG1 |
| |
| /* iterate over all limbs of bigint buffer for limbwise comparison of |
| buffer with the Modulus. After last loop cycle, FG1.C is set if bigint |
| in buffer is larger than Modulus */ |
| loop x30, 3 |
| |
| /* load limb of limb of Modulus to w3 */ |
| bn.lid x9, 0(x16++) |
| |
| /* load limb from bigint buffer to w2 */ |
| bn.movr x11, x8++ |
| |
| /* compare limb of flag with limb of Modulus */ |
| bn.cmpb w3, w2, FG1 |
| |
| /* restore pointers to bigint buffer in regfile */ |
| li x8, 4 |
| li x10, 4 |
| |
| /* restore dmem pointers for operand A and modulus */ |
| addi x16, x6, 0 |
| addi x19, x7, 0 |
| |
| /* conditionally subtract Modulus from buffer and store result in |
| dmem[x21] to dmem[x21+N] */ |
| jal x1, cond_sub_to_dmem |
| |
| /* restore dmem pointers for operand A and modulus */ |
| addi x16, x6, 0 |
| addi x19, x7, 0 |
| |
| ret |
| |
| |
| /** |
| * Constant-time Montgomery Modular Multiplication |
| * |
| * Returns: C = montmul(A,B) = A*B*R^(-1) mod M |
| * |
| * This implements the limb-by-limb interleaved Montgomery Modular |
| * Multiplication Algorithm. This is only a wrapper around the main loop body. |
| * For algorithmic implementation details see the mont_loop subroutine. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * @param[in] x16: dptr_M, dmem pointer to first limb of modulus M |
| * @param[in] x17: dptr_m0d, dmem pointer to Montgomery Constant m0' |
| * @param[in] x19: dptr_a, dmem pointer to first limb of operand A |
| * @param[in] x20: dptr_b, dmem pointer to first limb of operand B |
| * @param[in] w31: all-zero |
| * @param[in] x30: N, number of limbs |
| * @param[in] x31: N-1, number of limbs minus one |
| * @param[in] x9: pointer to temp reg, must be set to 3 |
| * @param[in] x10: pointer to temp reg, must be set to 4 |
| * @param[in] x11: pointer to temp reg, must be set to 2 |
| * @param[out] [w[4+N-1]:w4]: result C |
| * |
| * clobbered registers: x5, x6, x7, x8, x10, x12, x13, x16, x17, x19, x20, x21 |
| * w2, w3, w24 to w30, w4 to w[4+N-1] |
| * clobbered Flag Groups: FG0, FG1 |
| */ |
| montmul: |
| /* load Montgomery constant: w3 = dmem[x17] = dmem[dptr_m0d] = m0' */ |
| bn.lid x9, 0(x17) |
| |
| /* init regfile bigint buffer with zeros */ |
| bn.mov w2, w31 |
| loop x30, 1 |
| bn.movr x10++, x11 |
| |
| /* iterate over limbs of operand B */ |
| loop x30, 8 |
| |
| /* load limb of operand b */ |
| bn.lid x11, 0(x20++) |
| |
| /* save some regs */ |
| addi x5, x20, 0 |
| addi x6, x16, 0 |
| addi x7, x19, 0 |
| |
| /* Main loop body of Montgomery Multiplication algorithm */ |
| jal x1, mont_loop |
| |
| /* restore regs */ |
| addi x20, x5, 0 |
| addi x16, x6, 0 |
| addi x19, x7, 0 |
| |
| /* restore pointers */ |
| li x8, 4 |
| li x10, 4 |
| |
| ret |
| |
| |
| /** |
| * Conditionally overwrite bigint in dmem |
| * |
| * Depending on state of FG0.C overwrites a bigint in dmem with one from |
| * a buffer in the wide reg file. |
| * |
| * Flags: Does not set any flags, does not use flags except FG0.C |
| * |
| * @param[in] x21: dptr, pointer to first limb of bigint in dmem |
| * @param[in] x8: rptr, pointer to first limb of bigint in regfile buffer |
| * @param[in] FG.C: selection condition, overwrite dmem when FG0.C==1 |
| * @param[in] x30: number of limbs |
| * @param[in] x9: pointer to temp reg, must be set to 3 |
| * @param[in] x11: pointer to temp reg, must be set to 2 |
| * |
| * clobbered registers: x8, x21, w0, w2 |
| * clobbered Flag Groups: none |
| */ |
| sel_sqr_or_sqrmul: |
| /* iterate over all limbs */ |
| loop x30, 6 |
| |
| /* load limb from dmem */ |
| bn.lid x9, 0(x21) |
| |
| /* store limb to dmem */ |
| bn.sid x11, 0(x21) |
| |
| /* load limb from regfile buffer */ |
| bn.movr x11, x8++ |
| bn.mov w0, w2 |
| |
| /* conditional select: w2 = FG0.C?w[x8+i]:dmem[x21+i] */ |
| bn.sel w2, w0, w3, C |
| |
| /* store selected limb to dmem */ |
| bn.sid x11, 0(x21++) |
| |
| ret |
| |
| |
| /** |
| * Constant-time bigint modular exponentiation |
| * |
| * Returns: C = modexp(A,E) = A^E mod M |
| * |
| * This implements the square and multiply algorithm, i.e. for each bit of the |
| * exponent both the squared only and the squared with multiply results are |
| * computed but one result is discarded. |
| * Computation is carried out in the Montgomery domain, by using the montmul |
| * primitive. |
| * The squared Montgomery modulus RR and the Montgomery constant m0' have to |
| * be precomputed and provided at the appropriate locations in dmem. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * The base bignum A is expected in the input buffer, the exponent E in the |
| * exp buffer, the result C is written to the output buffer. |
| * Note, that the content of both, the input buffer and the exp buffer is |
| * modified during execution. |
| * |
| * @param[in] dmem[2] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[4] N: Number of limbs per bignum |
| * @param[in] dmem[8] dptr_m0d: pointer to m0' in dmem |
| * @param[in] dmem[12] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[16] dptr_m: pointer to first limb of modulus in dmem |
| * @param[in] dmem[20] dptr_in: pointer to input/base buffer |
| * @param[in] dmem[20] dptr_exp: pointer to exp buffer |
| * @param[in] dmem[28] dptr_out: pointer to output/result buffer |
| * |
| * clobbered registers: x3 to x13, x16 to x31 |
| * w0 to w3, w24 to w30 |
| * w4 to w[4+N-1] |
| * clobbered Flag Groups: FG0, FG1 |
| */ |
| modexp: |
| /* prepare pointers to temp regs */ |
| li x8, 4 |
| li x9, 3 |
| li x10, 4 |
| li x11, 2 |
| |
| /* load pointer to modulus */ |
| lw x16, 16(x0) |
| |
| /* load pointer to m0' */ |
| lw x17, 8(x0) |
| |
| /* load number of limbs */ |
| lw x30, 4(x0) |
| addi x31, x30, -1 |
| |
| /* convert to montgomery domain montmul(A,RR) |
| in = montmul(A,RR) montmul(A,RR) = C*R mod M */ |
| lw x19, 20(x0) |
| lw x20, 12(x0) |
| lw x21, 20(x0) |
| jal x1, montmul |
| /* Store result in dmem starting at dmem[dptr_c] */ |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* zeroize w2 and reset flags */ |
| bn.sub w2, w2, w2 |
| |
| /* initialize the output buffer with -M */ |
| lw x16, 16(x0) |
| lw x21, 28(x0) |
| loop x30, 3 |
| /* load limb from modulus */ |
| bn.lid x11, 0(x16++) |
| |
| /* subtract limb from 0 */ |
| bn.subb w2, w31, w2 |
| |
| /* store limb in dmem */ |
| bn.sid x11, 0(x21++) |
| |
| /* reload pointer to modulus */ |
| lw x16, 16(x0) |
| |
| /* compute bit length of current bigint size */ |
| slli x24, x30, 8 |
| |
| /* iterate over all bits of bigint */ |
| loop x24, 20 |
| /* square: out = montmul(out,out) */ |
| lw x19, 28(x0) |
| lw x20, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul |
| /* Store result in dmem starting at dmem[dptr_c] */ |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* multiply: out = montmul(in,out) */ |
| lw x19, 20(x0) |
| lw x20, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul |
| |
| /* w2 <= w2 << 1 */ |
| bn.add w2, w2, w2 |
| |
| /* the loop performs a 1-bit left shift of the exponent. Last MSB moves |
| to FG0.C, such that it can be used for selection */ |
| lw x20, 24(x0) |
| loop x30, 3 |
| bn.lid x11, 0(x20) |
| /* w2 <= w2 << 1 */ |
| bn.addc w2, w2, w2 |
| bn.sid x11, 0(x20++) |
| |
| /* select squared or squared+multiplied result */ |
| lw x21, 28(x0) |
| jal x1, sel_sqr_or_sqrmul |
| |
| nop |
| |
| /* convert back from montgomery domain */ |
| /* out = montmul(out,1) = out/R mod M */ |
| lw x19, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul_mul1 |
| |
| ret |
| |
| |
| /** |
| * Bigint modular exponentiation with fixed exponent of 65537 |
| * |
| * Returns: C = modexp(A,65537) = A^65537 mod M |
| * |
| * This implements the square and multiply algorithm for the fixed exponent |
| * of E=65537. Note that this implementation (in contrast to modexp) runs the |
| * multiplication step only for bits being actually set in the exponent. |
| * Since the exponent is fixed, this is inherently constant-time. |
| * |
| * The squared Montgomery modulus RR and the Montgomery constant m0' have to |
| * be precomputed and provided at the appropriate locations in dmem. |
| * |
| * Flags: The states of both FG0 and FG1 depend on intermediate values and are |
| * not usable after return. |
| * |
| * The base bignum A is expected in the input buffer, the result C is written |
| * to the output buffer. Note, that the content of the input buffer is |
| * modified during execution. |
| * |
| * @param[in] dmem[2] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[4] N: Number of limbs per bignum |
| * @param[in] dmem[8] dptr_m0d: pointer to m0' in dmem |
| * @param[in] dmem[12] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[16] dptr_m: pointer to first limb of modulus in dmem |
| * @param[in] dmem[20] dptr_in: pointer to input/base buffer |
| * @param[in] dmem[28] dptr_out: pointer to output/result buffer |
| * |
| * clobbered registers: x3 to x13, x16 to x31 |
| * w0 to w3, w24 to w30 |
| * w4 to w[4+N-1] |
| * clobbered Flag Groups: FG0, FG1 |
| */ |
| modexp_65537: |
| /* prepare pointers to temp regs */ |
| li x8, 4 |
| li x9, 3 |
| li x10, 4 |
| li x11, 2 |
| |
| /* load pointer to modulus */ |
| lw x16, 16(x0) |
| |
| /* load pointer to m0' */ |
| lw x17, 8(x0) |
| |
| /* load number of limbs */ |
| lw x30, 4(x0) |
| addi x31, x30, -1 |
| |
| /* convert to montgomery domain montmul(A,RR) |
| in = montmul(A,RR) montmul(A,RR) = C*R mod M */ |
| lw x19, 20(x0) |
| lw x20, 12(x0) |
| lw x21, 20(x0) |
| jal x1, montmul |
| /* Store result in dmem starting at dmem[dptr_c] */ |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* pointer to out buffer */ |
| lw x21, 28(x0) |
| |
| /* zeroize w2 and reset flags */ |
| bn.sub w2, w2, w2 |
| |
| /* this loop initializes the output buffer with -M */ |
| loop x30, 3 |
| /* load limb from modulus */ |
| bn.lid x11, 0(x16++) |
| |
| /* subtract limb from 0 */ |
| bn.subb w2, w31, w2 |
| |
| /* store limb in dmem */ |
| bn.sid x11, 0(x21++) |
| |
| /* reload pointer to 1st limb of modulus */ |
| lw x16, 16(x0) |
| |
| /* 65537 = 0b10000000000000001 |
| ^ sqr + mult |
| out = montmul(out,out) */ |
| lw x19, 28(x0) |
| lw x20, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul |
| /* Store result in dmem starting at dmem[dptr_c] */ |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* out = montmul(in,out) */ |
| lw x19, 20(x0) |
| lw x20, 28(x0) |
| lw x20, 28(x0) |
| jal x1, montmul |
| |
| /* store multiplication result in output buffer */ |
| lw x21, 28(x0) |
| li x8, 4 |
| loop x30, 2 |
| /* store selected limb to dmem */ |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* 65537 = 0b10000000000000001 |
| ^<< 16 x sqr >>^ */ |
| loopi 16, 8 |
| /* square: out = montmul(out, out) */ |
| lw x19, 28(x0) |
| lw x20, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul |
| /* Store result in dmem starting at dmem[dptr_c] */ |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| nop |
| |
| /* 65537 = 0b10000000000000001 |
| mult ^ |
| out = montmul(in,out) */ |
| lw x19, 20(x0) |
| lw x20, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul |
| |
| /* store multiplication result in output buffer */ |
| lw x21, 28(x0) |
| li x8, 4 |
| loop x30, 2 |
| bn.sid x8, 0(x21++) |
| addi x8, x8, 1 |
| |
| /* convert back from montgomery domain */ |
| /* out = montmul(out,1) = out/R mod M */ |
| lw x19, 28(x0) |
| lw x21, 28(x0) |
| jal x1, montmul_mul1 |
| |
| ret |
| |
| |
| /** |
| * Externally callable wrapper for computing Montgomery parameters |
| * |
| * Computes: |
| * - Montgomery Constant m0' |
| * - Squared Montgomery modulus RR mod N |
| * |
| * Needs to be executed once per constant Modulus. |
| * |
| * @param[in] dmem[2] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[4] N: Number of limbs per bignum |
| * @param[in] dmem[8] dptr_m0d: pointer to m0' in dmem |
| * @param[in] dmem[12] dptr_rr: pointer to RR in dmem |
| * @param[in] dmem[16] dptr_m: pointer to first limb of modulus in dmem |
| * @param[out] [dmem[dptr_m0d+31]:dmem[dptr_m0d]] computed m0' |
| * @parma[out] [dmem[dptr_RR+N*32-1]:dmem[dptr_RR]] computed RR |
| */ |
| modload: |
| |
| /* prepare all-zero reg */ |
| bn.xor w31, w31, w31 |
| |
| /* load pointer to modulus (dptr_m) */ |
| lw x16, 16(x0) |
| |
| /* load pointer to m0' (dptr_m0d) */ |
| lw x17, 8(x0) |
| |
| /* load pointer to RR (dptr_rr) */ |
| lw x18, 12(x0) |
| |
| /* load number of limbs (N) */ |
| lw x30, 4(x0) |
| |
| /* load lowest limb of modulus to w28 */ |
| li x8, 28 |
| bn.lid x8, 0(x16) |
| |
| /* Compute Montgomery constant */ |
| jal x1, m0inv |
| |
| /* Store Montgomery constant in dmem */ |
| li x9, 29 |
| bn.sid x9, 0(x17) |
| |
| /* Compute square of Montgomery modulus */ |
| jal x1, compute_rr |
| |
| ret |
