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140 lines
3.7 KiB
140 lines
3.7 KiB
#include "tommath_private.h"
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#ifdef BN_MP_ROOT_U32_C
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/* LibTomMath, multiple-precision integer library -- Tom St Denis */
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/* SPDX-License-Identifier: Unlicense */
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/* find the n'th root of an integer
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*
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* Result found such that (c)**b <= a and (c+1)**b > a
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*
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* This algorithm uses Newton's approximation
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* x[i+1] = x[i] - f(x[i])/f'(x[i])
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* which will find the root in log(N) time where
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* each step involves a fair bit.
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*/
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mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c)
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{
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mp_int t1, t2, t3, a_;
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mp_ord cmp;
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int ilog2;
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mp_err err;
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/* input must be positive if b is even */
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if (((b & 1u) == 0u) && (a->sign == MP_NEG)) {
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return MP_VAL;
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}
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if ((err = mp_init_multi(&t1, &t2, &t3, NULL)) != MP_OKAY) {
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return err;
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}
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/* if a is negative fudge the sign but keep track */
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a_ = *a;
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a_.sign = MP_ZPOS;
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/* Compute seed: 2^(log_2(n)/b + 2)*/
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ilog2 = mp_count_bits(a);
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/*
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If "b" is larger than INT_MAX it is also larger than
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log_2(n) because the bit-length of the "n" is measured
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with an int and hence the root is always < 2 (two).
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*/
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if (b > (uint32_t)(INT_MAX/2)) {
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mp_set(c, 1uL);
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c->sign = a->sign;
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err = MP_OKAY;
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goto LBL_ERR;
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}
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/* "b" is smaller than INT_MAX, we can cast safely */
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if (ilog2 < (int)b) {
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mp_set(c, 1uL);
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c->sign = a->sign;
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err = MP_OKAY;
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goto LBL_ERR;
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}
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ilog2 = ilog2 / ((int)b);
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if (ilog2 == 0) {
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mp_set(c, 1uL);
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c->sign = a->sign;
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err = MP_OKAY;
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goto LBL_ERR;
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}
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/* Start value must be larger than root */
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ilog2 += 2;
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if ((err = mp_2expt(&t2,ilog2)) != MP_OKAY) goto LBL_ERR;
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do {
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/* t1 = t2 */
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if ((err = mp_copy(&t2, &t1)) != MP_OKAY) goto LBL_ERR;
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/* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
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/* t3 = t1**(b-1) */
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if ((err = mp_expt_u32(&t1, b - 1u, &t3)) != MP_OKAY) goto LBL_ERR;
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/* numerator */
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/* t2 = t1**b */
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if ((err = mp_mul(&t3, &t1, &t2)) != MP_OKAY) goto LBL_ERR;
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/* t2 = t1**b - a */
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if ((err = mp_sub(&t2, &a_, &t2)) != MP_OKAY) goto LBL_ERR;
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/* denominator */
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/* t3 = t1**(b-1) * b */
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if ((err = mp_mul_d(&t3, b, &t3)) != MP_OKAY) goto LBL_ERR;
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/* t3 = (t1**b - a)/(b * t1**(b-1)) */
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if ((err = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY) goto LBL_ERR;
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if ((err = mp_sub(&t1, &t3, &t2)) != MP_OKAY) goto LBL_ERR;
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/*
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Number of rounds is at most log_2(root). If it is more it
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got stuck, so break out of the loop and do the rest manually.
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*/
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if (ilog2-- == 0) {
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break;
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}
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} while (mp_cmp(&t1, &t2) != MP_EQ);
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/* result can be off by a few so check */
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/* Loop beneath can overshoot by one if found root is smaller than actual root */
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for (;;) {
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if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR;
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cmp = mp_cmp(&t2, &a_);
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if (cmp == MP_EQ) {
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err = MP_OKAY;
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goto LBL_ERR;
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}
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if (cmp == MP_LT) {
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if ((err = mp_add_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR;
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} else {
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break;
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}
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}
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/* correct overshoot from above or from recurrence */
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for (;;) {
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if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR;
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if (mp_cmp(&t2, &a_) == MP_GT) {
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if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR;
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} else {
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break;
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}
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}
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/* set the result */
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mp_exch(&t1, c);
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/* set the sign of the result */
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c->sign = a->sign;
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err = MP_OKAY;
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LBL_ERR:
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mp_clear_multi(&t1, &t2, &t3, NULL);
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return err;
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}
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#endif
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