--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/libtommath/bn_mp_root_u32.c	Thu Oct 15 19:55:15 2020 +0800
@@ -0,0 +1,139 @@
+#include "tommath_private.h"
+#ifdef BN_MP_ROOT_U32_C
+/* LibTomMath, multiple-precision integer library -- Tom St Denis */
+/* SPDX-License-Identifier: Unlicense */
+
+/* find the n'th root of an integer
+ *
+ * Result found such that (c)**b <= a and (c+1)**b > a
+ *
+ * This algorithm uses Newton's approximation
+ * x[i+1] = x[i] - f(x[i])/f'(x[i])
+ * which will find the root in log(N) time where
+ * each step involves a fair bit.
+ */
+mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c)
+{
+   mp_int t1, t2, t3, a_;
+   mp_ord cmp;
+   int    ilog2;
+   mp_err err;
+
+   /* input must be positive if b is even */
+   if (((b & 1u) == 0u) && (a->sign == MP_NEG)) {
+      return MP_VAL;
+   }
+
+   if ((err = mp_init_multi(&t1, &t2, &t3, NULL)) != MP_OKAY) {
+      return err;
+   }
+
+   /* if a is negative fudge the sign but keep track */
+   a_ = *a;
+   a_.sign = MP_ZPOS;
+
+   /* Compute seed: 2^(log_2(n)/b + 2)*/
+   ilog2 = mp_count_bits(a);
+
+   /*
+     If "b" is larger than INT_MAX it is also larger than
+     log_2(n) because the bit-length of the "n" is measured
+     with an int and hence the root is always < 2 (two).
+   */
+   if (b > (uint32_t)(INT_MAX/2)) {
+      mp_set(c, 1uL);
+      c->sign = a->sign;
+      err = MP_OKAY;
+      goto LBL_ERR;
+   }
+
+   /* "b" is smaller than INT_MAX, we can cast safely */
+   if (ilog2 < (int)b) {
+      mp_set(c, 1uL);
+      c->sign = a->sign;
+      err = MP_OKAY;
+      goto LBL_ERR;
+   }
+   ilog2 =  ilog2 / ((int)b);
+   if (ilog2 == 0) {
+      mp_set(c, 1uL);
+      c->sign = a->sign;
+      err = MP_OKAY;
+      goto LBL_ERR;
+   }
+   /* Start value must be larger than root */
+   ilog2 += 2;
+   if ((err = mp_2expt(&t2,ilog2)) != MP_OKAY)                    goto LBL_ERR;
+   do {
+      /* t1 = t2 */
+      if ((err = mp_copy(&t2, &t1)) != MP_OKAY)                   goto LBL_ERR;
+
+      /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
+
+      /* t3 = t1**(b-1) */
+      if ((err = mp_expt_u32(&t1, b - 1u, &t3)) != MP_OKAY)       goto LBL_ERR;
+
+      /* numerator */
+      /* t2 = t1**b */
+      if ((err = mp_mul(&t3, &t1, &t2)) != MP_OKAY)               goto LBL_ERR;
+
+      /* t2 = t1**b - a */
+      if ((err = mp_sub(&t2, &a_, &t2)) != MP_OKAY)               goto LBL_ERR;
+
+      /* denominator */
+      /* t3 = t1**(b-1) * b  */
+      if ((err = mp_mul_d(&t3, b, &t3)) != MP_OKAY)               goto LBL_ERR;
+
+      /* t3 = (t1**b - a)/(b * t1**(b-1)) */
+      if ((err = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY)         goto LBL_ERR;
+
+      if ((err = mp_sub(&t1, &t3, &t2)) != MP_OKAY)               goto LBL_ERR;
+
+      /*
+          Number of rounds is at most log_2(root). If it is more it
+          got stuck, so break out of the loop and do the rest manually.
+       */
+      if (ilog2-- == 0) {
+         break;
+      }
+   }  while (mp_cmp(&t1, &t2) != MP_EQ);
+
+   /* result can be off by a few so check */
+   /* Loop beneath can overshoot by one if found root is smaller than actual root */
+   for (;;) {
+      if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY)            goto LBL_ERR;
+      cmp = mp_cmp(&t2, &a_);
+      if (cmp == MP_EQ) {
+         err = MP_OKAY;
+         goto LBL_ERR;
+      }
+      if (cmp == MP_LT) {
+         if ((err = mp_add_d(&t1, 1uL, &t1)) != MP_OKAY)          goto LBL_ERR;
+      } else {
+         break;
+      }
+   }
+   /* correct overshoot from above or from recurrence */
+   for (;;) {
+      if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY)            goto LBL_ERR;
+      if (mp_cmp(&t2, &a_) == MP_GT) {
+         if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY)          goto LBL_ERR;
+      } else {
+         break;
+      }
+   }
+
+   /* set the result */
+   mp_exch(&t1, c);
+
+   /* set the sign of the result */
+   c->sign = a->sign;
+
+   err = MP_OKAY;
+
+LBL_ERR:
+   mp_clear_multi(&t1, &t2, &t3, NULL);
+   return err;
+}
+
+#endif