Additional input checks and a test for b \cong 0 (mod a) in test_mp_sqrtmod_prime

to go along with it.

Author: czurnieden

demo/test.c

@@ -707,9 +707,9 @@ static int test_mp_sqrtmod_prime(void)
    };
 
    static struct mp_sqrtmod_prime_st sqrtmod_prime[] = {
-      { 5, 14, 3 },
-      { 7, 9, 4 },
-      { 113, 2, 62 }
+      { 5, 14, 3 },   /* 5 \cong 1 (mod 4) */
+      { 7, 9, 4 },    /* 7 \cong 3 (mod 4) */
+      { 113, 2, 62 }  /* 113 \cong 1 (mod 4) */
    };
    int i;
 
@@ -723,6 +723,14 @@ static int test_mp_sqrtmod_prime(void)
       DO(mp_sqrtmod_prime(&b, &a, &c));
       EXPECT(mp_cmp_d(&c, sqrtmod_prime[i].r) == MP_EQ);
    }
+   /* Check handling of wrong input (here: modulus is square and cong. 1 mod 4,24 ) */
+   mp_set_ul(&a, 25);
+   mp_set_ul(&b, 2);
+   EXPECT(mp_sqrtmod_prime(&b, &a, &c) == MP_VAL);
+   /* b \cong 0 (mod a) */
+   mp_set_ul(&a, 45);
+   mp_set_ul(&b, 3);
+   EXPECT(mp_sqrtmod_prime(&b, &a, &c) == MP_VAL);
 
    mp_clear_multi(&a, &b, &c, NULL);
    return EXIT_SUCCESS;

mp_sqrtmod_prime.c

@@ -13,28 +13,32 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
 {
    mp_err err;
    int legendre;
-   mp_int t1, C, Q, S, Z, M, T, R, two;
-   mp_digit i;
+   /* The type is "int" because of the types in the mp_int struct.
+      Don't forget to change them here when you change them there! */
+   int S, M, i;
+   mp_int t1, C, Q, Z, T, R, two;
 
    /* first handle the simple cases */
    if (mp_cmp_d(n, 0uL) == MP_EQ) {
       mp_zero(ret);
       return MP_OKAY;
    }
-   if (mp_cmp_d(prime, 2uL) == MP_EQ)                            return MP_VAL; /* prime must be odd */
-   if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY)        return err;
-   if (legendre == -1)                                           return MP_VAL; /* quadratic non-residue mod prime */
+   /* "prime" must be odd and > 2 */
+   if (mp_iseven(prime) || (mp_cmp_d(prime, 3uL) == MP_LT))      return MP_VAL;
+   if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY)     return err;
+   /* n \not\cong 0 (mod p) and n \cong r^2 (mod p) for some r \in N^+ */
+   if (legendre != 1)                                            return MP_VAL;
 
-   if ((err = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) {
+   if ((err = mp_init_multi(&t1, &C, &Q, &Z, &T, &R, &two, NULL)) != MP_OKAY) {
       return err;
    }
 
    /* SPECIAL CASE: if prime mod 4 == 3
     * compute directly: err = n^(prime+1)/4 mod prime
     * Handbook of Applied Cryptography algorithm 3.36
     */
-   if ((err = mp_mod_d(prime, 4uL, &i)) != MP_OKAY)               goto LBL_END;
-   if (i == 3u) {
+   /* x%4 == x&3 for x in N and x>0 */
+   if ((prime->dp[0] & 3u) == 3u) {
       if ((err = mp_add_d(prime, 1uL, &t1)) != MP_OKAY)           goto LBL_END;
       if ((err = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto LBL_END;
       if ((err = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto LBL_END;
@@ -49,12 +53,12 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
    if ((err = mp_copy(prime, &Q)) != MP_OKAY)                    goto LBL_END;
    if ((err = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY)                 goto LBL_END;
    /* Q = prime - 1 */
-   mp_zero(&S);
+   S = 0;
    /* S = 0 */
    while (mp_iseven(&Q)) {
       if ((err = mp_div_2(&Q, &Q)) != MP_OKAY)                    goto LBL_END;
       /* Q = Q / 2 */
-      if ((err = mp_add_d(&S, 1uL, &S)) != MP_OKAY)               goto LBL_END;
+      S++;
       /* S = S + 1 */
    }
 
@@ -63,6 +67,12 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
    /* Z = 2 */
    for (;;) {
       if ((err = mp_kronecker(&Z, prime, &legendre)) != MP_OKAY)     goto LBL_END;
+      /* If "prime" (p) is an odd prime Jacobi(k|p) = 0 for k \cong 0 (mod p) */
+      /* but there is at least one non-quadratic residue before k>=p if p is an odd prime. */
+      if (legendre == 0) {
+         err = MP_VAL;
+         goto LBL_END;
+      }
       if (legendre == -1) break;
       if ((err = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY)               goto LBL_END;
       /* Z = Z + 1 */
@@ -77,7 +87,7 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
    /* R = n ^ ((Q + 1) / 2) mod prime */
    if ((err = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY)          goto LBL_END;
    /* T = n ^ Q mod prime */
-   if ((err = mp_copy(&S, &M)) != MP_OKAY)                       goto LBL_END;
+   M = S;
    /* M = S */
    mp_set(&two, 2uL);
 
@@ -86,16 +96,21 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
       i = 0;
       for (;;) {
          if (mp_cmp_d(&t1, 1uL) == MP_EQ) break;
+         /*  No exponent in the range 0 < i < M found
+            (M is at least 1 in the first round because "prime" > 2) */
+         if (M == i) {
+            err = MP_VAL;
+            goto LBL_END;
+         }
          if ((err = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto LBL_END;
          i++;
       }
-      if (i == 0u) {
+      if (i == 0) {
          if ((err = mp_copy(&R, ret)) != MP_OKAY)                  goto LBL_END;
          err = MP_OKAY;
          goto LBL_END;
       }
-      if ((err = mp_sub_d(&M, i, &t1)) != MP_OKAY)                goto LBL_END;
-      if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY)             goto LBL_END;
+      mp_set_i32(&t1, M - i - 1);
       if ((err = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY)   goto LBL_END;
       /* t1 = 2 ^ (M - i - 1) */
       if ((err = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY)     goto LBL_END;
@@ -106,12 +121,12 @@ mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
       /* R = (R * t1) mod prime */
       if ((err = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY)        goto LBL_END;
       /* T = (T * C) mod prime */
-      mp_set(&M, i);
+      M = i;
       /* M = i */
    }
 
 LBL_END:
-   mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL);
+   mp_clear_multi(&t1, &C, &Q, &Z, &T, &R, &two, NULL);
    return err;
 }
 

tommath_class.h

@@ -872,12 +872,12 @@
 #   define MP_CMP_D_C
 #   define MP_COPY_C
 #   define MP_DIV_2_C
-#   define MP_DIV_D_C
 #   define MP_EXPTMOD_C
 #   define MP_INIT_MULTI_C
 #   define MP_KRONECKER_C
 #   define MP_MULMOD_C
 #   define MP_SET_C
+#   define MP_SET_I32_C
 #   define MP_SQRMOD_C
 #   define MP_SUB_D_C
 #   define MP_ZERO_C