BN_MOD_SQRT(3) | Library Functions Manual | BN_MOD_SQRT(3)

# NAME

`BN_mod_sqrt` — square root in a prime field

# SYNOPSIS

library “libcrypto”
`#include <openssl/bn.h>`

BIGNUM *
`BN_mod_sqrt`(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);

# DESCRIPTION

`BN_mod_sqrt`() solves

    r2≡a(modp)

for r in the prime field of characteristic p using the Tonelli-Shanks algorithm if needed and places one of the two solutions into r. The other solution is p - r.

The argument p is expected to be a prime number.

# RETURN VALUES

In case of success, `BN_mod_sqrt`() returns r, or a newly allocated BIGNUM object if the r argument is `NULL`.

In case of failure, `NULL` is returned. This for example happens if a is not a quadratic residue or if memory allocation fails.

# SEE ALSO

BN_CTX_new(3), BN_kronecker(3), BN_mod_sqr(3), BN_new(3)

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, Berlin, 1993, Algorithm 1.5.1.

# HISTORY

`BN_mod_sqrt`() first appeared in OpenSSL 0.9.7 and has been available since OpenBSD 3.2.

# CAVEATS

If p is not prime, `BN_mod_sqrt`() may succeed or fail. If it succeeds, the square of the returned value is congruent to a modulo p. If it fails, the reason reported by ERR_get_error(3) is often misleading. In particular, even if a is a perfect square, `BN_mod_sqrt`() often reports “not a square” instead of “p is not prime”.

June 8, 2025 | Debian