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=head1 NAME

BN_add, BN_sub, BN_mul, BN_sqr, BN_div, BN_mod, BN_nnmod, BN_mod_add,
BN_mod_sub, BN_mod_mul, BN_mod_sqr, BN_mod_sqrt, BN_exp, BN_mod_exp, BN_gcd -
arithmetic operations on BIGNUMs

=head1 SYNOPSIS

 #include <openssl/bn.h>

 int BN_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b);

 int BN_sub(BIGNUM *r, const BIGNUM *a, const BIGNUM *b);

 int BN_mul(BIGNUM *r, BIGNUM *a, BIGNUM *b, BN_CTX *ctx);

 int BN_sqr(BIGNUM *r, BIGNUM *a, BN_CTX *ctx);

 int BN_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *a, const BIGNUM *d,
            BN_CTX *ctx);

 int BN_mod(BIGNUM *rem, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);

 int BN_nnmod(BIGNUM *r, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);

 int BN_mod_add(BIGNUM *r, BIGNUM *a, BIGNUM *b, const BIGNUM *m,
                BN_CTX *ctx);

 int BN_mod_sub(BIGNUM *r, BIGNUM *a, BIGNUM *b, const BIGNUM *m,
                BN_CTX *ctx);

 int BN_mod_mul(BIGNUM *r, BIGNUM *a, BIGNUM *b, const BIGNUM *m,
                BN_CTX *ctx);

 int BN_mod_sqr(BIGNUM *r, BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);

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

 int BN_exp(BIGNUM *r, BIGNUM *a, BIGNUM *p, BN_CTX *ctx);

 int BN_mod_exp(BIGNUM *r, BIGNUM *a, const BIGNUM *p,
                const BIGNUM *m, BN_CTX *ctx);

 int BN_gcd(BIGNUM *r, BIGNUM *a, BIGNUM *b, BN_CTX *ctx);

=head1 DESCRIPTION

BN_add() adds I<a> and I<b> and places the result in I<r> (C<r=a+b>).
I<r> may be the same B<BIGNUM> as I<a> or I<b>.

BN_sub() subtracts I<b> from I<a> and places the result in I<r> (C<r=a-b>).
I<r> may be the same B<BIGNUM> as I<a> or I<b>.

BN_mul() multiplies I<a> and I<b> and places the result in I<r> (C<r=a*b>).
I<r> may be the same B<BIGNUM> as I<a> or I<b>.
For multiplication by powers of 2, use L<BN_lshift(3)>.

BN_sqr() takes the square of I<a> and places the result in I<r>
(C<r=a^2>). I<r> and I<a> may be the same B<BIGNUM>.
This function is faster than BN_mul(r,a,a).

BN_div() divides I<a> by I<d> and places the result in I<dv> and the
remainder in I<rem> (C<dv=a/d, rem=a%d>). Either of I<dv> and I<rem> may
be B<NULL>, in which case the respective value is not returned.
The result is rounded towards zero; thus if I<a> is negative, the
remainder will be zero or negative.
For division by powers of 2, use BN_rshift(3).

BN_mod() corresponds to BN_div() with I<dv> set to B<NULL>.

BN_nnmod() reduces I<a> modulo I<m> and places the non-negative
remainder in I<r>.

BN_mod_add() adds I<a> to I<b> modulo I<m> and places the non-negative
result in I<r>.

BN_mod_sub() subtracts I<b> from I<a> modulo I<m> and places the
non-negative result in I<r>.

BN_mod_mul() multiplies I<a> by I<b> and finds the non-negative
remainder respective to modulus I<m> (C<r=(a*b) mod m>). I<r> may be
the same B<BIGNUM> as I<a> or I<b>. For more efficient algorithms for
repeated computations using the same modulus, see
L<BN_mod_mul_montgomery(3)> and
L<BN_mod_mul_reciprocal(3)>.

BN_mod_sqr() takes the square of I<a> modulo B<m> and places the
result in I<r>.

BN_mod_sqrt() returns the modular square root of I<a> such that
C<in^2 = a (mod p)>. The modulus I<p> must be a
prime, otherwise an error or an incorrect "result" will be returned.
The result is stored into I<in> which can be NULL. The result will be
newly allocated in that case.

BN_exp() raises I<a> to the I<p>-th power and places the result in I<r>
(C<r=a^p>). This function is faster than repeated applications of
BN_mul().

BN_mod_exp() computes I<a> to the I<p>-th power modulo I<m> (C<r=a^p %
m>). This function uses less time and space than BN_exp(). Do not call this
function when B<m> is even and any of the parameters have the
B<BN_FLG_CONSTTIME> flag set.

BN_gcd() computes the greatest common divisor of I<a> and I<b> and
places the result in I<r>. I<r> may be the same B<BIGNUM> as I<a> or
I<b>.

For all functions, I<ctx> is a previously allocated B<BN_CTX> used for
temporary variables; see L<BN_CTX_new(3)>.

Unless noted otherwise, the result B<BIGNUM> must be different from
the arguments.

=head1 RETURN VALUES

The BN_mod_sqrt() returns the result (possibly incorrect if I<p> is
not a prime), or NULL.

For all remaining functions, 1 is returned for success, 0 on error. The return
value should always be checked (e.g., C<if (!BN_add(r,a,b)) goto err;>).
The error codes can be obtained by L<ERR_get_error(3)>.

=head1 SEE ALSO

L<ERR_get_error(3)>, L<BN_CTX_new(3)>,
L<BN_add_word(3)>, L<BN_set_bit(3)>

=head1 COPYRIGHT

Copyright 2000-2018 The OpenSSL Project Authors. All Rights Reserved.

Licensed under the OpenSSL license (the "License").  You may not use
this file except in compliance with the License.  You can obtain a copy
in the file LICENSE in the source distribution or at
L<https://www.openssl.org/source/license.html>.

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