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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_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);

 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>).

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_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)|BN_mod_mul_montgomery(3)> and
L<BN_mod_mul_reciprocal(3)|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_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().

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)|BN_CTX_new(3)>.

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

=head1 RETURN VALUES

For all 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)|ERR_get_error(3)>.

=head1 SEE ALSO

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

=head1 HISTORY

BN_add(), BN_sub(), BN_sqr(), BN_div(), BN_mod(), BN_mod_mul(),
BN_mod_exp() and BN_gcd() are available in all versions of SSLeay and
OpenSSL. The I<ctx> argument to BN_mul() was added in SSLeay
0.9.1b. BN_exp() appeared in SSLeay 0.9.0.
BN_nnmod(), BN_mod_add(), BN_mod_sub(), and BN_mod_sqr() were added in
OpenSSL 0.9.7.

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