.\" $OpenBSD: BN_add.3,v 1.20 2023/04/27 09:47:03 tb Exp $
.\" full merge up to: OpenSSL e9b77246 Jan 20 19:58:49 2017 +0100
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.\" Copyright (c) 2021 Ingo Schwarze <schwarze@openbsd.org>
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.\" and Bodo Moeller <bodo@openssl.org>.
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.Dd $Mdocdate: April 27 2023 $
.Dt BN_ADD 3
.Os
.Sh NAME
.Nm BN_add ,
.Nm BN_uadd ,
.Nm BN_sub ,
.Nm BN_usub ,
.Nm BN_mul ,
.Nm BN_sqr ,
.Nm BN_div ,
.Nm BN_mod ,
.Nm BN_nnmod ,
.Nm BN_mod_add ,
.Nm BN_mod_add_quick ,
.Nm BN_mod_sub ,
.Nm BN_mod_sub_quick ,
.Nm BN_mod_mul ,
.Nm BN_mod_sqr ,
.Nm BN_mod_lshift ,
.Nm BN_mod_lshift_quick ,
.Nm BN_mod_lshift1 ,
.Nm BN_mod_lshift1_quick ,
.Nm BN_exp ,
.Nm BN_mod_exp ,
.\" The following are public, but intentionally undocumented for now:
.\" .Nm BN_mod_exp_mont ,  r \(== a ^ p (mod m)
.\" .Nm BN_mod_exp_mont_consttime ,
.\" .Nm BN_mod_exp_mont_word ,
.\" .Nm BN_mod_exp_simple ,
.\" .Nm BN_mod_exp2_mont   r \(== (a1 ^ p1) * (a2 ^ p2) (mod m)
.\" Maybe they should be deleted from <openssl/bn.h>.
.Nm BN_gcd
.Nd arithmetic operations on BIGNUMs
.Sh SYNOPSIS
.In openssl/bn.h
.Ft int
.Fo BN_add
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fc
.Ft int
.Fo BN_uadd
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fc
.Ft int
.Fo BN_sub
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fc
.Ft int
.Fo BN_usub
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fc
.Ft int
.Fo BN_mul
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_sqr
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_div
.Fa "BIGNUM *dv"
.Fa "BIGNUM *rem"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *d"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod
.Fa "BIGNUM *rem"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_nnmod
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_add
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_add_quick
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "const BIGNUM *m"
.Fc
.Ft int
.Fo BN_mod_sub
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_sub_quick
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "const BIGNUM *m"
.Fc
.Ft int
.Fo BN_mod_mul
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_sqr
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_lshift
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "int n"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_lshift_quick
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "int n"
.Fa "const BIGNUM *m"
.Fc
.Ft int
.Fo BN_mod_lshift1
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_lshift1_quick
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *m"
.Fc
.Ft int
.Fo BN_exp
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *p"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_mod_exp
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *p"
.Fa "const BIGNUM *m"
.Fa "BN_CTX *ctx"
.Fc
.Ft int
.Fo BN_gcd
.Fa "BIGNUM *r"
.Fa "const BIGNUM *a"
.Fa "const BIGNUM *b"
.Fa "BN_CTX *ctx"
.Fc
.Sh DESCRIPTION
.Fn BN_add
adds
.Fa a
and
.Fa b
and places the result in
.Fa r
.Pq Li r=a+b .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
.Pp
.Fn BN_uadd
adds the absolute values of
.Fa a
and
.Fa b
and places the result in
.Fa r
.Pq Li r=|a|+|b|\& .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
.Pp
.Fn BN_sub
subtracts
.Fa b
from
.Fa a
and places the result in
.Fa r
.Pq Li r=a-b .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
.Pp
.Fn BN_usub
subtracts the absolute value of
.Fa b
from the absolute value of
.Fa a
and places the result in
.Fa r
.Pq Li r=|a|-|b|\& .
It requires the absolute value of
.Fa a
to be greater than the absolute value of
.Fa b ;
otherwise it will fail.
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
.Pp
.Fn BN_mul
multiplies
.Fa a
and
.Fa b
and places the result in
.Fa r
.Pq Li r=a*b .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
For multiplication by powers of 2, use
.Xr BN_lshift 3 .
.Pp
.Fn BN_sqr
takes the square of
.Fa a
and places the result in
.Fa r
.Pq Li r=a^2 .
.Fa r
and
.Fa a
may be the same
.Vt BIGNUM .
This function is faster than
.Fn BN_mul r a a .
.Pp
.Fn BN_div
divides
.Fa a
by
.Fa d
and places the result in
.Fa dv
and the remainder in
.Fa rem
.Pq Li dv=a/d , rem=a%d .
If the flag
.Dv BN_FLG_CONSTTIME
is set on
.Fa a
or
.Fa d ,
it operates in constant time.
Either of
.Fa dv
and
.Fa rem
may be
.Dv NULL ,
in which case the respective value is not returned.
The result is rounded towards zero; thus if
.Fa a
is negative, the remainder will be zero or negative.
For division by powers of 2, use
.Fn BN_rshift 3 .
.Pp
.Fn BN_mod
corresponds to
.Fn BN_div
with
.Fa dv
set to
.Dv NULL .
It is implemented as a macro.
.Pp
.Fn BN_nnmod
reduces
.Fa a
modulo
.Fa m
and places the non-negative remainder in
.Fa r .
.Pp
.Fn BN_mod_add
adds
.Fa a
to
.Fa b
modulo
.Fa m
and places the non-negative result in
.Fa r .
.Pp
.Fn BN_mod_add_quick
is a variant of
.Fn BN_mod_add
that requires
.Fa a
and
.Fa b
to both be non-negative and smaller than
.Fa m .
If any of these constraints are violated,
it silently produces wrong results.
.Pp
.Fn BN_mod_sub
subtracts
.Fa b
from
.Fa a
modulo
.Fa m
and places the non-negative result in
.Fa r .
.Pp
.Fn BN_mod_sub_quick
is a variant of
.Fn BN_mod_sub
that requires
.Fa a
and
.Fa b
to both be non-negative and smaller than
.Fa m .
If any of these constraints are violated,
it silently produces wrong results.
.Pp
.Fn BN_mod_mul
multiplies
.Fa a
by
.Fa b
and finds the non-negative remainder respective to modulus
.Fa m
.Pq Li r=(a*b)%m .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
For a more efficient algorithm for repeated computations using the same
modulus, see
.Xr BN_mod_mul_montgomery 3 .
.Pp
.Fn BN_mod_sqr
takes the square of
.Fa a
modulo
.Fa m
and places the result in
.Fa r .
.Pp
.Fn BN_mod_lshift
shifts
.Fa a
left by
.Fa n
bits, reduces the result modulo
.Fa m ,
and places the non-negative remainder in
.Fa r
.Pq Li r=a*2^n mod m .
.Pp
.Fn BN_mod_lshift1
shifts
.Fa a
left by one bit, reduces the result modulo
.Fa m ,
and places the non-negative remainder in
.Fa r
.Pq Li r=a*2 mod m .
.Pp
.Fn BN_mod_lshift_quick
and
.Fn BN_mod_lshift1_quick
are variants of
.Fn BN_mod_lshift
and
.Fn BN_mod_lshift1 ,
respectively, that require
.Fa a
to be non-negative and less than
.Fa m .
If either of these constraints is violated, they sometimes fail
and sometimes silently produce wrong results.
.Pp
.Fn BN_exp
raises
.Fa a
to the
.Fa p Ns -th
power and places the result in
.Fa r
.Pq Li r=a^p .
This function is faster than repeated applications of
.Fn BN_mul .
.Pp
.Fn BN_mod_exp
computes
.Fa a
to the
.Fa p Ns -th
power modulo
.Fa m
.Pq Li r=(a^p)%m .
If the flag
.Dv BN_FLG_CONSTTIME
is set on
.Fa p ,
it operates in constant time.
This function uses less time and space than
.Fn BN_exp .
.Pp
.Fn BN_gcd
computes the greatest common divisor of
.Fa a
and
.Fa b
and places the result in
.Fa r .
.Fa r
may be the same
.Vt BIGNUM
as
.Fa a
or
.Fa b .
.Pp
For all functions,
.Fa ctx
is a previously allocated
.Vt BN_CTX
used for temporary variables; see
.Xr BN_CTX_new 3 .
.Pp
Unless noted otherwise, the result
.Vt BIGNUM
must be different from the arguments.
.Sh RETURN VALUES
For all functions, 1 is returned for success, 0 on error.
The return value should always be checked, for example:
.Pp
.Dl if (!BN_add(r,a,b)) goto err;
.Pp
The error codes can be obtained by
.Xr ERR_get_error 3 .
.Sh SEE ALSO
.Xr BN_add_word 3 ,
.Xr BN_CTX_new 3 ,
.Xr BN_new 3 ,
.Xr BN_set_bit 3 ,
.Xr BN_set_flags 3 ,
.Xr BN_set_negative 3
.Sh HISTORY
.Fn BN_add ,
.Fn BN_sub ,
.Fn BN_mul ,
.Fn BN_sqr ,
.Fn BN_div ,
.Fn BN_mod ,
.Fn BN_mod_mul ,
.Fn BN_mod_exp ,
and
.Fn BN_gcd
first appeared in SSLeay 0.5.1.
.Fn BN_exp
first appeared in SSLeay 0.9.0.
All these functions have been available since
.Ox 2.4 .
.Pp
.Fn BN_uadd ,
.Fn BN_usub ,
and the
.Fa ctx
argument to
.Fn BN_mul
first appeared in SSLeay 0.9.1 and have been available since
.Ox 2.6 .
.Pp
.Fn BN_nnmod ,
.Fn BN_mod_add ,
.Fn BN_mod_add_quick ,
.Fn BN_mod_sub ,
.Fn BN_mod_sub_quick ,
.Fn BN_mod_sqr ,
.Fn BN_mod_lshift ,
.Fn BN_mod_lshift_quick ,
.Fn BN_mod_lshift1 ,
and
.Fn BN_mod_lshift1_quick
first appeared in OpenSSL 0.9.7 and have been available since
.Ox 3.2 .
.Sh BUGS
Even if the
.Dv BN_FLG_CONSTTIME
flag is set on
.Fa a
or
.Fa b ,
.Fn BN_gcd
neither fails nor operates in constant time, potentially allowing
timing side-channel attacks.
.Pp
Even if the
.Dv BN_FLG_CONSTTIME
flag is set on
.Fa p ,
if the modulus
.Fa m
is even,
.Fn BN_mod_exp
does not operate in constant time, potentially allowing
timing side-channel attacks.
.Pp
If
.Dv BN_FLG_CONSTTIME
is set on
.Fa p ,
.Fn BN_exp
fails instead of operating in constant time.