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=head1 NAME

BN_mod_mul_montgomery, BN_MONT_CTX_new, BN_MONT_CTX_init,
BN_MONT_CTX_free, BN_MONT_CTX_set, BN_MONT_CTX_copy,
BN_from_montgomery, BN_to_montgomery - Montgomery multiplication

=head1 SYNOPSIS

 #include <openssl/bn.h>

 BN_MONT_CTX *BN_MONT_CTX_new(void);
 void BN_MONT_CTX_init(BN_MONT_CTX *ctx);
 void BN_MONT_CTX_free(BN_MONT_CTX *mont);

 int BN_MONT_CTX_set(BN_MONT_CTX *mont, const BIGNUM *m, BN_CTX *ctx);
 BN_MONT_CTX *BN_MONT_CTX_copy(BN_MONT_CTX *to, BN_MONT_CTX *from);

 int BN_mod_mul_montgomery(BIGNUM *r, BIGNUM *a, BIGNUM *b,
         BN_MONT_CTX *mont, BN_CTX *ctx);

 int BN_from_montgomery(BIGNUM *r, BIGNUM *a, BN_MONT_CTX *mont,
         BN_CTX *ctx);

 int BN_to_montgomery(BIGNUM *r, BIGNUM *a, BN_MONT_CTX *mont,
         BN_CTX *ctx);

=head1 DESCRIPTION

These functions implement Montgomery multiplication. They are used
automatically when L<BN_mod_exp(3)|BN_mod_exp(3)> is called with suitable input,
but they may be useful when several operations are to be performed
using the same modulus.

BN_MONT_CTX_new() allocates and initializes a B<BN_MONT_CTX> structure.
BN_MONT_CTX_init() initializes an existing uninitialized B<BN_MONT_CTX>.

BN_MONT_CTX_set() sets up the I<mont> structure from the modulus I<m>
by precomputing its inverse and a value R.

BN_MONT_CTX_copy() copies the B<BN_MONT_CTX> I<from> to I<to>.

BN_MONT_CTX_free() frees the components of the B<BN_MONT_CTX>, and, if
it was created by BN_MONT_CTX_new(), also the structure itself.

BN_mod_mul_montgomery() computes Mont(I<a>,I<b>):=I<a>*I<b>*R^-1 and places
the result in I<r>.

BN_from_montgomery() performs the Montgomery reduction I<r> = I<a>*R^-1.

BN_to_montgomery() computes Mont(I<a>,R^2), i.e. I<a>*R.
Note that I<a> must be non-negative and smaller than the modulus.

For all functions, I<ctx> is a previously allocated B<BN_CTX> used for
temporary variables.

The B<BN_MONT_CTX> structure is defined as follows:

 typedef struct bn_mont_ctx_st
        {
        int ri;         /* number of bits in R */
        BIGNUM RR;      /* R^2 (used to convert to Montgomery form) */
        BIGNUM N;       /* The modulus */
        BIGNUM Ni;      /* R*(1/R mod N) - N*Ni = 1
                         * (Ni is only stored for bignum algorithm) */
        BN_ULONG n0;    /* least significant word of Ni */
        int flags;
        } BN_MONT_CTX;

BN_to_montgomery() is a macro.

=head1 RETURN VALUES

BN_MONT_CTX_new() returns the newly allocated B<BN_MONT_CTX>, and NULL
on error.

BN_MONT_CTX_init() and BN_MONT_CTX_free() have no return values.

For the other functions, 1 is returned for success, 0 on error.
The error codes can be obtained by L<ERR_get_error(3)|ERR_get_error(3)>.

=head1 WARNING

The inputs must be reduced modulo B<m>, otherwise the result will be
outside the expected range.

=head1 SEE ALSO

L<bn(3)|bn(3)>, L<ERR_get_error(3)|ERR_get_error(3)>, L<BN_add(3)|BN_add(3)>,
L<BN_CTX_new(3)|BN_CTX_new(3)>

=head1 HISTORY

BN_MONT_CTX_new(), BN_MONT_CTX_free(), BN_MONT_CTX_set(),
BN_mod_mul_montgomery(), BN_from_montgomery() and BN_to_montgomery()
are available in all versions of SSLeay and OpenSSL.

BN_MONT_CTX_init() and BN_MONT_CTX_copy() were added in SSLeay 0.9.1b.

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