=pod

=head1 NAME

bn - Multiprecision integer arithmetics

=head1 SYNOPSIS

 #include <openssl/bn.h>

 BIGNUM *BN_new(void);
 void BN_free(BIGNUM *a);
 void BN_init(BIGNUM *);
 void BN_clear(BIGNUM *a);
 void BN_clear_free(BIGNUM *a);

 BN_CTX *BN_CTX_new(void);
 void BN_CTX_init(BN_CTX *c);
 void BN_CTX_free(BN_CTX *c);

 BIGNUM *BN_copy(BIGNUM *a, const BIGNUM *b);
 BIGNUM *BN_dup(const BIGNUM *a);

 int BN_num_bytes(const BIGNUM *a);
 int BN_num_bits(const BIGNUM *a);
 int BN_num_bits_word(BN_ULONG w);

 int BN_add(BIGNUM *r, BIGNUM *a, 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_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *a, const BIGNUM *d,
         BN_CTX *ctx);
 int BN_sqr(BIGNUM *r, BIGNUM *a, BN_CTX *ctx);
 int BN_mod(BIGNUM *rem, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);
 int BN_mod_mul(BIGNUM *ret, BIGNUM *a, BIGNUM *b, 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);

 int BN_add_word(BIGNUM *a, BN_ULONG w);
 int BN_sub_word(BIGNUM *a, BN_ULONG w);
 int BN_mul_word(BIGNUM *a, BN_ULONG w);
 BN_ULONG BN_div_word(BIGNUM *a, BN_ULONG w);
 BN_ULONG BN_mod_word(const BIGNUM *a, BN_ULONG w);

 int BN_cmp(BIGNUM *a, BIGNUM *b);
 int BN_ucmp(BIGNUM *a, BIGNUM *b);
 int BN_is_zero(BIGNUM *a);
 int BN_is_one(BIGNUM *a);
 int BN_is_word(BIGNUM *a, BN_ULONG w);
 int BN_is_odd(BIGNUM *a);

 int BN_zero(BIGNUM *a);
 int BN_one(BIGNUM *a);
 BIGNUM *BN_value_one(void);
 int BN_set_word(BIGNUM *a, unsigned long w);
 unsigned long BN_get_word(BIGNUM *a);

 int BN_rand(BIGNUM *rnd, int bits, int top,int bottom);
 int BN_pseudo_rand(BIGNUM *rnd, int bits, int top,int bottom);

 BIGNUM *BN_generate_prime(BIGNUM *ret,int bits,int safe,BIGNUM *add,
         BIGNUM *rem,void (*callback)(int,int,void *),void *cb_arg);
 int BN_is_prime(const BIGNUM *p,int nchecks,void (*callback)(int,int,void *),
         BN_CTX *ctx,void *cb_arg);

 int BN_set_bit(BIGNUM *a, int n);
 int BN_clear_bit(BIGNUM *a, int n);
 int BN_is_bit_set(const BIGNUM *a, int n);
 int BN_mask_bits(BIGNUM *a, int n);
 int BN_lshift(BIGNUM *r, const BIGNUM *a, int n);
 int BN_lshift1(BIGNUM *r, BIGNUM *a);
 int BN_rshift(BIGNUM *r, BIGNUM *a, int n);
 int BN_rshift1(BIGNUM *r, BIGNUM *a);

 int BN_bn2bin(const BIGNUM *a, unsigned char *to);
 BIGNUM *BN_bin2bn(const unsigned char *s, int len, BIGNUM *ret);
 char *BN_bn2hex(const BIGNUM *a);
 char *BN_bn2dec(const BIGNUM *a);
 int BN_hex2bn(BIGNUM **a, const char *str);
 int BN_dec2bn(BIGNUM **a, const char *str);
 int BN_print(BIO *fp, const BIGNUM *a);
 int BN_print_fp(FILE *fp, const BIGNUM *a);
 int BN_bn2mpi(const BIGNUM *a, unsigned char *to);
 BIGNUM *BN_mpi2bn(unsigned char *s, int len, BIGNUM *ret);

 BIGNUM *BN_mod_inverse(BIGNUM *r, BIGNUM *a, const BIGNUM *n,
     BN_CTX *ctx);

 BN_RECP_CTX *BN_RECP_CTX_new(void);
 void BN_RECP_CTX_init(BN_RECP_CTX *recp);
 void BN_RECP_CTX_free(BN_RECP_CTX *recp);
 int BN_RECP_CTX_set(BN_RECP_CTX *recp, const BIGNUM *m, BN_CTX *ctx);
 int BN_mod_mul_reciprocal(BIGNUM *r, BIGNUM *a, BIGNUM *b,
        BN_RECP_CTX *recp, BN_CTX *ctx);

 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

This library performs arithmetic operations on integers of arbitrary
size. It was written for use in public key cryptography, such as RSA
and Diffie-Hellman.

It uses dynamic memory allocation for storing its data structures.
That means that there is no limit on the size of the numbers
manipulated by these functions, but return values must always be
checked in case a memory allocation error has occurred.

The basic object in this library is a B<BIGNUM>. It is used to hold a
single large integer. This type should be considered opaque and fields
should not be modified or accessed directly.

The creation of B<BIGNUM> objects is described in L<BN_new(3)|BN_new(3)>;
L<BN_add(3)|BN_add(3)> describes most of the arithmetic operations.
Comparison is described in L<BN_cmp(3)|BN_cmp(3)>; L<BN_zero(3)|BN_zero(3)>
describes certain assignments, L<BN_rand(3)|BN_rand(3)> the generation of
random numbers, L<BN_generate_prime(3)|BN_generate_prime(3)> deals with prime
numbers and L<BN_set_bit(3)|BN_set_bit(3)> with bit operations. The conversion
of B<BIGNUM>s to external formats is described in L<BN_bn2bin(3)|BN_bn2bin(3)>.

=head1 INTERNALS

The following description is based on the SSLeay documentation:

 typedef struct bignum_st
        {
        int top;        /* Index of last used d. */
        BN_ULONG *d;    /* Pointer to an array of 'BITS2' bit chunks. */
        int max;        /* Size of the d array. */
        int neg;
        } BIGNUM;

The big number is stored in B<d>, a malloc()ed array of B<BN_ULONG>s,
least significant first. A B<BN_ULONG> can be either 16, 32 or 64 bits
in size, depending on the 'number of bits' specified in bn.h. 

B<max> is the size of the B<d> array that has been allocated.  B<top>
is the 'last' entry being used, so for a value of 4, bn.d[0]=4 and
bn.top=1. B<neg> is 1 if the number is negative.  When a BIGNUM is
'0', the B<d> field can be NULL and B<top> == 0. Various routines in
this library require the use of temporary B<BIGNUM> variables during
their execution.  Since dynamic memory allocation to create B<BIGNUM>s
is rather expensive when used in conjunction with repeated subroutine
calls, the B<BN_CTX> structure is used.  This structure contains
B<BN_CTX_NUM> B<BIGNUM>s.  B<BN_CTX_NUM> is the maximum number of
temporary B<BIGNUM>s any publicly exported function will use.

 #define BN_CTX_NUM  12
 typedef struct bignum_ctx
        {
        int tos;                    /* top of stack */
        BIGNUM *bn[BN_CTX_NUM];     /* The variables */
        } BN_CTX;

B<tos> is the index of the first unused B<BIGNUM> in the B<bn> array.

=head1 SEE ALSO

L<dh(3)|dh(3)>, L<err(3)|err(3)>, L<rand(3)|rand(3)>, L<rsa(3)|rsa(3)>,
L<BN_new(3)|BN_new(3)>, L<BN_CTX_new(3)|BN_CTX_new(3)>,
L<BN_copy(3)|BN_copy(3)>, L<BN_num_bytes(3)|BN_num_bytes(3)>,
L<BN_add(3)|BN_add(3)>, L<BN_add_word(3)|BN_add_word(3)>,
L<BN_cmp(3)|BN_cmp(3)>, L<BN_zero(3)|BN_zero(3)>, L<BN_rand(3)|BN_rand(3)>,
L<BN_generate_prime(3)|BN_generate_prime(3)>, L<BN_set_bit(3)|BN_set_bit(3)>,
L<BN_bn2bin(3)|BN_bn2bin(3)>, L<BN_mod_inverse(3)|BN_mod_inverse(3)>,
L<BN_mod_mul_reciprocal(3)|BN_mod_mul_reciprocal(3)>,
L<BN_mod_mul_montgomery(3)|BN_mod_mul_montgomery(3)> 

=cut
