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libtommath/tommath.h
Daniel Mendler cf0042654e
define mp_isodd in terms of mp_iseven
2019-10-27 18:36:56 +01:00

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/* LibTomMath, multiple-precision integer library -- Tom St Denis */
/* SPDX-License-Identifier: Unlicense */
#ifndef TOMMATH_H_
#define TOMMATH_H_
#include <stddef.h>
#include <stdint.h>
#include <stdbool.h>
#ifndef MP_NO_FILE
# include <stdio.h>
#endif
#ifdef __cplusplus
extern "C" {
#endif
/* MS Visual C++ doesn't have a 128bit type for words, so fall back to 32bit MPI's (where words are 64bit) */
#if (defined(_MSC_VER) || defined(__LLP64__) || defined(__e2k__) || defined(__LCC__)) && !defined(MP_64BIT)
# define MP_32BIT
#endif
/* detect 64-bit mode if possible */
#if defined(__x86_64__) || defined(_M_X64) || defined(_M_AMD64) || \
defined(__powerpc64__) || defined(__ppc64__) || defined(__PPC64__) || \
defined(__s390x__) || defined(__arch64__) || defined(__aarch64__) || \
defined(__sparcv9) || defined(__sparc_v9__) || defined(__sparc64__) || \
defined(__ia64) || defined(__ia64__) || defined(__itanium__) || defined(_M_IA64) || \
defined(__LP64__) || defined(_LP64) || defined(__64BIT__)
# if !(defined(MP_64BIT) || defined(MP_32BIT) || defined(MP_16BIT))
# if defined(__GNUC__) && !defined(__hppa)
/* we support 128bit integers only via: __attribute__((mode(TI))) */
# define MP_64BIT
# else
/* otherwise we fall back to MP_32BIT even on 64bit platforms */
# define MP_32BIT
# endif
# endif
#endif
#ifdef MP_DIGIT_BIT
# error Defining MP_DIGIT_BIT is disallowed, use MP_16/31/32/64BIT
#endif
/* some default configurations.
*
* A "mp_digit" must be able to hold MP_DIGIT_BIT + 1 bits
* A "mp_word" must be able to hold 2*MP_DIGIT_BIT + 1 bits
*
* At the very least a mp_digit must be able to hold 7 bits
* [any size beyond that is ok provided it doesn't overflow the data type]
*/
#if defined(MP_16BIT)
typedef uint16_t mp_digit;
# define MP_DIGIT_BIT 15
#elif defined(MP_64BIT)
typedef uint64_t mp_digit;
# define MP_DIGIT_BIT 60
#else
typedef uint32_t mp_digit;
# ifdef MP_31BIT
/*
* This is an extension that uses 31-bit digits.
* Please be aware that not all functions support this size, especially s_mp_mul_digs_fast
* will be reduced to work on small numbers only:
* Up to 8 limbs, 248 bits instead of up to 512 limbs, 15872 bits with MP_28BIT.
*/
# define MP_DIGIT_BIT 31
# else
/* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */
# define MP_DIGIT_BIT 28
# define MP_28BIT
# endif
#endif
#define MP_MASK ((((mp_digit)1)<<((mp_digit)MP_DIGIT_BIT))-((mp_digit)1))
#define MP_DIGIT_MAX MP_MASK
/* Primality generation flags */
#define MP_PRIME_BBS 0x0001 /* BBS style prime */
#define MP_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */
#define MP_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */
typedef enum {
MP_ZPOS = 0, /* positive */
MP_NEG = 1 /* negative */
} mp_sign;
typedef enum {
MP_LT = -1, /* less than */
MP_EQ = 0, /* equal */
MP_GT = 1 /* greater than */
} mp_ord;
typedef enum {
MP_OKAY = 0, /* no error */
MP_ERR = -1, /* unknown error */
MP_MEM = -2, /* out of mem */
MP_VAL = -3, /* invalid input */
MP_ITER = -4, /* maximum iterations reached */
MP_BUF = -5 /* buffer overflow, supplied buffer too small */
} mp_err;
typedef enum {
MP_LSB_FIRST = -1,
MP_MSB_FIRST = 1
} mp_order;
typedef enum {
MP_LITTLE_ENDIAN = -1,
MP_NATIVE_ENDIAN = 0,
MP_BIG_ENDIAN = 1
} mp_endian;
/* tunable cutoffs */
#ifndef MP_FIXED_CUTOFFS
extern int
MP_KARATSUBA_MUL_CUTOFF,
MP_KARATSUBA_SQR_CUTOFF,
MP_TOOM_MUL_CUTOFF,
MP_TOOM_SQR_CUTOFF;
#endif
/* define this to use lower memory usage routines (exptmods mostly) */
/* #define MP_LOW_MEM */
#if defined(__GNUC__) && __GNUC__ >= 4
# define MP_NULL_TERMINATED __attribute__((sentinel))
#else
# define MP_NULL_TERMINATED
#endif
/*
* MP_WUR - warn unused result
* ---------------------------
*
* The result of functions annotated with MP_WUR must be
* checked and cannot be ignored.
*
* Most functions in libtommath return an error code.
* This error code must be checked in order to prevent crashes or invalid
* results.
*
* If you still want to avoid the error checks for quick and dirty programs
* without robustness guarantees, you can `#define MP_WUR` before including
* tommath.h, disabling the warnings.
*/
#ifndef MP_WUR
# if defined(__GNUC__) && __GNUC__ >= 4
# define MP_WUR __attribute__((warn_unused_result))
# else
# define MP_WUR
# endif
#endif
#if defined(__GNUC__) && (__GNUC__ * 100 + __GNUC_MINOR__ >= 405)
# define MP_DEPRECATED(x) __attribute__((deprecated("replaced by " #x)))
# define PRIVATE_MP_DEPRECATED_PRAGMA(s) _Pragma(#s)
# define MP_DEPRECATED_PRAGMA(s) PRIVATE_MP_DEPRECATED_PRAGMA(GCC warning s)
#elif defined(_MSC_VER) && _MSC_VER >= 1500
# define MP_DEPRECATED(x) __declspec(deprecated("replaced by " #x))
# define MP_DEPRECATED_PRAGMA(s) __pragma(message(s))
#else
# define MP_DEPRECATED(s)
# define MP_DEPRECATED_PRAGMA(s)
#endif
/* the infamous mp_int structure */
typedef struct {
int used, alloc;
mp_sign sign;
mp_digit *dp;
} mp_int;
/* error code to char* string */
const char *mp_error_to_string(mp_err code) MP_WUR;
/* ---> init and deinit bignum functions <--- */
/* init a bignum */
mp_err mp_init(mp_int *a) MP_WUR;
/* free a bignum */
void mp_clear(mp_int *a);
/* init a null terminated series of arguments */
mp_err mp_init_multi(mp_int *mp, ...) MP_NULL_TERMINATED MP_WUR;
/* clear a null terminated series of arguments */
void mp_clear_multi(mp_int *mp, ...) MP_NULL_TERMINATED;
/* exchange two ints */
void mp_exch(mp_int *a, mp_int *b);
/* shrink ram required for a bignum */
mp_err mp_shrink(mp_int *a) MP_WUR;
/* grow an int to a given size */
mp_err mp_grow(mp_int *a, int size) MP_WUR;
/* init to a given number of digits */
mp_err mp_init_size(mp_int *a, int size) MP_WUR;
/* ---> Basic Manipulations <--- */
#define mp_iszero(a) ((a)->used == 0)
#define mp_isneg(a) ((a)->sign != MP_ZPOS)
#define mp_iseven(a) (((a)->used == 0) || (((a)->dp[0] & 1u) == 0u))
#define mp_isodd(a) (!mp_iseven(a))
/* set to zero */
void mp_zero(mp_int *a);
/* get and set doubles */
double mp_get_double(const mp_int *a) MP_WUR;
mp_err mp_set_double(mp_int *a, double b) MP_WUR;
/* get integer, set integer and init with integer (int32_t) */
int32_t mp_get_i32(const mp_int *a) MP_WUR;
void mp_set_i32(mp_int *a, int32_t b);
mp_err mp_init_i32(mp_int *a, int32_t b) MP_WUR;
/* get integer, set integer and init with integer, behaves like two complement for negative numbers (uint32_t) */
#define mp_get_u32(a) ((uint32_t)mp_get_i32(a))
void mp_set_u32(mp_int *a, uint32_t b);
mp_err mp_init_u32(mp_int *a, uint32_t b) MP_WUR;
/* get integer, set integer and init with integer (int64_t) */
int64_t mp_get_i64(const mp_int *a) MP_WUR;
void mp_set_i64(mp_int *a, int64_t b);
mp_err mp_init_i64(mp_int *a, int64_t b) MP_WUR;
/* get integer, set integer and init with integer, behaves like two complement for negative numbers (uint64_t) */
#define mp_get_u64(a) ((uint64_t)mp_get_i64(a))
void mp_set_u64(mp_int *a, uint64_t b);
mp_err mp_init_u64(mp_int *a, uint64_t b) MP_WUR;
/* get magnitude */
uint32_t mp_get_mag_u32(const mp_int *a) MP_WUR;
uint64_t mp_get_mag_u64(const mp_int *a) MP_WUR;
unsigned long mp_get_mag_ul(const mp_int *a) MP_WUR;
unsigned long long mp_get_mag_ull(const mp_int *a) MP_WUR;
/* get integer, set integer (long) */
long mp_get_l(const mp_int *a) MP_WUR;
void mp_set_l(mp_int *a, long b);
mp_err mp_init_l(mp_int *a, long b) MP_WUR;
/* get integer, set integer (unsigned long) */
#define mp_get_ul(a) ((unsigned long)mp_get_l(a))
void mp_set_ul(mp_int *a, unsigned long b);
mp_err mp_init_ul(mp_int *a, unsigned long b) MP_WUR;
/* get integer, set integer (long long) */
long long mp_get_ll(const mp_int *a) MP_WUR;
void mp_set_ll(mp_int *a, long long b);
mp_err mp_init_ll(mp_int *a, long long b) MP_WUR;
/* get integer, set integer (unsigned long long) */
#define mp_get_ull(a) ((unsigned long long)mp_get_ll(a))
void mp_set_ull(mp_int *a, unsigned long long b);
mp_err mp_init_ull(mp_int *a, unsigned long long b) MP_WUR;
/* set to single unsigned digit, up to MP_DIGIT_MAX */
void mp_set(mp_int *a, mp_digit b);
mp_err mp_init_set(mp_int *a, mp_digit b) MP_WUR;
/* copy, b = a */
mp_err mp_copy(const mp_int *a, mp_int *b) MP_WUR;
/* inits and copies, a = b */
mp_err mp_init_copy(mp_int *a, const mp_int *b) MP_WUR;
/* trim unused digits */
void mp_clamp(mp_int *a);
/* unpack binary data */
mp_err mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, mp_endian endian,
size_t nails, const void *op) MP_WUR;
/* pack binary data */
size_t mp_pack_count(const mp_int *a, size_t nails, size_t size) MP_WUR;
mp_err mp_pack(void *rop, size_t maxcount, size_t *written, mp_order order, size_t size,
mp_endian endian, size_t nails, const mp_int *op) MP_WUR;
/* ---> digit manipulation <--- */
/* right shift by "b" digits */
void mp_rshd(mp_int *a, int b);
/* left shift by "b" digits */
mp_err mp_lshd(mp_int *a, int b) MP_WUR;
/* c = a / 2**b, implemented as c = a >> b */
mp_err mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d) MP_WUR;
/* b = a/2 */
mp_err mp_div_2(const mp_int *a, mp_int *b) MP_WUR;
/* a/3 => 3c + d == a */
mp_err mp_div_3(const mp_int *a, mp_int *c, mp_digit *d) MP_WUR;
/* c = a * 2**b, implemented as c = a << b */
mp_err mp_mul_2d(const mp_int *a, int b, mp_int *c) MP_WUR;
/* b = a*2 */
mp_err mp_mul_2(const mp_int *a, mp_int *b) MP_WUR;
/* c = a mod 2**b */
mp_err mp_mod_2d(const mp_int *a, int b, mp_int *c) MP_WUR;
/* computes a = 2**b */
mp_err mp_2expt(mp_int *a, int b) MP_WUR;
/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(const mp_int *a) MP_WUR;
/* I Love Earth! */
/* makes a pseudo-random mp_int of a given size */
mp_err mp_rand(mp_int *a, int digits) MP_WUR;
/* use custom random data source instead of source provided the platform */
void mp_rand_source(mp_err(*source)(void *out, size_t size));
/* ---> binary operations <--- */
/* c = a XOR b (two complement) */
mp_err mp_xor(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = a OR b (two complement) */
mp_err mp_or(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = a AND b (two complement) */
mp_err mp_and(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* b = ~a (bitwise not, two complement) */
mp_err mp_complement(const mp_int *a, mp_int *b) MP_WUR;
/* right shift with sign extension */
mp_err mp_signed_rsh(const mp_int *a, int b, mp_int *c) MP_WUR;
/* ---> Basic arithmetic <--- */
/* b = -a */
mp_err mp_neg(const mp_int *a, mp_int *b) MP_WUR;
/* b = |a| */
mp_err mp_abs(const mp_int *a, mp_int *b) MP_WUR;
/* compare a to b */
mp_ord mp_cmp(const mp_int *a, const mp_int *b) MP_WUR;
/* compare |a| to |b| */
mp_ord mp_cmp_mag(const mp_int *a, const mp_int *b) MP_WUR;
/* c = a + b */
mp_err mp_add(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = a - b */
mp_err mp_sub(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = a * b */
mp_err mp_mul(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* b = a*a */
mp_err mp_sqr(const mp_int *a, mp_int *b) MP_WUR;
/* a/b => cb + d == a */
mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) MP_WUR;
/* c = a mod b, 0 <= c < b */
mp_err mp_mod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* Increment "a" by one like "a++". Changes input! */
#define mp_incr(a) mp_add_d((a), 1, (a))
/* Decrement "a" by one like "a--". Changes input! */
#define mp_decr(a) mp_sub_d((a), 1, (a))
/* ---> single digit functions <--- */
/* compare against a single digit */
mp_ord mp_cmp_d(const mp_int *a, mp_digit b) MP_WUR;
/* c = a + b */
mp_err mp_add_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR;
/* c = a - b */
mp_err mp_sub_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR;
/* c = a * b */
mp_err mp_mul_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR;
/* a/b => cb + d == a */
mp_err mp_div_d(const mp_int *a, mp_digit b, mp_int *c, mp_digit *d) MP_WUR;
/* c = a mod b, 0 <= c < b */
mp_err mp_mod_d(const mp_int *a, mp_digit b, mp_digit *c) MP_WUR;
/* ---> number theory <--- */
/* d = a + b (mod c) */
mp_err mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR;
/* d = a - b (mod c) */
mp_err mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR;
/* d = a * b (mod c) */
mp_err mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR;
/* c = a * a (mod b) */
mp_err mp_sqrmod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = 1/a (mod b) */
mp_err mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* c = (a, b) */
mp_err mp_gcd(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* produces value such that U1*a + U2*b = U3 */
mp_err mp_exteuclid(const mp_int *a, const mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) MP_WUR;
/* c = [a, b] or (a*b)/(a, b) */
mp_err mp_lcm(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR;
/* finds one of the b'th root of a, such that |c|**b <= |a|
*
* returns error if a < 0 and b is even
*/
mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c) MP_WUR;
/* special sqrt algo */
mp_err mp_sqrt(const mp_int *arg, mp_int *ret) MP_WUR;
/* special sqrt (mod prime) */
mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret) MP_WUR;
/* is number a square? */
mp_err mp_is_square(const mp_int *arg, bool *ret) MP_WUR;
/* computes the Kronecker symbol c = (a | p) (like jacobi() but with {a,p} in Z */
mp_err mp_kronecker(const mp_int *a, const mp_int *p, int *c) MP_WUR;
/* used to setup the Barrett reduction for a given modulus b */
mp_err mp_reduce_setup(mp_int *a, const mp_int *b) MP_WUR;
/* Barrett Reduction, computes a (mod b) with a precomputed value c
*
* Assumes that 0 < x <= m*m, note if 0 > x > -(m*m) then you can merely
* compute the reduction as -1 * mp_reduce(mp_abs(x)) [pseudo code].
*/
mp_err mp_reduce(mp_int *x, const mp_int *m, const mp_int *mu) MP_WUR;
/* setups the montgomery reduction */
mp_err mp_montgomery_setup(const mp_int *n, mp_digit *rho) MP_WUR;
/* computes a = B**n mod b without division or multiplication useful for
* normalizing numbers in a Montgomery system.
*/
mp_err mp_montgomery_calc_normalization(mp_int *a, const mp_int *b) MP_WUR;
/* computes x/R == x (mod N) via Montgomery Reduction */
mp_err mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) MP_WUR;
/* returns 1 if a is a valid DR modulus */
bool mp_dr_is_modulus(const mp_int *a) MP_WUR;
/* sets the value of "d" required for mp_dr_reduce */
void mp_dr_setup(const mp_int *a, mp_digit *d);
/* reduces a modulo n using the Diminished Radix method */
mp_err mp_dr_reduce(mp_int *x, const mp_int *n, mp_digit k) MP_WUR;
/* returns true if a can be reduced with mp_reduce_2k */
bool mp_reduce_is_2k(const mp_int *a) MP_WUR;
/* determines k value for 2k reduction */
mp_err mp_reduce_2k_setup(const mp_int *a, mp_digit *d) MP_WUR;
/* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
mp_err mp_reduce_2k(mp_int *a, const mp_int *n, mp_digit d) MP_WUR;
/* returns true if a can be reduced with mp_reduce_2k_l */
bool mp_reduce_is_2k_l(const mp_int *a) MP_WUR;
/* determines k value for 2k reduction */
mp_err mp_reduce_2k_setup_l(const mp_int *a, mp_int *d) MP_WUR;
/* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
mp_err mp_reduce_2k_l(mp_int *a, const mp_int *n, const mp_int *d) MP_WUR;
/* Y = G**X (mod P) */
mp_err mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y) MP_WUR;
/* ---> Primes <--- */
/* performs one Fermat test of "a" using base "b".
* Sets result to 0 if composite or 1 if probable prime
*/
mp_err mp_prime_fermat(const mp_int *a, const mp_int *b, bool *result) MP_WUR;
/* performs one Miller-Rabin test of "a" using base "b".
* Sets result to 0 if composite or 1 if probable prime
*/
mp_err mp_prime_miller_rabin(const mp_int *a, const mp_int *b, bool *result) MP_WUR;
/* This gives [for a given bit size] the number of trials required
* such that Miller-Rabin gives a prob of failure lower than 2^-96
*/
int mp_prime_rabin_miller_trials(int size) MP_WUR;
/* performs one strong Lucas-Selfridge test of "a".
* Sets result to 0 if composite or 1 if probable prime
*/
mp_err mp_prime_strong_lucas_selfridge(const mp_int *a, bool *result) MP_WUR;
/* performs one Frobenius test of "a" as described by Paul Underwood.
* Sets result to 0 if composite or 1 if probable prime
*/
mp_err mp_prime_frobenius_underwood(const mp_int *N, bool *result) MP_WUR;
/* performs t random rounds of Miller-Rabin on "a" additional to
* bases 2 and 3. Also performs an initial sieve of trial
* division. Determines if "a" is prime with probability
* of error no more than (1/4)**t.
* Both a strong Lucas-Selfridge to complete the BPSW test
* and a separate Frobenius test are available at compile time.
* With t<0 a deterministic test is run for primes up to
* 318665857834031151167461. With t<13 (abs(t)-13) additional
* tests with sequential small primes are run starting at 43.
* Is Fips 186.4 compliant if called with t as computed by
* mp_prime_rabin_miller_trials();
*
* Sets result to 1 if probably prime, 0 otherwise
*/
mp_err mp_prime_is_prime(const mp_int *a, int t, bool *result) MP_WUR;
/* finds the next prime after the number "a" using "t" trials
* of Miller-Rabin.
*
* bbs_style = true means the prime must be congruent to 3 mod 4
*/
mp_err mp_prime_next_prime(mp_int *a, int t, bool bbs_style) MP_WUR;
/* makes a truly random prime of a given size (bits),
*
* Flags are as follows:
*
* MP_PRIME_BBS - make prime congruent to 3 mod 4
* MP_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies MP_PRIME_BBS)
* MP_PRIME_2MSB_ON - make the 2nd highest bit one
*
* You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can
* have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself
* so it can be NULL
*
*/
mp_err mp_prime_rand(mp_int *a, int t, int size, int flags) MP_WUR;
/* Integer logarithm to integer base */
mp_err mp_log_u32(const mp_int *a, uint32_t base, uint32_t *c) MP_WUR;
/* c = a**b */
mp_err mp_expt_u32(const mp_int *a, uint32_t b, mp_int *c) MP_WUR;
/* ---> radix conversion <--- */
int mp_count_bits(const mp_int *a) MP_WUR;
size_t mp_ubin_size(const mp_int *a) MP_WUR;
mp_err mp_from_ubin(mp_int *a, const unsigned char *buf, size_t size) MP_WUR;
mp_err mp_to_ubin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) MP_WUR;
size_t mp_sbin_size(const mp_int *a) MP_WUR;
mp_err mp_from_sbin(mp_int *a, const unsigned char *buf, size_t size) MP_WUR;
mp_err mp_to_sbin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) MP_WUR;
mp_err mp_read_radix(mp_int *a, const char *str, int radix) MP_WUR;
mp_err mp_to_radix(const mp_int *a, char *str, size_t maxlen, size_t *written, int radix) MP_WUR;
mp_err mp_radix_size(const mp_int *a, int radix, size_t *size) MP_WUR;
#ifndef MP_NO_FILE
mp_err mp_fread(mp_int *a, int radix, FILE *stream) MP_WUR;
mp_err mp_fwrite(const mp_int *a, int radix, FILE *stream) MP_WUR;
#endif
#define mp_to_binary(M, S, N) mp_to_radix((M), (S), (N), NULL, 2)
#define mp_to_octal(M, S, N) mp_to_radix((M), (S), (N), NULL, 8)
#define mp_to_decimal(M, S, N) mp_to_radix((M), (S), (N), NULL, 10)
#define mp_to_hex(M, S, N) mp_to_radix((M), (S), (N), NULL, 16)
#ifdef __cplusplus
}
#endif
#endif
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