ulong_extras.h – arithmetic and number-theoretic functions for single-word integers

Random functions

ulong n_randlimb(flint_rand_t state)

Returns a uniformly pseudo random limb.

The algorithm generates two random half limbs \(s_j\), \(j = 0, 1\), by iterating respectively \(v_{i+1} = (v_i a + b) \bmod{p_j}\) for some initial seed \(v_0\), randomly chosen values \(a\) and \(b\) and p_0 = 4294967311 = nextprime(2^32) on a 64-bit machine and p_0 = nextprime(2^16) on a 32-bit machine and p_1 = nextprime(p_0).

ulong n_randbits(flint_rand_t state, unsigned int bits)

Returns a uniformly pseudo random number with the given number of bits. The most significant bit is always set, unless zero is passed, in which case zero is returned.

ulong n_randtest_bits(flint_rand_t state, int bits)

Returns a uniformly pseudo random number with the given number of bits. The most significant bit is always set, unless zero is passed, in which case zero is returned. The probability of a value with a sparse binary representation being returned is increased. This function is intended for use in test code.

ulong n_randint(flint_rand_t state, ulong limit)

Returns a uniformly pseudo random number up to but not including the given limit. If zero is passed as a parameter, an entire random limb is returned.

ulong n_urandint(flint_rant_t state, ulong limit)

Returns a uniformly pseudo random number up to but not including the given limit. If zero is passed as a parameter, an entire random limb is returned. This function provides somewhat better randomness as compared to n_randint, especially for larger values of limit.

ulong n_randtest(flint_rand_t state)

Returns a pseudo random number with a random number of bits, from \(0\) to FLINT_BITS. The probability of the special values \(0\), \(1\), COEFF_MAX and WORD_MAX is increased as is the probability of a value with sparse binary representation. This random function is mainly used for testing purposes. This function is intended for use in test code.

ulong n_randtest_not_zero(flint_rand_t state)

As for n_randtest(), but does not return \(0\). This function is intended for use in test code.

ulong n_randprime(flint_rand_t state, unsigned slong bits, int proved)

Returns a random prime number (proved = 1) or probable prime (proved = 0) with bits bits, where bits must be at least 2 and at most FLINT_BITS.

ulong n_randtest_prime(flint_rand_t state, int proved)

Returns a random prime number (proved = 1) or probable prime (proved = 0) with size randomly chosen between 2 and FLINT_BITS bits. This function is intended for use in test code.

Basic arithmetic

ulong n_pow(ulong n, ulong exp)

Returns n^exp. No checking is done for overflow. The exponent may be zero. We define \(0^0 = 1\).

The algorithm simply uses a for loop. Repeated squaring is unlikely to speed up this algorithm.

ulong n_flog(ulong n, ulong b)

Returns \(\floor{\log_b x}\).

Assumes that \(x \geq 1\) and \(b \geq 2\).

ulong n_clog(ulong n, ulong b)

Returns \(\ceil{\log_b x}\).

Assumes that \(x \geq 1\) and \(b \geq 2\).

Miscellaneous

ulong n_revbin(ulong n, ulong b)

Returns the binary reverse of \(n\), assuming it is \(b\) bits in length, e.g. n_revbin(10110, 6) will return 110100.

int n_sizeinbase(ulong n, int base)

Returns the exact number of digits needed to represent \(n\) as a string in base base assumed to be between 2 and 36. Returns 1 when \(n = 0\).

Basic arithmetic with precomputed inverses

ulong n_preinvert_limb(ulong n)

Returns a precomputed inverse of \(n\), as defined in citep{GraMol2010}. This precomputed inverse can be used with all of the functions that take a precomputed inverse whose names are suffixed by _preinv.

We require \(n > 0\).

double n_precompute_inverse(ulong n)

Returns a precomputed inverse of \(n\) with double precision value \(1/n\). This precomputed inverse can be used with all of the functions that take a precomputed inverse whose names are suffixed by _precomp.

We require \(n > 0\).

ulong n_mod_precomp(ulong a, ulong n, double ninv)

Returns \(a \bmod{n}\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse(). We require n < 2^FLINT_D_BITS and a < 2^(FLINT_BITS-1) and \(0 \leq a < n^2\).

We assume the processor is in the standard round to nearest mode. Thus ninv is correct to \(53\) binary bits, the least significant bit of which we shall call a place, and can be at most half a place out. When \(a\) is multiplied by \(ninv\), the binary representation of \(a\) is exact and the mantissa is less than \(2\), thus we see that a * ninv can be at most one out in the mantissa. We now truncate a * ninv to the nearest integer, which is always a round down. Either we already have an integer, or we need to make a change down of at least \(1\) in the last place. In the latter case we either get precisely the exact quotient or below it as when we rounded the product to the nearest place we changed by at most half a place. In the case that truncating to an integer takes us below the exact quotient, we have rounded down by less than \(1\) plus half a place. But as the product is less than \(n\) and \(n\) is less than \(2^{53}\), half a place is less than \(1\), thus we are out by less than \(2\) from the exact quotient, i.e.the quotient we have computed is the quotient we are after or one too small. That leaves only the case where we had to round up to the nearest place which happened to be an integer, so that truncating to an integer didn’t change anything. But this implies that the exact quotient \(a/n\) is less than \(2^{-54}\) from an integer. We deal with this rare case by subtracting 1 from the quotient. Then the quotient we have computed is either exactly what we are after, or one too small.

ulong n_mod2_precomp(ulong a, ulong n, double ninv)

Returns \(a \bmod{n}\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse(). There are no restrictions on \(a\) or on \(n\).

As for n_mod_precomp() for \(n < 2^{53}\) and \(a < n^2\) the computed quotient is either what we are after or one too large or small. We deal with these cases. Otherwise we can be sure that the top \(52\) bits of the quotient are computed correctly. We take the remainder and adjust the quotient by multiplying the remainder by ninv to compute another approximate quotient as per mod_precomp. Now the remainder may be either negative or positive, so the quotient we compute may be one out in either direction.

ulong n_divrem2_preinv(ulong * q, ulong a, ulong n, ulong ninv)

Returns \(a \bmod{n}\) and sets \(q\) to the quotient of \(a\) by \(n\), given a precomputed inverse of \(n\) computed by n_preinvert_limb(). There are no restrictions on \(a\) and the only restriction on \(n\) is that it be nonzero.

This uses the algorithm of Granlund and M”oller citep{GraMol2010}. First \(n\) is normalised and \(a\) is shifted into two limbs to compensate. Then their algorithm is applied verbatim and the remainder shifted back.

ulong n_div2_preinv(ulong a, ulong n, ulong ninv)

Returns the Euclidean quotient of \(a\) by \(n\) given a precomputed inverse of \(n\) computed by n_preinvert_limb(). There are no restrictions on \(a\) and the only restriction on \(n\) is that it be nonzero.

This uses the algorithm of Granlund and M”oller citep{GraMol2010}. First \(n\) is normalised and \(a\) is shifted into two limbs to compensate. Then their algorithm is applied verbatim.

ulong n_mod2_preinv(ulong a, ulong n, ulong ninv)

Returns \(a \bmod{n}\) given a precomputed inverse of \(n\) computed by n_preinvert_limb(). There are no restrictions on \(a\) and the only restriction on \(n\) is that it be nonzero.

This uses the algorithm of Granlund and M”oller citep{GraMol2010}. First \(n\) is normalised and \(a\) is shifted into two limbs to compensate. Then their algorithm is applied verbatim and the result shifted back.

% Torbjorn Granlund and Niels Moller % Improved Division by Invariant Integers % https://gmplib.org/ tege/division-paper.pdf

ulong n_divrem2_precomp(ulong * q, ulong a, ulong n, double npre)

Returns \(a \bmod{n}\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse() and sets \(q\) to the quotient. There are no restrictions on \(a\) or on \(n\).

This is as for n_mod2_precomp() with some additional care taken to retain the quotient information. There are also special cases to deal with the case where \(a\) is already reduced modulo \(n\) and where \(n\) is \(64\) bits and \(a\) is not reduced modulo \(n\).

ulong n_ll_mod_preinv(ulong a_hi, ulong a_lo, ulong n, ulong ninv)

Returns \(a \bmod{n}\) given a precomputed inverse of \(n\) computed by n_preinvert_limb(). There are no restrictions on \(a\), which will be two limbs (a_hi, a_lo), or on \(n\).

The old version of this function merely reduced the top limb a_hi modulo \(n\) so that udiv_qrnnd_preinv() could be used.

The new version reduces the top limb modulo \(n\) as per n_mod2_preinv() and then the algorithm of Granlund and M”oller citep{GraMol2010} is used again to reduce modulo \(n\).

% Torbjorn Granlund and Niels Moller % Improved Division by Invariant Integers % http://www.lysator.liu.se/ nisse/archive/draft-division-paper.pdf

ulong n_lll_mod_preinv(ulong a_hi, ulong a_mi, ulong a_lo, ulong n, ulong ninv)

Returns \(a \bmod{n}\), where \(a\) has three limbs (a_hi, a_mi, a_lo), given a precomputed inverse of \(n\) computed by n_preinvert_limb(). It is assumed that a_hi is reduced modulo \(n\). There are no restrictions on \(n\).

This function uses the algorithm of Granlund and M”oller citep{GraMol2010} to first reduce the top two limbs modulo \(n\), then does the same on the bottom two limbs.

% Torbjorn Granlund and Niels Moller % Improved Division by Invariant Integers % http://www.lysator.liu.se/ nisse/archive/draft-division-paper.pdf

ulong n_mulmod_precomp(ulong a, ulong b, ulong n, double ninv)

Returns \(a b \bmod{n}\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse(). We require n < 2^FLINT_D_BITS and \(0 \leq a, b < n\).

We assume the processor is in the standard round to nearest mode. Thus ninv is correct to \(53\) binary bits, the least significant bit of which we shall call a place, and can be at most half a place out. The product of \(a\) and \(b\) is computed with error at most half a place. When a * b is multiplied by \(ninv\) we find that the exact quotient and computed quotient differ by less than two places. As the quotient is less than \(n\) this means that the exact quotient is at most \(1\) away from the computed quotient. We truncate this quotient to an integer which reduces the value by less than \(1\). We end up with a value which can be no more than two above the quotient we are after and no less than two below. However an argument similar to that for n_mod_precomp() shows that the truncated computed quotient cannot be two smaller than the truncated exact quotient. In other words the computed integer quotient is at most two above and one below the quotient we are after.

ulong n_mulmod2_preinv(ulong a, ulong b, ulong n, ulong ninv)

Returns \(a b \bmod{n}\) given a precomputed inverse of \(n\) computed by\ n_preinvert_limb(). There are no restrictions on \(a\), \(b\) or on \(n\). This is implemented by multiplying using umul_ppmm() and then reducing using n_ll_mod_preinv().

ulong n_mulmod2(ulong a, ulong b, ulong n)

Returns \(a b \bmod{n}\). There are no restrictions on \(a\), \(b\) or on \(n\). This is implemented by multiplying using umul_ppmm() and then reducing using n_ll_mod_preinv() after computing a precomputed inverse.

ulong n_mulmod_preinv(ulong a, ulong b, ulong n, ulong ninv, ulong norm)

Returns \(a b \pmod{n}\) given a precomputed inverse of \(n\) computed by\ n_preinvert_limb(), assuming \(a\) and \(b\) are reduced modulo \(n\) and \(n\) is normalised, i.e. with most significant bit set. There are no other restrictions on \(a\), \(b\) or \(n\).

The value norm is provided for convenience. As \(n\) is required to be normalised, it may be that \(a\) and \(b\) have been shifted to the left by norm bits before calling the function. Their product then has an extra factor of \(2^\text{norm}\). Specifying a nonzero norm will shift the product right by this many bits before reducing it.

The algorithm use is that of Granlund and M”oller citep{GraMol2010}.

% Torbjorn Granlund and Niels Moller % Improved Division by Invariant Integers % http://www.lysator.liu.se/ nisse/archive/draft-division-paper.pdf

Greatest common divisor

ulong n_gcd(ulong x, ulong y)

Returns the greatest common divisor \(g\) of \(x\) and \(y\). No assumptions are made about the values \(x\) and \(y\).

The algorithm is a slight embelishment of the Euclidean algorithm which uses some branches to avoid most divisions.

One wishes to compute the quotient and remainder of \(u_3 / v_3\) without division where possible. This is accomplished when \(u_3 < 4 v_3\), i.e.the quotient is either \(1\), \(2\) or \(3\).

We first compute \(s = u_3 - v_3\). If \(s < v_3\), i.e.\(u_3 < 2 v_3\), we know the quotient is \(1\), else if \(s < 2 v_3\), i.e.\(u_3 < 3 v_3\) we know the quotient is \(2\). In the remaining cases, the quotient must be \(3\). When the quotient is \(4\) or above, we use division. However this happens rarely for generic inputs.

ulong n_gcd_full(ulong x, ulong y)

Returns the greatest common divisor \(g\) of \(x\) and \(y\). No assumptions are made about \(x\) and \(y\).

This function is deprecated.

ulong n_gcdinv(ulong * a, ulong x, ulong y)

Returns the greatest common divisor \(g\) of \(x\) and \(y\) and computes \(a\) such that \(0 \leq a < y\) and \(a x = \gcd(x, y) \bmod{y}\), when this is defined. We require \(x < y\).

When \(y = 1\) the greatest common divisor is set to \(1\) and \(a\) is set to \(0\).

This is merely an adaption of the extended Euclidean algorithm computing just one cofactor and reducing it modulo \(y\).

ulong n_xgcd(ulong * a, ulong * b, ulong x, ulong y)

Returns the greatest common divisor \(g\) of \(x\) and \(y\) and unsigned values \(a\) and \(b\) such that \(a x - b y = g\). We require \(x \geq y\).

We claim that computing the extended greatest common divisor via the Euclidean algorithm always results in cofactor \(\abs{a} < x/2\), \(\abs{b} < x/2\), with perhaps some small degenerate exceptions.

We proceed by induction.

Suppose we are at some step of the algorithm, with \(x_n = q y_n + r\) with \(r \geq 1\), and suppose \(1 = s y_n - t r\) with \(s < y_n / 2\), \(t < y_n / 2\) by hypothesis.

Write \(1 = s y_n - t (x_n - q y_n) = (s + t q) y_n - t x_n\).

It suffices to show that \((s + t q) < x_n / 2\) as \(t < y_n / 2 < x_n / 2\), which will complete the induction step.

But at the previous step in the backsubstitution we would have had \(1 = s r - c d\) with \(s < r/2\) and \(c < r/2\).

Then \(s + t q < r/2 + y_n / 2 q = (r + q y_n)/2 = x_n / 2\).

See the documentation of n_gcd() for a description of the branching in the algorithm, which is faster than using division.

Jacobi and Kronecker symbols

int n_jacobi(mp_limb_signed_t x, ulong y)

Computes the Jacobi symbol of \(x \bmod{y}\). Assumes that \(y\) is positive and odd, and for performance reasons that \(\gcd(x, y) = 1\).

This is just a straightforward application of the law of quadratic reciprocity. For performance, divisions are replaced with some comparisons and subtractions where possible.

int n_jacobi_unsigned(ulong x, ulong y)

Computes the Jacobi symbol, allowing \(x\) to go up to a full limb.

Modular Arithmetic

ulong n_addmod(ulong a, ulong b, ulong n)

Returns \((a + b) \bmod{n}\).

ulong n_submod(ulong a, ulong b, ulong n)

Returns \((a - b) \bmod{n}\).

ulong n_invmod(ulong x, ulong y)

Returns the inverse of \(x\) modulo \(y\), if it exists. Otherwise an exception is thrown.

This is merely an adaption of the extended Euclidean algorithm with appropriate normalisation.

ulong n_powmod_precomp(ulong a, mp_limb_signed_t exp, ulong n, double npre)

Returns a^exp modulo \(n\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse(). We require \(n < 2^{53}\) and \(0 \leq a < n\). There are no restrictions on exp, i.e.it can be negative.

This is implemented as a standard binary powering algorithm using repeated squaring and reducing modulo \(n\) at each step.

ulong n_powmod_ui_precomp(ulong a, ulong exp, ulong n, double npre)

Returns a^exp modulo \(n\) given a precomputed inverse of \(n\) computed by\ n_precompute_inverse(). We require \(n < 2^{53}\) and \(0 \leq a < n\). The exponent exp is unsigned and so can be larger than allowed by n_powmod_precomp.

This is implemented as a standard binary powering algorithm using repeated squaring and reducing modulo \(n\) at each step.

ulong n_powmod(ulong a, mp_limb_signed_t exp, ulong n)

Returns a^exp modulo \(n\). We require n < 2^FLINT_D_BITS and \(0 \leq a < n\). There are no restrictions on exp, i.e.it can be negative.

This is implemented by precomputing an inverse and calling the precomp version of this function.

ulong n_powmod2_preinv(ulong a, mp_limb_signed_t exp, ulong n, ulong ninv)

Returns (a^exp) % n given a precomputed inverse of \(n\) computed by n_preinvert_limb(). We require \(0 \leq a < n\), but there are no restrictions on \(n\) or on exp, i.e.it can be negative.

This is implemented as a standard binary powering algorithm using repeated squaring and reducing modulo \(n\) at each step.

If exp is negative but \(a\) is not invertible modulo \(n\), an exception is raised.

ulong n_powmod2(ulong a, mp_limb_signed_t exp, ulong n)

Returns (a^exp) % n. We require \(0 \leq a < n\), but there are no restrictions on \(n\) or on exp, i.e.it can be negative.

This is implemented by precomputing an inverse limb and calling the preinv version of this function.

If exp is negative but \(a\) is not invertible modulo \(n\), an exception is raised.

ulong n_powmod2_ui_preinv(ulong a, ulong exp, ulong n, ulong ninv)

Returns (a^exp) % n given a precomputed inverse of \(n\) computed by n_preinvert_limb(). We require \(0 \leq a < n\), but there are no restrictions on \(n\). The exponent exp is unsigned and so can be larger than allowed by n_powmod2_preinv.

This is implemented as a standard binary powering algorithm using repeated squaring and reducing modulo \(n\) at each step.

ulong n_powmod2_fmpz_preinv(ulong a, const fmpz_t exp, ulong n, ulong ninv)

Returns (a^exp) % n given a precomputed inverse of \(n\) computed by n_preinvert_limb(). We require \(0 \leq a < n\), but there are no restrictions on \(n\). The exponent exp must not be negative.

This is implemented as a standard binary powering algorithm using repeated squaring and reducing modulo \(n\) at each step.

ulong n_sqrtmod(ulong a, ulong p)

If \(p\) is prime, compute a square root of \(a\) modulo \(p\) if \(a\) is a quadratic residue modulo \(p\), otherwise return \(0\).

If \(p\) is not prime the result is with high probability \(0\), indicating that \(p\) is not prime, or \(a\) is not a square modulo \(p\). Otherwise the result is meaningless.

Assumes that \(a\) is reduced modulo \(p\).

slong n_sqrtmod_2pow(ulong ** sqrt, ulong a, slong exp)

Computes all the square roots of a modulo 2^exp. The roots are stored in an array which is created and whose address is stored in the location pointed to by sqrt. The array of roots is allocated by the function but must be cleaned up by the user by calling flint_free. The number of roots is returned by the function. If a is not a quadratic residue modulo 2^exp then 0 is returned by the function and the location sqrt points to is set to NULL.

slong n_sqrtmod_primepow(ulong ** sqrt, ulong a, ulong p, slong exp)

Computes all the square roots of a modulo p^exp. The roots are stored in an array which is created and whose address is stored in the location pointed to by sqrt. The array of roots is allocated by the function but must be cleaned up by the user by calling flint_free. The number of roots is returned by the function. If a is not a quadratic residue modulo p^exp then 0 is returned by the function and the location sqrt points to is set to NULL.

slong n_sqrtmodn(ulong ** sqrt, ulong a, n_factor_t * fac)

Computes all the square roots of a modulo m given the factorisation of m in fac. The roots are stored in an array which is created and whose address is stored in the location pointed to by sqrt. The array of roots is allocated by the function but must be cleaned up by the user by calling flint_free. The number of roots is returned by the function. If a is not a quadratic residue modulo m then 0 is returned by the function and the location sqrt points to is set to NULL.

mp_limb_t n_mulmod_shoup(mp_limb_t w, mp_limb_t t, mp_limb_t w_precomp, mp_limb_t p)

Returns \(w t \bmod{p}\) given a precomputed scaled approximation of \(w / p\) computed by n_mulmod_precomp_shoup(). The value of \(p\) should be less than \(2^{\mathtt{FLINT_BITS} - 1}\). \(w\) and \(t\) should be less than \(p\). Works faster than n_mulmod2_preinv() if \(w\) fixed and \(t\) from array (for example, scalar multiplication of vector).

mp_limb_t n_mulmod_precomp_shoup(mp_limb_t w, mp_limb_t p)

Returns \(w'\), scaled approximation of \(w / p\). \(w'\) is equal to the integer part of \(w * 2^{\mathtt{FLINT_BITS}} / p\).

Prime number generation and counting

void n_primes_init(n_primes_t iter)

Initialises the prime number iterator iter for use.

void n_primes_clear(n_primes_t iter)

Clears memory allocated by the prime number iterator iter.

ulong n_primes_next(n_primes_t iter)

Returns the next prime number and advances the state of iter. The first call returns 2.

Small primes are looked up from flint_small_primes. When this table is exhausted, primes are generated in blocks by calling n_primes_sieve_range.

void n_primes_jump_after(n_primes_t iter, ulong n)

Changes the state of iter to start generating primes after \(n\) (excluding \(n\) itself).

void n_primes_extend_small(n_primes_t iter, ulong bound)

Extends the table of small primes in iter to contain at least two primes larger than or equal to bound.

void n_primes_sieve_range(n_primes_t iter, ulong a, ulong b)

Sets the block endpoints of iter to the smallest and largest odd numbers between \(a\) and \(b\) inclusive, and sieves to mark all odd primes in this range. The iterator state is changed to point to the first number in the sieved range.

void n_compute_primes(ulong num_primes)

Precomputes at least num_primes primes and their double precomputed inverses and stores them in an internal cache. Assuming that FLINT has been built with support for thread-local storage, each thread has its own cache.

const ulong * n_primes_arr_readonly(ulong num_primes)

Returns a pointer to a read-only array of the first num_primes prime numbers. The computed primes are cached for repeated calls. The pointer is valid until the user calls n_cleanup_primes in the same thread.

const double * n_prime_inverses_arr_readonly(ulong n)

Returns a pointer to a read-only array of inverses of the first num_primes prime numbers. The computed primes are cached for repeated calls. The pointer is valid until the user calls n_cleanup_primes in the same thread.

void n_cleanup_primes()

Frees the internal cache of prime numbers used by the current thread. This will invalidate any pointers returned by n_primes_arr_readonly or n_prime_inverses_arr_readonly.

ulong n_nextprime(ulong n, int proved)

Returns the next prime after \(n\). Assumes the result will fit in an ulong. If proved is \(0\), i.e.false, the prime is not proven prime, otherwise it is.

ulong n_prime_pi(ulong n)

Returns the value of the prime counting function \(\pi(n)\), i.e.the number of primes less than or equal to \(n\). The invariant n_prime_pi(n_nth_prime(n)) == n.

Currently, this function simply extends the table of cached primes up to an upper limit and then performs a binary search.

void n_prime_pi_bounds(ulong *lo, ulong *hi, ulong n)

Calculates lower and upper bounds for the value of the prime counting function lo <= pi(n) <= hi. If lo and hi point to the same location, the high value will be stored.

This does a table lookup for small values, then switches over to some proven bounds.

The upper approximation is \(1.25506 n / \ln n\), and the lower is \(n / \ln n\). These bounds are due to Rosser and Schoenfeld citep{RosSch1962} and valid for \(n \geq 17\).

We use the number of bits in \(n\) (or one less) to form an approximation to \(\ln n\), taking care to use a value too small or too large to maintain the inequality.

ulong n_nth_prime(ulong n)

Returns the \(n`th prime number `p_n\), using the mathematical indexing convention \(p_1 = 2, p_2 = 3, \dotsc\).

This function simply ensures that the table of cached primes is large enough and then looks up the entry.

void n_nth_prime_bounds(ulong *lo, ulong *hi, ulong n)

Calculates lower and upper bounds for the \(n`th prime number `p_n\), lo <= p_n <= hi. If lo and hi point to the same location, the high value will be stored. Note that this function will overflow for sufficiently large \(n\).

We use the following estimates, valid for \(n > 5\): begin{align*} p_n & > n (ln n + ln ln n - 1) \ p_n & < n (ln n + ln ln n) \ p_n & < n (ln n + ln ln n - 0.9427) quad (n geq 15985) end{align*}

The first inequality was proved by Dusart citep{Dus1999}, and the last is due to Massias and Robin citep{MasRob1996}. For a further overview, see url{http://primes.utm.edu/howmany.shtml}.

We bound \(\ln n\) using the number of bits in \(n\) as in n_prime_pi_bounds(), and estimate \(\ln \ln n\) to the nearest integer; this function is nearly constant.

% P. Dusart, “The kth prime is greater than k(ln k+ln ln k-1) for k> 2,” % Math. Comp., 68:225 (January 1999) 411–415.

% J. Massias and G. Robin, “Bornes effectives pour certaines fonctions % concernant les nombres premiers,” J. Theorie Nombres Bordeaux, 8 % (1996) 215-242.

Primality testing

int n_is_oddprime_small(ulong n)

Returns \(1\) if \(n\) is an odd prime smaller than FLINT_ODDPRIME_SMALL_CUTOFF. Expects \(n\) to be odd and smaller than the cutoff.

This function merely uses a lookup table with one bit allocated for each odd number up to the cutoff.

int n_is_oddprime_binary(ulong n)

This function performs a simple binary search through the table of cached primes for \(n\). If it exists in the array it returns \(1\), otherwise \(0\). For the algorithm to operate correctly \(n\) should be odd and at least \(17\).

Lower and upper bounds are computed with n_prime_pi_bounds(). Once we have bounds on where to look in the table, we refine our search with a simple binary algorithm, taking the top or bottom of the current interval as necessary.

int n_is_prime_pocklington(ulong n, ulong iterations)

Tests if \(n\) is a prime using the Pocklington–Lehmer primality test. If \(1\) is returned \(n\) has been proved prime. If \(0\) is returned \(n\) is composite. However \(-1\) may be returned if nothing was proved either way due to the number of iterations being too small.

The most time consuming part of the algorithm is factoring \(n - 1\). For this reason n_factor_partial() is used, which uses a combination of trial factoring and Hart’s one line factor algorithm citep{Har2009} to try to quickly factor \(n - 1\). Additionally if the cofactor is less than the square root of \(n - 1\) the algorithm can still proceed.

One can also specify a number of iterations if less time should be taken. Simply set this to `` WORD(0)`` if this is irrelevant. In most cases a greater number of iterations will not significantly affect timings as most of the time is spent factoring.

See url{http://mathworld.wolfram.com/PocklingtonsTheorem.html} for a description of the algorithm.

int n_is_prime_pseudosquare(ulong n)

Tests if \(n\) is a prime according to citep[Theorem 2.7]{LukPatWil1996}.

% “Some results on pseudosquares” by Lukes, Patterson and Williams, % Math. Comp. vol 65, No. 213. pp 361-372. See % http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00678-3/ % S0025-5718-96-00678-3.pdf

We first factor \(N\) using trial division up to some limit \(B\). In fact, the number of primes used in the trial factoring is at most FLINT_PSEUDOSQUARES_CUTOFF.

Next we compute \(N/B\) and find the next pseudosquare \(L_p\) above this value, using a static table as per url{http://oeis.org/A002189/b002189.txt}.

As noted in the text, if \(p\) is prime then Step 3 will pass. This test rejects many composites, and so by this time we suspect that \(p\) is prime. If \(N\) is \(3\) or \(7\) modulo \(8\), we are done, and \(N\) is prime.

We now run a probable prime test, for which no known counterexamples are known, to reject any composites. We then proceed to prove \(N\) prime by executing Step 4. In the case that \(N\) is \(1\) modulo \(8\), if Step 4 fails, we extend the number of primes \(p_i\) at Step 3 and hope to find one which passes Step 4. We take the test one past the largest \(p\) for which we have pseudosquares \(L_p\) tabulated, as this already corresponds to the next \(L_p\) which is bigger than \(2^{64}\) and hence larger than any prime we might be testing.

As explained in the text, Condition 4 cannot fail if \(N\) is prime.

The possibility exists that the probable prime test declares a composite prime. However in that case an error is printed, as that would be of independent interest.

int n_is_prime(ulong n)

Tests if \(n\) is a prime. This first sieves for small prime factors, then simply calls n_is_probabprime(). This has been checked against the tables of Feitsma and Galway url{http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html} and thus constitutes a check for primality (rather than just pseudoprimality) up to \(2^64\).

In future, this test may produce and check a certificate of primality. This is likely to be significantly slower for prime inputs.

int n_is_strong_probabprime_precomp(ulong n, double npre, ulong a, ulong d)

Tests if \(n\) is a strong probable prime to the base \(a\). We require that \(d\) is set to the largest odd factor of \(n - 1\) and npre is a precomputed inverse of \(n\) computed with n_precompute_inverse(). We also require that \(n < 2^{53}\), \(a\) to be reduced modulo \(n\) and not \(0\) and \(n\) to be odd.

If we write \(n - 1 = 2^s d\) where \(d\) is odd then \(n\) is a strong probable prime to the base \(a\), i.e.an \(a\)-SPRP, if either \(a^d = 1 \pmod n\) or \((a^d)^{2^r} = -1 \pmod n\) for some \(r\) less than \(s\).

A description of strong probable primes is given here: url{http://mathworld.wolfram.com/StrongPseudoprime.html}

int n_is_strong_probabprime2_preinv(ulong n, ulong ninv, ulong a, ulong d)

Tests if \(n\) is a strong probable prime to the base \(a\). We require that \(d\) is set to the largest odd factor of \(n - 1\) and npre is a precomputed inverse of \(n\) computed with n_preinvert_limb(). We require a to be reduced modulo \(n\) and not \(0\) and \(n\) to be odd.

If we write \(n - 1 = 2^s d\) where \(d\) is odd then \(n\) is a strong probable prime to the base \(a\) (an \(a\)-SPRP) if either \(a^d = 1 \pmod n\) or \((a^d)^{2^r} = -1 \pmod n\) for some \(r\) less than \(s\).

A description of strong probable primes is given here: url{http://mathworld.wolfram.com/StrongPseudoprime.html}

int n_is_probabprime_fermat(ulong n, ulong i)

Returns \(1\) if \(n\) is a base \(i\) Fermat probable prime. Requires \(1 < i < n\) and that \(i\) does not divide \(n\).

By Fermat’s Little Theorem if \(i^{n-1}\) is not congruent to \(1\) then \(n\) is not prime.

int n_is_probabprime_fibonacci(ulong n)

Let \(F_j\) be the \(j`th element of the Fibonacci sequence `0, 1, 1, 2, 3, 5, \dotsc\), starting at \(j = 0\). Then if \(n\) is prime we have \(F_{n - (n/5)} = 0 \pmod n\), where \((n/5)\) is the Jacobi symbol.

For further details, see citep[pp. 142]{CraPom2005}.

We require that \(n\) is not divisible by \(2\) or \(5\).

int n_is_probabprime_BPSW(ulong n)

Implements a Baillie–Pomerance–Selfridge–Wagstaff probable primality test. This is a variant of the usual BPSW test (which only uses strong base-2 probable prime and Lucas-Selfridge tests, see Baillie and Wagstaff citep{BaiWag1980}).

This implementation makes use of a weakening of the usual Baillie-PSW test given in citep{Chen2003}, namely replacing the Lucas test with a Fibonacci test when \(n \equiv 2, 3 \pmod{5}\), (see also the comment on page 143 of citep{CraPom2005}) regarding Fibonacci pseudoprimes.

There are no known counterexamples to this being a primality test.

Up to \(2^64\) the test we use has been checked against tables of pseudoprimes. Thus it is a primality test up to this limit.

int n_is_probabprime_lucas(ulong n)

For details on Lucas pseudoprimes, see citep[pp. 143]{CraPom2005}.

We implement a variant of the Lucas pseudoprime test similar to that described by Baillie and Wagstaff citep{BaiWag1980}.

int n_is_probabprime(ulong n)

Tests if \(n\) is a probable prime. Up to FLINT_ODDPRIME_SMALL_CUTOFF this algorithm uses n_is_oddprime_small() which uses a lookup table.

Next it calls n_compute_primes() with the maximum table size and uses this table to perform a binary search for \(n\) up to the table limit.

Then up to \(1050535501\) it uses a number of strong probable prime tests, n_is_strong_probabprime_preinv(), etc., for various bases. The output of the algorithm is guaranteed to be correct up to this bound due to exhaustive tables, described at url{http://uucode.com/obf/dalbec/alg.html}.

Beyond that point the BPSW probabilistic primality test is used, by calling the function n_is_probabprime_BPSW(). There are no known counterexamples, and it has been checked against the tables of Feitsma and Galway and up to the accuracy of those tables, this is an exhaustive check up to \(2^64\), i.e. there are no counterexamples.

Square root and perfect power testing

ulong n_sqrt(ulong a)

Computes the integer truncation of the square root of \(a\).

The implementation uses a call to the IEEE floating point sqrt function. The integer itself is represented by the nearest double and its square root is computed to the nearest place. If \(a\) is one below a square, the rounding may be up, whereas if it is one above a square, the rounding will be down. Thus the square root may be one too large in some instances which we then adjust by checking if we have the right value. We also have to be careful when the square of this too large value causes an overflow. The same assumptions hold for a single precision float provided the square root itself can be represented in a single float, i.e.for \(a < 281474976710656 = 2^{46}\).

ulong n_sqrtrem(ulong * r, ulong a)

Computes the integer truncation of the square root of \(a\).

The integer itself is represented by the nearest double and its square root is computed to the nearest place. If \(a\) is one below a square, the rounding may be up, whereas if it is one above a square, the rounding will be down. Thus the square root may be one too large in some instances which we then adjust by checking if we have the right value. We also have to be careful when the square of this too large value causes an overflow. The same assumptions hold for a single precision float provided the square root itself can be represented in a single float, i.e. for \(a < 281474976710656 = 2^{46}\).

The remainder is computed by subtracting the square of the computed square root from \(a\).

int n_is_square(ulong x)

Returns \(1\) if \(x\) is a square, otherwise \(0\).

This code first checks if \(x\) is a square modulo \(64\), \(63 = 3 \times 3 \times 7\) and \(65 = 5 \times 13\), using lookup tables, and if so it then takes a square root and checks that the square of this equals the original value.

int n_is_perfect_power235(ulong n)

Returns \(1\) if \(n\) is a perfect square, cube or fifth power.

This function uses a series of modular tests to reject most non 235-powers. Each modular test returns a value from 0 to 7 whose bits respectively indicate whether the value is a square, cube or fifth power modulo the given modulus. When these are logically AND-ed together, this gives a powerful test which will reject most non-235 powers.

If a bit remains set indicating it may be a square, a standard square root test is performed. Similarly a cube root or fifth root can be taken, if indicated, to determine whether the power of that root is exactly equal to \(n\).

int n_is_perfect_power(ulong * root, ulong n)

If \(n = r^k\), return \(k\) and set root to \(r\). Note that \(0\) and \(1\) are considered squares. No guarantees are made about \(r\) or \(k\) being the minimum possible value.

ulong n_rootrem(ulong* remainder, ulong n, ulong root)

This function uses the Newton iteration method to calculate the nth root of a number. First approximation is calculated by an algorithm mentioned in this article : url{http://en.wikipedia.org/wiki/Fast_inverse_square_root}. Instead of the inverse square root, the nth root is calculated.

Returns the integer part of n ^ 1/root. Remainder is set as n - base^root. In case \(n < 1\) or root < 1, \(0\) is returned.

ulong n_cbrt(ulong n)

This function returns the integer truncation of the cube root of \(n\). First approximation is calculated by an algorithm mentioned in this article : url{http://en.wikipedia.org/wiki/Fast_inverse_square_root}. Instead of the inverse sqare root, the cube root is calculated. This functions uses different algorithms to calculate the cube root, depending upon the size of \(n\). For numbers greater than \(2^46\), it uses n_cbrt_chebyshev_approx(). Otherwise, it makes use of the iteration, \(x \leftarrow x - (x*x*x - a)*x/(2*x*x*x + a)\) for getting a good estimate, as mentioned in the paper by W. Kahan citep{Kahan1991}.

ulong n_cbrt_newton_iteration(ulong n)

This function returns the integer truncation of the cube root of \(n\). Makes use of Newton iterations to get a close value, and then adjusts the estimate so as to get the correct value.

ulong n_cbrt_binary_search(ulong n)

This function returns the integer truncation of the cube root of \(n\). Uses binary search to get the correct value.

ulong n_cbrt_chebyshef_approx(ulong n)

This function returns the integer truncation of the cube root of \(n\). The number is first expressed in the form x * 2^exp. This ensures \(x\) is in the range [0.5, 1]. Cube root of x is calculated using Chebyshev’s approximation polynomial for the function \(y = x^1/3\). The values of the coefficient are calculated from the python module mpmath, url{http://mpmath.org}, using the function chebyfit. x is multiplied by 2^exp and the cube root of 1, 2 or 4 (according to exp%3).

ulong n_cbrtrem(ulong* remainder, ulong n)

This function returns the integer truncation of the cube root of \(n\). Remainder is set as \(n\) minus the cube of the value returned.

Factorisation

int n_remove(ulong * n, ulong p)

Removes the highest possible power of \(p\) from \(n\), replacing \(n\) with the quotient. The return value is that highest power of \(p\) that divided \(n\). Assumes \(n\) is not \(0\).

For \(p = 2\) trailing zeroes are counted. For other primes \(p\) is repeatedly squared and stored in a table of powers with the current highest power of \(p\) removed at each step until no higher power can be removed. The algorithm then proceeds down the power tree again removing powers of \(p\) until none remain.

int n_remove2_precomp(ulong * n, ulong p, double ppre)

Removes the highest possible power of \(p\) from \(n\), replacing \(n\) with the quotient. The return value is that highest power of \(p\) that divided \(n\). Assumes \(n\) is not \(0\). We require ppre to be set to a precomputed inverse of \(p\) computed with n_precompute_inverse().

For \(p = 2\) trailing zeroes are counted. For other primes \(p\) we make repeated use of n_divrem2_precomp() until division by \(p\) is no longer possible.

void n_factor_insert(n_factor_t * factors, ulong p, ulong exp)

Inserts the given prime power factor p^exp into the n_factor_t factors. See the documentation for n_factor_trial() for a description of the n_factor_t type.

The algorithm performs a simple search to see if \(p\) already exists as a prime factor in the structure. If so the exponent there is increased by the supplied exponent. Otherwise a new factor p^exp is added to the end of the structure.

There is no test code for this function other than its use by the various factoring functions, which have test code.

ulong n_factor_trial_range(n_factor_t * factors, ulong n, ulong start, ulong num_primes)

Trial factor \(n\) with the first num_primes primes, but starting at the prime with index start (counting from zero).

One requires an initialised n_factor_t structure, but factors will be added by default to an already used n_factor_t. Use the function n_factor_init() defined in ulong_extras if initialisation has not already been completed on factors.

Once completed, num will contain the number of distinct prime factors found. The field \(p\) is an array of ulong’s containing the distinct prime factors, exp an array containing the corresponding exponents.

The return value is the unfactored cofactor after trial factoring is done.

The function calls n_compute_primes() automatically. See the documentation for that function regarding limits.

The algorithm stops when the current prime has a square exceeding \(n\), as no prime factor of \(n\) can exceed this unless \(n\) is prime.

The precomputed inverses of all the primes computed by n_compute_primes() are utilised with the n_remove2_precomp() function.

ulong n_factor_trial(n_factor_t * factors, ulong n, ulong num_primes)

This function calls n_factor_trial_range(), with the value of \(0\) for start. By default this adds factors to an already existing n_factor_t or to a newly initialised one.

ulong n_factor_power235(ulong *exp, ulong n)

Returns \(0\) if \(n\) is not a perfect square, cube or fifth power. Otherwise it returns the root and sets exp to either \(2\), \(3\) or \(5\) appropriately.

This function uses a series of modular tests to reject most non 235-powers. Each modular test returns a value from 0 to 7 whose bits respectively indicate whether the value is a square, cube or fifth power modulo the given modulus. When these are logically AND-ed together, this gives a powerful test which will reject most non-235 powers.

If a bit remains set indicating it may be a square, a standard square root test is performed. Similarly a cube root or fifth root can be taken, if indicated, to determine whether the power of that root is exactly equal to \(n\).

ulong n_factor_one_line(ulong n, ulong iters)

This implements Bill Hart’s one line factoring algorithm citep{Har2009}. It is a variant of Fermat’s algorithm which cycles through a large number of multipliers instead of incrementing the square root. It is faster than SQUFOF for \(n\) less than about \(2^{40}\).

ulong n_factor_lehman(ulong n)

Lehman’s factoring algorithm. Currently works up to \(10^{16}\), but is not particularly efficient and so is not used in the general factor function. Always returns a factor of \(n\).

ulong n_factor_SQUFOF(ulong n, ulong iters)

Attempts to split \(n\) using the given number of iterations of SQUFOF. Simply set iters to `` WORD(0)`` for maximum persistence.

The version of SQUFOF imlemented here is as described by Gower and Wagstaff citep{GowWag2008}.

% Jason Gower and Sam Wagstaff % “Square form factoring” % Math. Comp. 77, 2008, pp 551-588, see: % http://www.ams.org/mcom/2008-77-261/S0025-5718-07-02010-8/home.html

We start by trying SQUFOF directly on \(n\). If that fails we multiply it by each of the primes in flint_primes_small in turn. As this multiplication may result in a two limb value we allow this in our implementation of SQUFOF. As SQUFOF works with values about half the size of \(n\) it only needs single limb arithmetic internally.

If SQUFOF fails to factor \(n\) we return \(0\), however with iters large enough this should never happen.

void n_factor(n_factor_t * factors, ulong n, int proved)

Factors \(n\) with no restrictions on \(n\). If the prime factors are required to be checked with a primality test, one may set proved to \(1\), otherwise set it to \(0\), and they will only be probable primes. N.B: at the present there is no difference because the probable prime tests have been exhaustively tested up to \(2^64\).

However, in future, this flag may produce and separately check a primality certificate. This may be quite slow (and probably no less reliable in practice).

For details on the n_factor_t structure, see n_factor_trial().

This function first tries trial factoring with a number of primes specified by the constant FLINT_FACTOR_TRIAL_PRIMES. If the cofactor is \(1\) or prime the function returns with all the factors.

Otherwise, the cofactor is placed in the array factor_arr. Whilst there are factors remaining in there which have not been split, the algorithm continues. At each step each factor is first checked to determine if it is a perfect power. If so it is replaced by the power that has been found. Next if the factor is small enough and composite, in particular, less than FLINT_FACTOR_ONE_LINE_MAX then n_factor_one_line() is called with FLINT_FACTOR_ONE_LINE_ITERS to try and split the factor. If that fails or the factor is too large for n_factor_one_line() then n_factor_SQUFOF() is called, with FLINT_FACTOR_SQUFOF_ITERS. If that fails an error results and the program aborts. However this should not happen in practice.

ulong n_factor_trial_partial(n_factor_t * factors, ulong n, ulong * prod, ulong num_primes, ulong limit)

Attempts trial factoring of \(n\) with the first num_primes primes, but stops when the product of prime factors so far exceeds limit.

One requires an initialised n_factor_t structure, but factors will be added by default to an already used n_factor_t. Use the function n_factor_init() defined in ulong_extras if initialisation has not already been completed on factors.

Once completed, num will contain the number of distinct prime factors found. The field \(p\) is an array of ulong’s containing the distinct prime factors, exp an array containing the corresponding exponents.

The return value is the unfactored cofactor after trial factoring is done. The value prod will be set to the product of the factors found.

The function calls n_compute_primes() automatically. See the documentation for that function regarding limits.

The algorithm stops when the current prime has a square exceeding \(n\), as no prime factor of \(n\) can exceed this unless \(n\) is prime.

The precomputed inverses of all the primes computed by n_compute_primes() are utilised with the n_remove2_precomp() function.

ulong n_factor_partial(n_factor_t * factors, ulong n, ulong limit, int proved)

Factors \(n\), but stops when the product of prime factors so far exceeds limit.

One requires an initialised n_factor_t structure, but factors will be added by default to an already used n_factor_t. Use the function n_factor_init() defined in ulong_extras if initialisation has not already been completed on factors.

On exit, num will contain the number of distinct prime factors found. The field \(p\) is an array of ulong’s containing the distinct prime factors, exp an array containing the corresponding exponents.

The return value is the unfactored cofactor after factoring is done.

The factors are proved prime if proved is \(1\), otherwise they are merely probably prime.

ulong n_factor_pp1(ulong n, ulong B1, ulong c)

Factors \(n\) using Williams’ \(p + 1\) factoring algorithm, with prime limit set to \(B1\). We require \(c\) to be set to a random value. Each trial of the algorithm with a different value of \(c\) gives another chance to factor \(n\), with roughly exponentially decreasing chance of finding a missing factor. If \(p + 1\) (or \(p - 1\)) is not smooth for any factor \(p\) of \(n\), the algorithm will never succeed. The value \(c\) should be less than \(n\) and greater than \(2\).

If the algorithm succeeds, it returns the factor, otherwise it returns \(0\) or \(1\) (the trivial factors modulo \(n\)).

int n_factor_pollard_brent_single(mp_limb_t *factor, mp_limb_t n, mp_limb_t ninv, mp_limb_t ai, mp_limb_t xi, mp_limb_t normbits, mp_limb_t max_iters)

Pollard Rho algorithm (with Brent modification) for integer factorization. Assumes that the \(n\) is not prime. \(factor\) is set as the factor if found. It is not assured that the factor found will be prime. Does not compute the complete factorization, just one factor. Returns 1 if factorization is successfull (non trivial factor is found), else returns 0. Assumes \(n\) is normalized, (shifted by normbits bits), and takes as input a precomputed inverse of \(n\) as computed by n_preinvert_limb(). \(ai\) and \(xi\) should also be shifted left by \(normbits\).

\(ai\) is the constant of the polynomial used, \(xi\) is the initial value. \(max_iters\) is the number of iterations tried in process of finding the cycle.

The algorithm used is a modification of the original Pollard Rho algorithm, suggested by Richard Brent in the paper, availible at http://maths-people.anu.edu.au/ brent/pd/rpb051i.pdf

int n_factor_pollard_brent(mp_limb_t *factor, flint_rand_t state, mp_limb_t n_in, mp_limb_t max_tries, mp_limb_t max_iters)

Pollard Rho algorithm, modified as suggested by Richar Brent. Makes a call to n_factor_pollard_brent_single. The input parameters ai and xi for n_factor_pollard_brent_single are selected at random.

If the algorithm fails to find a non trivial factor in one call, it tries again (this time with a different set of random values). This process is repeated a maximum of \(max_tries\) times.

Assumes \(n\) is not prime. \(factor\) is set as the factor found, if factorization is successful. In such a case, 1 is returned. Otherwise, 0 is returned. Factor discovered is not necessarily prime.

Arithmetic functions

int n_moebius_mu(ulong n)

Computes the Moebius function \(\mu(n)\), which is defined as \(\mu(n) = 0\) if \(n\) has a prime factor of multiplicity greater than \(1\), \(\mu(n) = -1\) if \(n\) has an odd number of distinct prime factors, and \(\mu(n) = 1\) if \(n\) has an even number of distinct prime factors. By convention, \(\mu(0) = 0\).

For even numbers, we use the identities \(\mu(4n) = 0\) and \(\mu(2n) = - \mu(n)\). Odd numbers up to a cutoff are then looked up from a precomputed table storing \(\mu(n) + 1\) in groups of two bits.

For larger \(n\), we first check if \(n\) is divisible by a small odd square and otherwise call n_factor() and count the factors.

void n_moebius_mu_vec(int * mu, ulong len)

Computes \(\mu(n)\) for n = 0, 1, ..., len - 1. This is done by sieving over each prime in the range, flipping the sign of \(\mu(n)\) for every multiple of a prime \(p\) and setting \(\mu(n) = 0\) for every multiple of \(p^2\).

int n_is_squarefree(ulong n)

Returns \(0\) if \(n\) is divisible by some perfect square, and \(1\) otherwise. This simply amounts to testing whether \(\mu(n) \neq 0\). As special cases, \(1\) is considered squarefree and \(0\) is not considered squarefree.

ulong n_euler_phi(ulong n)

Computes the Euler totient function \(\phi(n)\), counting the number of positive integers less than or equal to \(n\) that are coprime to \(n\).

Factorials

ulong n_factorial_fast_mod2_preinv(ulong n, ulong p, ulong pinv)

Returns \(n! \bmod p\) given a precomputed inverse of \(p\) as computed by n_preinvert_limb(). \(p\) is not required to be a prime, but no special optimisations are made for composite \(p\). Uses fast multipoint evaluation, running in about \(O(n^{1/2})\) time.

ulong n_factorial_mod2_preinv(ulong n, ulong p, ulong pinv)

Returns \(n! \bmod p\) given a precomputed inverse of \(p\) as computed by n_preinvert_limb(). \(p\) is not required to be a prime, but no special optimisations are made for composite \(p\).

Uses a lookup table for small \(n\), otherwise computes the product if \(n\) is not too large, and calls the fast algorithm for extremely large \(n\).

Primitive Roots and Discrete Logarithms

ulong n_primitive_root_prime_prefactor(ulong p, n_factor_t * factors)

Returns a primitive root for the multiplicative subgroup of \(\Z/p\Z\) where \(p\) is prime given the factorisation (\(factors\)) of \(p - 1\).

ulong n_primitive_root_prime(ulong p)

Returns a primitive root for the multiplicative subgroup of \(\Z/p\Z\) where \(p\) is prime.

ulong n_discrete_log_bsgs(ulong b, ulong a, ulong n)

Returns the discrete logarithm of \(b\) with respect to \(a\) in the multiplicative subgroup of \(\Z/n\Z\) when \(\Z/n\Z\) is cyclic That is, it returns an number \(x\) such that \(a^x = b \bmod n\). The multiplicative subgroup is only cyclic when \(n\) is \(2\), \(4\), \(p^k\), or \(2p^k\) where \(p\) is an odd prime and \(k\) is a positive integer.

Elliptic curve method for factorization of code{mp_limb_t}

void n_factor_ecm_double(mp_limb_t *x, mp_limb_t *z, mp_limb_t x0, mp_limb_t z0, mp_limb_t n, n_ecm_t n_ecm_inf);

Sets the point \((x : z)\) to two times \((x_0 : z_0)\) modulo \(n\) according to the formula

x = (x_0 + z_0)^2 \cdot (x_0 - z_0)^2 \mod n,

z = 4 x_0 z_0 \left((x_0 - z_0)^2 + 4a_{24}x_0z_0\right) \mod n.

This group doubling is valid only for points expressed in Montgomery projective coordinates.

void n_factor_ecm_add(mp_limb_t *x, mp_limb_t *z, mp_limb_t x1, mp_limb_t z1, mp_limb_t x2, mp_limb_t z2, mp_limb_t x0, mp_limb_t z0, mp_limb_t n, n_ecm_t n_ecm_inf)

Sets the point \((x : z)\) to the sum of \((x_1 : z_1)\) and \((x_2 : z_2)\) modulo \(n\), given the difference \((x_0 : z_0)\) according to the formula

This group doubling is valid only for points expressed in Montgomery projective coordinates.

void n_factor_ecm_mul_montgomery_ladder(mp_limb_t *x, mp_limb_t *z, mp_limb_t x0, mp_limb_t z0, mp_limb_t k, mp_limb_t n, n_ecm_t n_ecm_inf)

Montgomery ladder algorithm for scalar multiplication of elliptic points.

Sets the point \((x : z)\) to \(k(x_0 : z_0)\) modulo \(n\).

Valid only for points expressed in Montgomery projective coordinates.

int n_factor_ecm_select_curve(mp_limb_t *f, mp_limb_t sigma, mp_limb_t n, n_ecm_t n_ecm_inf)

Selects a random elliptic curve given a random integer sigma, according to Suyama’s parameterization. If the factor is found while selecting the curve, \(1\) is returned. In case the curve found is not suitable, \(0\) is returned.

Also selects the initial point \(x_0\), and the value of \((a + 2)/4\), where \(a\) is a curve parameter. Sets \(z_0\) as \(1\) (shifted left by n_ecm_inf->normbits. All these are stored in the n_ecm_t struct.

The curve selected is of Montgomery form, the points selected satisfy the curve and are projective coordinates.

int n_factor_ecm_stage_I(mp_limb_t *f, const mp_limb_t *prime_array, mp_limb_t num, mp_limb_t B1, mp_limb_t n, n_ecm_t n_ecm_inf)

StageI implementation of the ECM algorithm.

f is set as the factor if found. num is number of prime numbers \(<=\) the bound B1. prime_array is an array of first B1 primes. \(n\) is the number being factored.

If the factor is found, \(1\) is returned, otherwise \(0\).

int n_factor_ecm_stage_II(mp_limb_t *f, mp_limb_t B1, mp_limb_t B2, mp_limb_t P, mp_limb_t n, n_ecm_t n_ecm_inf)

StageII implementation of the ECM algorithm.

f is set as the factor if found. B1, B2 are the two bounds. P is the primorial (approximately equal to \(\sqrt{B2}\)). \(n\) is the number being factored.

If the factor is found, \(1\) is returned, otherwise \(0\).

int n_factor_ecm(mp_limb_t *f, mp_limb_t curves, mp_limb_t B1, mp_limb_t B2, flint_rand_t state, mp_limb_t n)

Outer wrapper function for the ECM algorithm. It factors \(n\) which must fit into a mp_limb_t.

The function calls stageI andII, and the precomputations (builds prime_array for stageI, GCD_table and prime_table for stageII).

f is set as the factor if found. curves is the number of random curves being tried. B1, B2 are the two bounds or stageI and stageII. \(n\) is the number being factored.

If a factor is found in stageI, \(1\) is returned. If a factor is found in stageII, \(2\) is returned. If a factor is found while selecting the curve, \(-1\) is returned. Otherwise \(0\) is returned.