# qsieve.h – Quadratic sieve

ulong qsieve_knuth_schroeppel(qs_t qs_inf)

  Return the Knuth-Schroeppel multiplier for the \(n\), integer to be factored based upon the Knuth-Schroeppel function.

ulong qsieve_primes_init(qs_t qs_inf)

  Compute the factor base prime along with there inverse for \(kn\), where \(k\) is Knuth-Schroeppel multiplier and \(n\) is the integer to be factored. It also computes the square root of \(kn\) modulo factor base primes.

ulong qsieve_primes_increment(qs_t qs_inf, ulong delta)

  It increase the number of factor base primes by amount ‘delta’ and calculate inverse of those primes along with the square root of \(kn\) modulo those primes.

void qsieve_init_A0(qs_t qs_inf)

  First it chooses the possible range of factor of \(A _0\), based on the number of bits in optimal value of \(A _0\). It tries to select range such that we have plenty of primes to choose from as well as number of factor in \(A _0\) are sufficient. For input of size less than 130 bit, this selection method doesn’t work therefore we randomly generate 2 or 3-subset of all the factor base prime as the factor of \(A _0\). Otherwise, if we have to select \(s\) factor for \(A _0\), we generate \(s - 1\)- subset from odd indices of the possible range of factor and then search last factor using binary search from the even indices of possible range of factor such that value of \(A _0\) is close to it’s optimal value.

void qsieve_next_A0(qs_t qs_inf)

  Find next candidate for \(A _0\) as follows: generate next lexicographic \(s - 1\)-subset from the odd indices of possible range of factor base and choose the last factor from even indices using binary search so that value \(A _0\) is close to it’s optimal value.

void qsieve_compute_pre_data(qs_t qs_inf)

  Precompute all the data associated with factor’s of \(A _0\), since \(A _0\) is going to be fixed for several \(A\).

void qsieve_init_poly_first(qs_t qs_inf)

  Initializes the value of \(A = q _0 * A _0\), where \(q _0\) is non-factor base prime. precompute the data necessary for generating different \(B\) value using grey code formula. Combine the data calculated for the factor of \(A _0\) along with the parameter \(q _0\) to obtain data as for factor of \(A\). It also calculates the sieve offset for all the factor base prime, for first polynomial.

void qsieve_init_poly_next(qs_t qs_inf, slong i)

  Generate next polynomial or next \(B\) value for particular \(A\) and also updates the sieve offsets for all the factor base prime, for this \(B\) value.

void qsieve_compute_C(fmpz_t C, qs_t qs_inf, qs_poly_t poly)

  Given \(A\) and \(B\), calculate \(C = (B ^2 - A) / N\).

void qsieve_do_sieving(qs_t qs_inf, unsigned char *sieve, qs_poly_t poly)

  First initialize the sieve array to zero, then for each \(p \in\) `factor base`, add \(\log_2(p)\) to the locations \(\operatorname{soln1} _p + i * p\) and \(\operatorname{soln2} _p + i * p\) for \(i = 0, 1, 2,\dots\), where \(\operatorname{soln1} _p\) and \(\operatorname{soln2} _p\) are the sieve offsets calculated for \(p\).

void qsieve_do_sieving2(qs_t qs_inf, unsigned char *sieve, qs_poly_t poly)

  Perform the same task as above but instead of sieving over whole array at once divide the array in blocks and then sieve over each block for all the primes in factor base.

slong qsieve_evaluate_candidate(qs_t qs_inf, ulong i, unsigned char *sieve, qs_poly_t poly)

  For location \(i\) in sieve array value at which, is greater than sieve threshold, check the value of \(Q(x)\) at position \(i\) for smoothness. If value is found to be smooth then store it for later processing, else check the residue for the partial if it is found to be partial then store it for late processing.

slong qsieve_evaluate_sieve(qs_t qs_inf, unsigned char *sieve, qs_poly_t poly)

  Scan the sieve array for location at, which accumulated value is greater than sieve threshold.

slong qsieve_collect_relations(qs_t qs_inf, unsigned char *sieve)

  Call for initialization of polynomial, sieving, and scanning of sieve for all the possible polynomials for particular hypercube i.e. \(A\).

void qsieve_write_to_file(qs_t qs_inf, ulong prime, const fmpz_t Y, const qs_poly_t poly)

  Write a relation to the file in a binary format as follows. First, write large prime of size `sizeof(ulong)`, in case of full relation it is 1. After this, write the number of small primes with size `sizeof(slong)`. Then, write the small primes, with a total size of `number_of_small_primes * sizeof(slong)`. Then, write the number of factors with a size of `sizeof(slong)`. After that, write the factors and their exponents in the format `factor_1, exponent_1, factor_2, ...`, all with a total size of `2 * number_of_factors * sizeof(slong)`. Then write `Y` with the size of `Y` first (size `sizeof(slong)`, that may be negative), and then its limbs (size `Y_size * sizeof(ulong)`).

hash_t *qsieve_get_table_entry(qs_t qs_inf, ulong prime)

  Return the pointer to the location of ‘prime’ is hash table if it exist, else create and entry for it in hash table and return pointer to that.

void qsieve_add_to_hashtable(qs_t qs_inf, ulong prime)

  Add ‘prime’ to the hast table.

relation_t qsieve_parse_relation(qs_t qs_inf)

  Read a relation from the file associated with qs_inf and parse it to obtain all the parameters of the relation.

relation_t qsieve_merge_relation(qs_t qs_inf, relation_t a, relation_t b)

  Given two partial relation having same large prime, merge them to obtain a full relation.

int qsieve_compare_relation(const void *a, const void *b)

  Compare two relation based on, first large prime, then number of factor and then offsets of factor in factor base.

int qsieve_remove_duplicates(relation_t *rel_list, slong num_relations)

  Remove duplicate from given list of relations by sorting relations in the list.

void qsieve_insert_relation2(qs_t qs_inf, relation_t *rel_list, slong num_relations)

  Given a list of relations, insert each relation from the list into the matrix for further processing.

int qsieve_process_relation(qs_t qs_inf)

  After we have accumulated required number of relations, first process the file by reading all the relations, removes singleton. Then merge all the possible partial to obtain full relations.

void qsieve_factor(fmpz_factor_t factors, const fmpz_t n)

  Factor \(n\) using the quadratic sieve method. It is required that \(n\) is not a prime and not a perfect power. There is no guarantee that the factors found will be prime, or distinct.