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90 lines
3.2 KiB
90 lines
3.2 KiB
// Number theoretic operations.
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#ifndef _CL_NUMTHEORY_H
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#define _CL_NUMTHEORY_H
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#include "cln/number.h"
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#include "cln/integer.h"
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#include "cln/modinteger.h"
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#include "cln/condition.h"
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namespace cln {
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// jacobi(a,b) returns the Jacobi symbol
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// ( a )
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// ( --- )
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// ( b )
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// a, b must be integers, b > 0, b odd. The result is 0 iff gcd(a,b) > 1.
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extern int jacobi (sintV a, sintV b);
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extern int jacobi (const cl_I& a, const cl_I& b);
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// isprobprime(n), n integer > 0,
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// returns true when n is probably prime.
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// This is pretty quick, but no caching is done.
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extern cl_boolean isprobprime (const cl_I& n);
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// nextprobprime(x) returns the smallest probable prime >= x.
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extern const cl_I nextprobprime (const cl_R& x);
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#if 0
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// primitive_root(R) of R = Z/pZ, with p a probable prime,
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// returns
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// either a generator of (Z/pZ)^*, assuming p is prime, or
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// a proof that p is not prime, maybe even a non-trivial factor of p.
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struct primitive_root_t {
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cl_composite_condition* condition;
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cl_MI gen;
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// Constructors.
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primitive_root_t (cl_composite_condition* c) : condition (c) {}
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primitive_root_t (const cl_MI& g) : condition (NULL), gen (g) {}
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};
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extern const primitive_root_t primitive_root (const cl_modint_ring& R);
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#endif
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// sqrt_mod_p(R,x) where x is an element of R = Z/pZ, with p a probable prime,
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// returns
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// either the square roots of x in R, assuming p is prime, or
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// a proof that p is not prime, maybe even a non-trivial factor of p.
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struct sqrt_mod_p_t {
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cl_composite_condition* condition;
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// If no condition:
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int solutions; // 0,1,2
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cl_I factor; // zero or non-trivial factor of p
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cl_MI solution[2]; // max. 2 solutions
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// Constructors.
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sqrt_mod_p_t () {}
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sqrt_mod_p_t (cl_composite_condition* c) : condition (c) {}
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sqrt_mod_p_t (int s) : condition (NULL), solutions (s) {}
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sqrt_mod_p_t (int s, const cl_MI& x0) : condition (NULL), solutions (s)
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{ solution[0] = x0; }
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sqrt_mod_p_t (int s, const cl_MI& x0, const cl_MI& x1) : condition (NULL), solutions (s)
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{ solution[0] = x0; solution[1] = x1; }
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};
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extern const sqrt_mod_p_t sqrt_mod_p (const cl_modint_ring& R, const cl_MI& x);
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// cornacchia1(d,p) solves x^2 + d*y^2 = p.
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// cornacchia4(d,p) solves x^2 + d*y^2 = 4*p.
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// d is an integer > 0, p is a probable prime.
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// It returns
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// either a nonnegative solution (x,y), if it exists, assuming p is prime, or
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// a proof that p is not prime, maybe even a non-trivial factor of p.
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struct cornacchia_t {
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cl_composite_condition* condition;
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// If no condition:
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int solutions; // 0,1
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// If solutions=1 and d > 4 (d > 64 for cornacchia4):
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// All solutions are (x,y), (-x,y), (x,-y), (-x,-y).
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cl_I solution_x; // x >= 0
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cl_I solution_y; // y >= 0
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// Constructors.
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cornacchia_t () {}
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cornacchia_t (cl_composite_condition* c) : condition (c) {}
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cornacchia_t (int s) : condition (NULL), solutions (s) {}
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cornacchia_t (int s, const cl_I& x, const cl_I& y) : condition (NULL), solutions (s), solution_x (x), solution_y (y) {}
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};
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extern const cornacchia_t cornacchia1 (const cl_I& d, const cl_I& p);
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extern const cornacchia_t cornacchia4 (const cl_I& d, const cl_I& p);
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} // namespace cln
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#endif /* _CL_NUMTHEORY_H */
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