7.2 Functions on modular integers

Given a modular integer ring R, the following members can be used.

cl_I R->modulus

This is the ring’s modulus, normalized to be nonnegative: abs(N).

cl_MI R->zero()

This returns 0 mod N.

cl_MI R->one()

This returns 1 mod N.

cl_MI R->canonhom (const cl_I& x)

This returns x mod N.

cl_I R->retract (const cl_MI& x)

This is a partial inverse function to R->canonhom. It returns the standard representative (>=0, <N) of x.

cl_MI R->random(random_state& randomstate)

cl_MI R->random()

This returns a random integer modulo N.

The following operations are defined on modular integers.

cl_modint_ring x.ring ()

Returns the ring to which the modular integer x belongs.

cl_MI operator+ (const cl_MI&, const cl_MI&)

Returns the sum of two modular integers. One of the arguments may also be a plain integer.

cl_MI operator- (const cl_MI&, const cl_MI&)

Returns the difference of two modular integers. One of the arguments may also be a plain integer.

cl_MI operator- (const cl_MI&)

Returns the negative of a modular integer.

cl_MI operator* (const cl_MI&, const cl_MI&)

Returns the product of two modular integers. One of the arguments may also be a plain integer.

cl_MI square (const cl_MI&)

Returns the square of a modular integer.

cl_MI recip (const cl_MI& x)

Returns the reciprocal x^-1 of a modular integer x. x must be coprime to the modulus, otherwise an error message is issued.

cl_MI div (const cl_MI& x, const cl_MI& y)

Returns the quotient x*y^-1 of two modular integers x, y. y must be coprime to the modulus, otherwise an error message is issued.

cl_MI expt_pos (const cl_MI& x, const cl_I& y)

y must be > 0. Returns x^y.

cl_MI expt (const cl_MI& x, const cl_I& y)

Returns x^y. If y is negative, x must be coprime to the modulus, else an error message is issued.

cl_MI operator<< (const cl_MI& x, const cl_I& y)

Returns x*2^y.

cl_MI operator>> (const cl_MI& x, const cl_I& y)

Returns x*2^-y. When y is positive, the modulus must be odd, or an error message is issued.

bool operator== (const cl_MI&, const cl_MI&)

bool operator!= (const cl_MI&, const cl_MI&)

Compares two modular integers, belonging to the same modular integer ring, for equality.

bool zerop (const cl_MI& x)

Returns true if x is 0 mod N.

The following output functions are defined (see also the chapter on input/output).

void fprint (std::ostream& stream, const cl_MI& x)

std::ostream& operator<< (std::ostream& stream, const cl_MI& x)

Prints the modular integer x on the stream. The output may depend on the global printer settings in the variable default_print_flags.

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