On Degrees of Modular Common Divisors and the Big Prime Algorithm

Algorithm 2

Big prime modular gcd algorithm.

Input: non-zero polynomials   .

Calculate their greatest common divisor  gcd.

(01) Calculate cont, cont in the Euclidean domain , choose their signs so that

    and .

(02) Set and .

(03) Calculate in the Euclidean domain by (25).

(04) Set and (they are positive by our selection of signs for and ).

(05) Calculate the positive in the Euclidean domain .

(06) Set .

(07) Compute the Landau-Mignotte bound by (20).

(08) Choose a new prime number .

(09) Apply the reduction to calculate the modular images .

(10) Calculate in the Euclidean domain .

(11) If

(12)  go to step (08);

(13) else

(14)  choose a such that the ;

(15)  call Algorithm 1 to calculate the preimage of ;

(16)  calculate in the Euclidean domain ;

(17)  set ;

(18)    if and

(19)      go to step (23);

(20)    else

(21)      set ;

(22)      go to step (08).

(23)  Output the result: .