On Degrees of Modular Common Divisors and the Big Prime Algorithm

Algorithm 4

Big prime modular gcd algorithm with a preliminary estimate on divisor degree.

Input: non-zero polynomials .

Calculate their greatest common divisor  .

(01) Calculate , 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) Choose a prime number .

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

(09) Calculate in the Euclidean domain .

(10) Set .

(11) Calculate by (40) using the value of .

(12) Choose a new prime number .

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

(14) Calculate in the Euclidean domain .

(15) If

(16)  go to step (12).

(17) else

(18)  choose a such that the ;

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

(20)  calculate in the Euclidean domain ;

(21)  set ;

(22)    if and

(23)      go to step (27);

(24)     else

(25)      set ;

(26)      go to step (12).

(27) Output the result: .