On Degrees of Modular Common Divisors and the Big Prime Algorithm

Algorithm 5

Big prime modular gcd algorithm with preliminary estimates on divisor degrees by multiple primes.

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) Compute the bound for polynomials by (32).

(04) Find the maximal for which .

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

(06) Set and (they are positive by our selection of signs

   for and ).

(07) Calculate the positive in the Euclidean domain .

(08) Set .

(09) Set .

(10) While

(11)  choose a new prime ;

(12)  apply the reduction to calculate the modular images ;

(13)  calculate in the Euclidean domain ;

(14)  if

(15)    set ;

(16)    if

(17)      set ;

(18)      go to step (32);

(19)  set .

(20) Calculate by (43) using the value of .

(21) Choose a new prime number .

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

(23) Calculate in the Euclidean domain .

(24) If

(25)  choose a such that the ;

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

(27)  calculate in the Euclidean domain ;

(28)  set ;

(29)  go to step (32);

(30) else

(31)  go to step (21).

(32) Output the result: .