On Degrees of Modular Common Divisors and the Big Prime Algorithm

Algorithm 3

Coprime polynomials detection modular algorithm.

Input: non-zero polynomials .

Detect if and are coprime.

(01) Calculate , in the Euclidean domain .

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

(03) If

(04)  output the result: and are not coprime and stop.

(05) Set and .

(06) Calculate in the Euclidean domain .

(07) Set and .

(08) Compute the bound for polynomials by (32).

(09) Find the maximal for which .

(10)  Set .

(11)  While

(12)  choose a new prime ;

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

(14)  calculate in the Euclidean domain ;

(15)    if

(16)      output the result: and are coprime and stop.

(17)  set .

(18)  If

(19)  go to step (12).

(20)  else

(21)  output the result: and are not coprime.