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We consider a few modifications of the Big prime modular $\gcd$ algorithm forpolynomials in $\Z[x]$. Our modifications are based on bounds of degrees ofmodular common divisors of polynomials, on estimates of the number of primedivisors of a resultant and on finding preliminary bounds on degrees of commondivisors using auxiliary primes. These modifications are used to suggestimproved algorithms for $\gcd$ calculation and for coprime polynomialsdetection. To illustrate the ideas we apply the constructed algorithms oncertain polynomials, in particular, on polynomials from Knuth's example ofintermediate expression swell.

On degrees of modular common divisors and the Big prime gcd algorithm