On primitivity tests for polynomials

Primitive polynomials over finite fields are an important class of polynomials which find many applications in coding theory and cryptography. This paper considers algorithms for primitivity tests of polynomials over finite fields. The algorithms are optimized for the case that the polynomials tested turn out to be primitive. As the main contribution, we suggest a careful selection among many known algorithms for exponentiation. The results provide a primitivity test whose implementation confirmed that it is significantly faster than other known algorithms.