A recent survey of permutation binomials over finite fields

Permutation polynomials over finite fields are an important research area in which significant progress has been made. Some special polynomials with fewer terms serve more effective applications than a general permutation polynomial. We review the recent substantial contributions to the development of permutation binomials over finite fields. Significant results and unique methodologies are emphasized. The paper is divided into two parts: the existence and nonexistence of permutation binomials.