The Reducibility of a Special Binary Pentanomial

Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree , where , and exactly one of？？ ,？？and？？ is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2. 1. Introduction Irreducible polynomials of low weight over a finite field are frequently used in many applications such as coding theory and cryptography due to efficient arithmetic implementation in an extension field and, thus, it is important to determine the irreducibility of such polynomials. The weight of a polynomial means the number of its nonzero coefficients. Characterization of the parity of the number of irreducible factors of a given polynomial is of significance in this context. If a polynomial has an even number of irreducible factors, then it is reducible and, thus, the study on the parity of this number can give a necessary condition for irreducibility. Swan [1] gives the first result determining the parity of the number of irreducible factors of trinomials over . Vishne [2] extends Swan’s theorem to trinomials over an even-dimensional extension of . Many Swan-like results focus on determining the reducibility of higher weight polynomials over ; see for example [3, 4]. Some researchers obtain the results on the reducibility of polynomials over a finite field of odd characteristic. We refer to [5, 6]. On the other hand, Ahmadi and Menezes [7] estimate the number of trace-one elements on the trinomial and pentanomial bases for a fast and low-cost implementation of trace computation. They also present a table of irreducible pentanomials whose corresponding polynomial bases have exactly one trace-one element. Each pentanomial of even degree in this table is of the form , where , and exactly one of ,？？and？？ is odd. In this work, we characterize the parity of the number of irreducible factors of this pentanomial. We describe some preliminary results related to Swan-like results in Section 2 and determine the reducibility of the pentanomial mentioned above in Section 3. 2. Preliminaries In this section, we recall Swan’s theorem determining the parity of the number of irreducible factors of a polynomial over and some results about the discriminant and the resultant of polynomials. Let be a field and let , where are the roots of in an extension of . The discriminant of is defined by From the definition, it is clear that has a repeated