Optimal extension fields for XTR

Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security [6]. XTR is a new method to represent elements of a subgroup of a multiplicative group of a finite field GF(p⁶) and it can be generalized to the field GF(p^6m) [6,9]. This paper proposes optimal extension fields for XTR among Galois fields GF(p^6m) which can be applied to XTR. In order to select such fields, we introduce a new notion of Generalized Optimal Extension Fields(GOEFs) and suggest a condition of prime p, a defining polynomial of GF(p^2m) and a fast method of multiplication in GF(p^2m) to achieve fast finite field arithmetic in GF(p^2m). From our implementation results, GF(p³⁶) → GF(p¹²) is the most efficient extension fields for XTR and computing Tr(gⁿ) given Tr(g) in GF(p¹²) is on average more than twice faster than that of the XTR system[6,10] on Pentium III/700MHz which has 32-bit architecture.