Power Functions over Finite Fields with Low c-Differential Uniformity
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Perfect nonlinear (PN) functions have important applications in cryptography, coding theory, and sequence design. During the past three decades, PN functions have been extensively studied. It is known that PN functions do not exist over the finite fields GF(2^n). Very recently, a new concept called multiplicative differential (and the corresponding c-differential uniformity) was introduced by Ellingsen et al<cit.>. Specifically, a function F(x) over finite field GF(p^n) to itself is said to be c-differential uniformity δ, or equivalent, F(x) is (c,δ)-differential uniform, when the maximum number of solutions x∈GF(p^n) of F(x+a)-F(cx)=b, a,b,c∈GF(p^n), c≠1 if a=0, is equal to δ. It turns out that perfect c-nonlinear (PcN) functions exist over GF(2^n). The objective of this paper is to study power function F(x)=x^d over finite fields with c-differential uniformity. Some power functions are shown to be perfect c-nonlinear or almost perfect c-nonlinear.
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