Permutation polynomials over Galois fields of characteristic

In this paper, a class of permutation polynomial known as o-polynomial over Galois fields of characteristic 2 was studied. A necessary and sufficients condition for a monomial 𝑥2k to be an o-polynomial over F2t is given and two results obtained by Gupta and Sharma (2016) were deduced.

Image encryption based on permutation polynomials over finite fields

In this paper, we propose an image encryption algorithm based on a permutation polynomial over finite fields proposed by the authors. The proposed image encryption process consists of four stages: i) a mapping from pixel gray-levels into finite field, ii) a pre-scrambling of pixels’ positions based on the parameterized permutation polynomial, iii) a symmetric matrix transform over finite fields which completes the operation of diffusion and, iv) a post-scrambling based on the permutation polynomial with different parameters. The parameters used for the polynomial parameterization and for constructing the symmetric matrix are used as cipher keys. Theoretical analysis and simulation demonstrate that the proposed image encryption scheme is feasible with a high efficiency and a strong ability of resisting various common attacks. In addition, there are not any round-off errors in computation over finite fields, thus guaranteeing a strictly lossless image encryption. Due to the intrinsic nonlinearity of permutation polynomials in finite fields, the proposed image encryption system is nonlinear and can resist known-plaintext and chosen-plaintext attacks.

A NEW PROOF OF THE CARLITZ–LUTZ THEOREM

A polynomial $f$ over a finite field $\mathbb{F}_{q}$ can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers $f^{e}$ for each $e$ from $1$ to $q-2$, as well as the existence of a unique solution to $f(x)=0$ in $\mathbb{F}_{q}$. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.