Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Some of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding binary cyclic codes from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687, arXiv:1206.437

The differential properties of certain permutation polynomials over finite fields

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their possible applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very interesting connection between the $c$-differential uniformity and boomerang uniformity when $c=-1$ was pointed out, showing that that they are the same for an odd APN permutations. This makes the construction of functions with low $c$-differential uniformity an intriguing problem. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. As a byproduct, our proofs shows the permutation property of these classes. To solve the involved equations over finite fields, we use various techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest

P℘\wpN functions, complete mappings and quasigroup difference sets

We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that $F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of $\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$N function $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P$\wp$N functions, respectively, the difference sets in the corresponding quasigroup

Efficient Bit-parallel Multiplication with Subquadratic Space Complexity in Binary Extension Field

Bit-parallel multiplication in GF(2^n) with subquadratic space complexity has been explored in recent years due to its lower area cost compared with traditional parallel multiplications. Based on \u27divide and conquer\u27 technique, several algorithms have been proposed to build subquadratic space complexity multipliers. Among them, Karatsuba algorithm and its generalizations are most often used to construct multiplication architectures with significantly improved efficiency. However, recursively using one type of Karatsuba formula may not result in an optimal structure for many finite fields. It has been shown that improvements on multiplier complexity can be achieved by using a combination of several methods. After completion of a detailed study of existing subquadratic multipliers, this thesis has proposed a new algorithm to find the best combination of selected methods through comprehensive search for constructing polynomial multiplication over GF(2^n). Using this algorithm, ameliorated architectures with shortened critical path or reduced gates cost will be obtained for the given value of n, where n is in the range of [126, 600] reflecting the key size for current cryptographic applications. With different input constraints the proposed algorithm can also yield subquadratic space multiplier architectures optimized for trade-offs between space and time. Optimized multiplication architectures over NIST recommended fields generated from the proposed algorithm are presented and analyzed in detail. Compared with existing works with subquadratic space complexity, the proposed architectures are highly modular and have improved efficiency on space or time complexity. Finally generalization of the proposed algorithm to be suitable for much larger size of fields discussed