In card-based cryptography, a deck of physical cards is used to achieve secure computation. A shuffle, which randomly permutes a card-sequence along with some probability distribution, ensures the security of a card-based protocol. The authors proposed a new class of shuffles called graph shuffles, which randomly permutes a card-sequence by an automorphism of a directed graph (New Generation Computing 2022). For a directed graph G with n vertices and m edges, such a shuffle could be implemented with pile-scramble shuffles with 2(n + m) cards. In this paper, we study graph shuffles and give an implementation, an application, and a slight generalization. First, we propose a new protocol for graph shuffles with 2n + m cards. Second, as a new application of graph shuffles, we show that any cyclic group shuffle, which is a shuffle over a cyclic group, is a graph shuffle associated with some graph. Third, we define a hypergraph shuffle, which is a shuffle by an automorphism of a hypergraph, and show that any hypergraph shuffle can also be implemented with pile-scramble shuffles.

Computing the weight distribution of a code is a challenging problem in coding theory. In this paper, the weight distributions of (256, k) extended binary primitive BCH codes with k≤71 and k≥187 are given. The weight distributions of the codes with k≤63 and k≥207 have already been obtained in our previous work. Affine permutation and trellis structure are used to reduce the computing time. Computer programs in C language which use recent CPU instructions, such as SIMD, are developed. These programs can be deployed even on an entry model workstation to obtain the new results in this paper.

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices; we call them echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic. The first algorithm is composed of elementary methods and is appropriate for low-rate codes. The second algorithm is based on a novel formula and has smaller computational complexity than the first one for high-rate codes with the number of orbits (cyclic parts) less than half of the code length. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements.