Abstract. Recently, some collisions have been exposed for a variety of cryptographic hash functions [19] including some of the most widely used today. Many other hash functions using similar constrcutions can however still be considered secure. Nevertheless, this has drawn attention on the need for new hash function designs. In this article is presented a familly of secure hash functions, whose security is directly related to the syndrome decoding problem from the theory of error-correcting codes. Taking into account the analysis by Coron and Joux [4] based on Wagner’s generalized birthday algorithm [18] we study the asymptotical security of our functions. We demonstrate that this attack is always exponential in terms of the length of the hash value. We also study the work-factor of this attack, along with other attacks from coding theory, for non asymptotic range, i.e. for practical values. Accordingly, we propose a few sets of parameters giving a good security and either a faster hashing or a shorter desciption for the function. Key Words: cryptographic hash functions, provable security, syndrome decoding, NP-completeness, Wagner’s generalized birthday problem.

Submitted to IEEE Transactions on Information Theory A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudorandom nature, and the new codes are quasi-cyclic. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 were found. Upper bounds on the minimum Hamming distance are found both analytically and algorithmically. The bounds indicate that the minimum distance grows with block length. Near-codewords are one of the causes for error floors in LDPC codes; the new construction provides a good framework for studying near-codewords in LDPC codes. Nine example codes are given, and computer simulation results show the excellent error performance of these codes. Finally, connections are made between this new LDPC construction and turbo codes using interleavers generated by quadratic permutation polynomials.

This paper extends results that have been reported in [1],[6], and [5] for algebraic construction of a low-density matrix to be used as a parity-check matrix for an error-correcting code. After describing the basic idea and the results we finish with a performance diagram that suggests that these codes are as good as random constructions in the area of low SNR and perform better than an algebraic construction tested in [7] in the area of higher SNR. As has been noted in [8], codes with algebraic structure may be easier to implement. The flexibility of the construction we describe may also be advantageous. 1 Details of the Construction For a positive integer R we will use the notation R for {1,..., R}. Let G be an abelian group and let R and S be positive integers. We construct a bipartite graph with vertex sets R × G and S × G and edges determined by translation and dilation maps on G. The binary adjacency matrix for the bipartite graph can be used as a check matrix for an error correcting code. Definition 1.1 Let G be an abelian group. For each c ∈ G, the translation defined by c, is the map g ↦ → g + c on G. For each automorphism α of G and each b ∈ G we call the map g ↦ → g α + b a dilation of G. We will be using α in the center of the automorphism group of G. We construct the bipartite graph as follows. Construction 1.2 Let G a finite abelian group, let R, S and T positive integers, and let ρ: T − → R and σ: T − → S be mappings. Furthermore, for each t ∈ T fix a dilation θt of G. The reader may agree on the notation v ∼ w if vertices v, w ∈ V are connected by an edge in the underlying graph. We define a bipartite graph (V, E) with V: = R × G ∪ S × G, and E: = {(ρ(t), g), (σ(t) ∼ θt(g)) | t ∈ T, g ∈ G}. In order to make the construction more transparent we give two examples that also help to characterize previous work. Example 1.3 (a) Let G = Z/p × Z/q for p and q primes. Let R = 1, and let T = 3S. For t ∈ T let σ(t) = ⌈t/3 ⌉ and let θt: (x, y) ↦ → (x + ct, aty + bt) where at is nonzero.

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 Gröbner basis of modules, there has been no algorithm that computes Gröbner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Gröbner 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.

Simple cycles are the easiest cycles to reveal in Low-Density Parity Check (LDPC) codes. All minimum matrices of simple cycles with the same length will be equivalent after row or column permutations.In this paper,we analysis the structure of simple cycles and show all figures of minimal matrices of simple cycles. Firstly, we introduce a more general definition of cycle and investigate simple cycles in LDPC codes. Secondly, we present the number of simple cycles of arbitrary length and all the minimal matrices of simple cycles. Finally, we have proved that the number of all minimal matrices of 2k-simple cycles ( k −1)! k is. Pk, for k ≥ 3.

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 Gröbner basis of modules, there has been no algorithm that computes Gröbner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Gröbner basis for GQC codes from their parity check matrices: 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 the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. 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. Keywords: automorphism group, Buchberger’s algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes. 1 1