Quadratic permutation polynomials (QPPs) have been widely studied and used as interleavers in turbo codes. However, less attention has been given to cubic permutation polynomials (CPPs). This paper proves a theorem which states sufficient and necessary conditions for a cubic permutation polynomial to be a null permutation polynomial. The result is used to reduce the search complexity of CPP int...

In Part VII, we proved that the range LX of the big J-function in permutation-equivariant genus-0 quantum K-theory is an overruled cone, and gave its adelic characterization. Here we show that the ruling spaces are Dq-modules in Novikov’s variables, and moreover, that the whole cone LX is invariant under a large group of symmetries of LX defined in terms of q-difference operators. We employ thi...

This paper presents two new algorithms for inline transforming an integer array ‘a’ into its own sorting permutation that is: after performing either of these algorithms, a[i] is the index in the unsorted input array ‘a’ of its i’th largest element (i=0,1..n-1). The difference between the two IPS (Inline Permutation Substitution) algorithms is that the first and fastest generates an unstable pe...

A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ωn (the set of n-tuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their group-theoretic properties have been studied, and links with graded algebras, Ramsey theory, topological dynamics, an...

A split complete weight enumerator in six variables is used to study the colored designs held by codewords of xed composition in Type III codes containing the all one codeword In particular the ternary Golay code contains colored designs We conjecture that every weight class in a Type III code with the all one codeword holds colored designs Introduction Colored designs were introduced in to stu...

We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane (walks avoid a half line) that was first solved by Bousquet-Mélou and Schaeffer. We also solve a problem about walks in the half plane avoiding a half line by s...

We prove that a (balanced) 10-round Feistel network is indifferentiable from a random permutation. In a previous seminal result, Holenstein et al. [17] had established indifferentiability of Feistel at 14 rounds. Our simulator achieves security O(q/2) and query complexity O(q), where n is half the block length, similarly to the 14-round simulator of [17], so that our result is a strict (and als...

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Also, the recently proposed quadratic permutation polynomial (QPP) based interleavers by Sun and Takeshita (IEEE Trans. Inf. Theory, Jan. 2005) provide excellent performan...

Function clones are sets of functions on a fixed domain that are closed under composition and contain the projections. They carry a natural algebraic structure, provided by the laws of composition which hold in them, as well as a natural topological structure, provided by the topology of pointwise convergence, under which composition of functions becomes continuous. Inspired by recent results i...

In this paper we provide a simple, concrete and improved security analysis of Parallelizable Message Authentication Code or PMAC. In particular, we show that the advantage of any distinguisher A at distinguishing PMAC from a random function is at most (5qσ − 3.5q)/2. Here, σ is the total number of message blocks in all q queries made by A and PMAC is based on a random permutation over {0, 1}. I...