Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the number of primitive $\sigma$-LFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question of Niederreiter (1995) on the enumeration of splitting subspaces of a given dimension.Comment: Version 2 with some minor changes; to appear in Designs, Codes and Cryptography

Enumeration of Linear Transformation Shift Registers

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.Comment: 16 page

The Splitting Subspace Conjecture

We answer a question by Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields by proving a conjecture by Ghorpade and Ram.Comment: 12 pages, Corrected factor of (1-q) in L=R expression in proof of Theorem 3.

The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

A Trivium-Inspired Pseudorandom Number Generator with a Statistical Comparison to the Randomness of SecureRandom and Trivium

A pseudorandom number generator (PRNG) is an algorithm that produces a sequence of numbers which emulates the characteristics of a random sequence. In comparison to its genuine counterpart, PRNGs are considered more suitable for computing devices in that they do not consume a lot of resources (in terms of memory) and their portability; they can also be used on a wide range of devices. Cryptographically Secure PRNGs (CSPRNGs) are the only type of PRNGs suitable for cryptographic applications. They are specially designed to withstand security attacks. In this thesis, we provide descriptions of two CSPRNGs: Trivium, a hardware-based stream cipher designed for constrained computing environments, and OpenJDK SecureRandom, a traditional CSPRNG recommended for Java programs that include a cryptographic algorithm. Our contributions are Quadrivium, a PRNG inspired by Trivium and analysis results comparing statistical properties of SecureRandom, Trivium and Quadrivium

Coordinated Science Laboratory progress report for December 1965, January, and February 1966

Studies in mechanical damping in possible gyro materials, electron scattering from surface of tungsten, and control system