Andrew Klapper

Feedback shift registers, 2-adic span, and combiners with memory

Feedback shift registers with carry operation (FCSRs) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSRs) are presented, including a synthesis algorithm (analogous to the Berlekamp-Massey algorithm for LFSRs) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the Marsaglia-Zaman random number generator. Possible variations on the FCSR architecture are indicated at the end.

A new index for polytopes

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.

d-form sequences: families of sequences with low correlation values and large linear spans

Large families of binary sequences with low correlation values and large linear span are critical for spread-spectrum communication systems. The author describes a method for constructing such families from families of homogeneous functions over finite fields, satisfying certain properties. He then uses this general method to construct specific families of sequences with optimal correlations and exponentially better linear span than No sequences (No. 1988). >

Cascaded GMW sequences

Pseudorandom binary sequences with high linear complexity and low correlation function values are sought in many applications of modern communication systems. A new family of pseudorandom binary sequences, cascaded GMW sequences, is constructed. These sequences are shown to share many desirable correlation properties with the GMW sequences of B. Gordon, W.A. Mills, and L.R. Welch (1962)-for example, high-shifted autocorrelation values and, in many cases, three-valued cross-correlation values with m-sequences. It is shown, moreover, that in many cases the linear complexities of cascaded GMW sequences are far greater than those of GMW sequences. >

Algebraic Shift Register Sequences

1. Introduction Part I. Algebraically Defined Sequences: 2. Sequences 3. Linear feedback shift registers and linear recurrences 4. Feedback with carry shift registers and multiply with carry sequences 5. Algebraic feedback shift registers 6. d-FCSRs 7. Galois mode, linear registers, and related circuits Part II. Pseudo-Random and Pseudo-Noise Sequences: 8. Measures of pseudo-randomness 9. Shift and add sequences 10. M-sequences 11. Related sequences and their correlations 12. Maximal period function field sequences 13. Maximal period FCSR sequences 14. Maximal period d-FCSR sequences Part III. Register Synthesis and Security Measures: 15. Register synthesis and LFSR synthesis 16. FCSR synthesis 17. AFSR synthesis 18. Average and asymptotic behavior of security measures Part IV. Algebraic Background: A. Abstract algebra B. Fields C. Finite local rings and Galois rings D. Algebraic realizations of sequences Bibliography Index.

Arithmetic crosscorrelations of feedback with carry shift register sequences

An arithmetic version of the crosscorrelation of two sequences is defined, generalizing Mandelbaums (1967) arithmetic autocorrelations. Large families of sequences are constructed with ideal (vanishing) arithmetic crosscorrelations. These sequences are decimations of the 2-adic expansions of rational numbers p/q such that 2 is a primitive root module q.

Cryptoanalysis Based on 2-Adic Rational Approximation

This paper presents a new algorithm for cryptanalytically attacking stream ciphers. There is an associated measure of security, the 2-adic span. In order for a stream cipher to be secure, its 2-adic span must be large. This attack exposes a weakness of Rueppel and Masseys summation combiner. The algorithm, based on De Weger and Mahlers rational approximation theory for 2-adic numbers, synthesizes a shortest feedback with carry shift register that outputs a particular key stream, given a small number of bits of the key stream. It is adaptive in that it does not neeed to know the number of available bits beforehand.

Algebraic feedback shift registers

A general framework for the design of feedback registers based on algebra over complete rings is described. These registers generalize linear feedback shift registers and feedback with carry shift registers. Basic properties of the output sequences are studied: relations to the algebra of the underlying ring; synthesis of the register from the sequence (which has implications for cryptanalysis); and basic statistical properties. These considerations lead to security measures for stream ciphers, analogous to the notion of linear complexity that arises from linear feedback shift registers. We also show that when the underlying ring is a polynomial ring over a finite field, the new registers can be simulated by linear feedback shift registers with small nonlinear filters.

Register Synthesis for Algebraic Feedback Shift Registers Based on Non-Primes

In this paper, we describe a solution to the register synthesis problem for a class of sequence generators known as algebraic feedback shift registers (AFSRs). These registers are based on the algebra of π-adic numbers, where π is an element in a ring R, and produce sequences of elements in R/(π). We give several cases where the register synthesis problem can be solved by an efficient algorithm. Consequently, any keystreams over R/(π) used in stream ciphers must be unable to be generated by a small register in these classes. This paper extends the analyses of feedback with carry shift registers and algebraic feedback shift registers by Goresky, Klapper, and Xu.

Feedback registers based on ramified extensions of the 2-adic numbers

A new class of feedback register, based on ramified extensions of the 2-adic numbers, is described. An algebraic framework for the analysis of these registers and the sequences they output is given. This framework parallels that of linear feedback shift registers. As one consequence of this, a method for cracking summation ciphers is given. These registers give rise to new measures of cryptologic security.