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**1 - 5**of**5**### Expected π-Adic Security Measures of Sequences

"... Various measures of security of stream ciphers have been studied that are based on the problem of finding a minimum size generator for the keystream in some special class of generators. These include linear and p-adic spans, as well as π-adic span, which is based on a choice of an element π in a fin ..."

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Various measures of security of stream ciphers have been studied that are based on the problem of finding a minimum size generator for the keystream in some special class of generators. These include linear and p-adic spans, as well as π-adic span, which is based on a choice of an element π in a finite extension of the integers. The corresponding sequence generators are known as linear feedback shift registers, feedback with carry shift registers, and the more general algebraic feedback shift registers, respectively. In this paper the average behavior of such security measures when π d = p> 0 or π 2 = −p < 0 is studied. In these cases, if Z[π] is the ring of integers in its fraction field and is a UFD, it is shown that the average π-adic span is n − O(log(n)) for sequences with period n.

### Algebraic Feedback Shift Registers Based on Function Fields

"... We study algebraic feedback shift registers (AFSRs) based on quotients of polynomial rings in several variables over a finite field. These registers are natural generalizations of linear feedback shift registers. We describe conditions under which such AFSRs produce sequences with various ideal ran ..."

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We study algebraic feedback shift registers (AFSRs) based on quotients of polynomial rings in several variables over a finite field. These registers are natural generalizations of linear feedback shift registers. We describe conditions under which such AFSRs produce sequences with various ideal randomness properties. We also show that there is an efficient algorithm which, given a prefix of a sequence, synthesizes a minimal such AFSR that outputs the sequence.

### The Asymptotic Behavior of N-Adic Complexity

"... We study the asymptotic behavior of stream cipher security measures associated with classes of sequence generators such as linear feedback shift registers and feedback with carry shift registers. For nonperiodic sequences we consider normalized measures and study the set of accumulation points for a ..."

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We study the asymptotic behavior of stream cipher security measures associated with classes of sequence generators such as linear feedback shift registers and feedback with carry shift registers. For nonperiodic sequences we consider normalized measures and study the set of accumulation points for a fixed sequence. We see that the the set of accumulation points is always a closed subinterval of [0, 1]. For binary or ternary FCSRs we see that this interval is of the form [B, 1 − B], a result that is an analog of an earlier result by Dai, Jiang, Imamura, and Gong for

### A New Class of Pseudonoise Sequences

, 2003

"... We apply the framework of pi-adic algebra and algebraic feedback shift registers to polynomial rings over finite fields. We give a construction of new pseudorandom sequences over a non-prime finite field that satisfy Golomb's randomness criteria. ..."

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We apply the framework of pi-adic algebra and algebraic feedback shift registers to polynomial rings over finite fields. We give a construction of new pseudorandom sequences over a non-prime finite field that satisfy Golomb's randomness criteria.

### Research Summary

"... models for answering questions on the existence of secure families of sequence generators. 5. Design and analysis of families of sequences for secure spread-spectrum communications. These sequences include geometric sequences and d-form sequences (the latter invented by me). ..."

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models for answering questions on the existence of secure families of sequence generators. 5. Design and analysis of families of sequences for secure spread-spectrum communications. These sequences include geometric sequences and d-form sequences (the latter invented by me).