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Title:  Index tables of finite fields and modular golomb rulers 
Authors:  Salagean, A.M. Gardner, David Phan, Raphael C.W. 
Keywords:  Finite fields Modular Golomb Rulers Galois LFSR Nonlinear filter generator 
Issue Date:  2012 
Publisher:  © Springer Verlag 
Citation:  SALAGEAN, A.M., GARDNER, D. and PHAN, R., 2012. Index tables of finite fields and modular golomb rulers. Sequences and Their Applications (SETA 2012), Lecture Notes in Computer Science, 7280 pp. 136  147. 
Series/Report no.:  Lecture Notes in Computer Science;7280 
Abstract:  For a Galois field GF(2 n ) defined by a primitive element α with minimal polynomial f, the index table contains in row i the coordinates of α i in the polynomial basis α n − 1, α n − 2,…, α, 1. Each column i in this table equals the msequence with characteristic polynomial f, shifted cyclically by some offset h i .
In this paper we show that the set of the n shifts h i contains large subsets which are modular Golomb rulers modulo 2 n − 1 (i.e. all the differences are different). Let D be the set of integers j such that the coefficient of x j in f is nonzero. We prove that the set H D of shifts corresponding to columns j ∈ D can be partitioned into two subsets (the columns in the left half of the table and the ones in the right half) each of which is a modular Golomb ruler. Based on this result and on computational data, we conjecture that in fact the whole set H D is a modular Golomb ruler.
We give a polynomial time algorithm for deciding if given a subset of column positions, the corresponding shifts are a modular Golomb ruler. These results are applied to filter generators used in the design of stream ciphers. Golić recommends that in order to withstand his inversion attack, one of the design requirements should be that the inputs of the nonlinear filtering function are taken from positions of a Fibonacci LFSR which form a Golomb ruler. We propose using a Galois LFSR instead and selecting positions such that the corresponding shifts form a modular Golomb ruler. This would allow for a larger number of inputs to be selected (roughly n/2 rather than 2n − − √ ) while still satisfying Golić’s requirement. 
Description:  This paper was presented at Sequences and Their Applications  SETA 2012, 7th International Conference, Waterloo, Canada, June 48, 2012 and published in the series Lecture Notes in Computer Science, Vol. 7280 [© Springer Verlag]. 
Version:  Accepted for publication 
DOI:  10.1007/9783642306150_13 
URI:  https://dspace.lboro.ac.uk/2134/12398 
Publisher Link:  http://dx.doi.org/10.1007/9783642306150_13 
ISBN:  9783642306143 
Appears in Collections:  Conference Papers and Presentations (Computer Science)

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