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Index tables of finite fields and modular golomb rulers

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journal contribution
posted on 06.06.2013 by Ana Salagean, David Gardner, Raphael C.-W. Phan
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 m-sequence 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 non-zero. 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 non-linear 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.



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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.


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Publication date



This paper was presented at Sequences and Their Applications - SETA 2012, 7th International Conference, Waterloo, Canada, June 4-8, 2012 and published in the series Lecture Notes in Computer Science, Vol. 7280 [© Springer Verlag].



Book series

Lecture Notes in Computer Science;7280