Index Tables of Finite Fields and Modular Golomb Rulers

  • Ana Sălăgean
  • David Gardner
  • Raphael Phan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)


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 \(\sqrt{2n}\)) while still satisfying Golić’s requirement.


Boolean Function Finite Field Stream Cipher Polynomial Basis Primitive Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blackburn, S.R.: Increasing the Rate of Output of m-Sequences. Information Processing Letters 51, 73–77 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Giuliani, K., Gong, G.: New LFSR-Based Cryptosystems and the Trace Discrete Log Problem (trace-DLP). In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 298–312. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Golić, J.D.: On the Security of Nonlinear Filter Generators. In: Gollmann, D. (ed.) FSE 1996. LNCS, vol. 1039, pp. 173–188. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  4. 4.
    Graham, R.L., Sloane, N.J.A.: On Additive Bases and Harmonious Graphs. Siam Journal on Algebraic and Discrete Methods 1, 382–404 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press (1994)Google Scholar
  6. 6.
    Smyshlyaev, S.V.: Perfectly Balanced Boolean Functions and Golić Conjecture. Journal of Cryptology, 1–20 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ana Sălăgean
    • 1
  • David Gardner
    • 1
  • Raphael Phan
    • 1
  1. 1.Loughborough UniversityUK

Personalised recommendations