On a Conjecture about Binary Strings Distribution

  • Jean-Pierre Flori
  • Hugues Randriam
  • Gérard Cohen
  • Sihem Mesnager
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)


It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many good cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true:

Conjecture 0.1. Let S t,k be the following set:

$$ {S_{t,k}}= \left\{{(a,b)} {\in} \left( {{\mathbb Z} / {(2^k-1)} {\mathbb Z}} \right)^2 | a + b = t ~{\rm and }~ w(a) + w(b) < k \right\}. $$


$$ |{S_{t,k}}| \leq 2^{k-1}. $$

The main contribution of the present paper is the reformulation of the problem in terms of carries which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of \(\left|{S_{t,k}}\right|\) for numbers whose binary expansion is made of one block, a proof that the conjecture is asymptotically true and a proof that a family of numbers (whose binary expansion has a high number of 1s and isolated 0s) reaches the bound of the conjecture. We also conjecture that the numbers in that family are the only ones reaching the bound.


Boolean Function Binary String Stream Cipher Linear Feedback Shift Register Algebraic Degree 
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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-Pierre Flori
    • 1
  • Hugues Randriam
    • 1
  • Gérard Cohen
    • 1
  • Sihem Mesnager
    • 2
  1. 1.Institut TélécomTélécom ParisTech, CNRS LTCIParis Cedex 13France
  2. 2.LAGA (Laboratoire Analyse, Géometrie et Applications), UMR 7539, CNRS, Department of MathematicsUniversity of Paris XIII and University of Paris VIIISaint-Denis CedexFrance

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