Eurocode ’92 pp 185-200 | Cite as

Rational Interval Maps and Cryptography

  • S. Harari
  • P. Liardet
Part of the International Centre for Mechanical Sciences book series (CISM, volume 339)


In Eurocrypt 1991 Habutsu, Nishio, Sasase, Mori [4] introduced a cryptosystem using tent maps (denoted HNSM in the sequel). Though the functions that are used have good chaotic properties this system seems to has many weaknesses [7]. A general framework for such cryptosystems can be described as follow. Let ε = {E i ; i = 0,..., M} be a family of so-called plaintext-spaces and for each i = 1,..., M, let L i be an integer ≥ 2 and let T i = {T i,ℓ ; = 1,...,L i } be a family of cipher-maps T i,ℓ : E iℒ1E i . We assume that there exist maps S i : E i E i−1 such that for all ∈ {1,..., L i } one has S i o T i,ℓ = Id Ei−1. The key is given by {S 1,..., S M ) and the cryptogram of a given paintext yE 0 is any element in
where the union is taken over all finite sequences ( 1,..., M ) in . Usually all the plaintext-spaces are identical to a same space E and we choose the same family of enciphering maps. Therefore, the system is given by a deciphering map Σ: E → E and a family of right inverse maps of Σ: T 1,..., T L , L ≥ 2. The integer M corresponds to the number of iterations and the key is (Σ, M). To avoid attacks on the enciphering algorithm, the choice of the cryptogram depends on a random process which produces a uniform-like distribution of the ciphertext in E M . It seem that the firt use of such a scheme was considered in terms of cellular automata by S. Wolfram in Crypto’85 [8] and recently H. Gutowitz in [3] proposed a scheme, according to this model, of an enciphering/deciphering system at high rate. The HNSM system uses tent maps Φ t for keys, namely t is a parameter in]0,1[ and Φ t is the piecewise linear map defined by Φ t (x) = x/t for 0 ≤ xt and for t < x ≤ 1.


Cellular Automaton Rational Number Unit Interval Ergodic Measure Ergodic Property 
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  1. [1]
    ADLER R. and FLATTO L.: Geodesic flows,universal maps,and symbolic dynamics, Bulletin of the ANS, 25, No 2 (1991), 229–334.Google Scholar
  2. [2]
    DENKER M., GRILLENBER.GER C.: SIGMUND K.: Ergodic theory on compact spaces,Lecture Notes in Math., 525 (1976), Springer-Verlag.Google Scholar
  3. [3]
    C,UTOWITZ H.: A cellular automaton cryptosystem; specification and call for attack, preprint 1992.Google Scholar
  4. [4]
    HABUTSU T., NISHIO Y., SASASE I., MORI S.: A secret key cryptosystem by iterating chaotic maps. Eurocrypt’ 91.Google Scholar
  5. [5]
    KNUTH D.: Semi numerical algorithms,Adisson Wesley (1980).Google Scholar
  6. [6]
    PARRY W.: Symbolic Dynamics and transformations of the unit intervals. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    RUMP SESSION, Eurocrypt 91: Biham’s attack.Google Scholar
  8. [8]
    SCHWEIGER F.: Ergodic properties of piecewice fractional linear maps, Arbeitsbericht, Math. Institut Universität Salzburg (1980), 24–32.Google Scholar
  9. [9]
    SCHWEIGER F.: Ergodic properties of fibered systems, Institut für Mathematik der Univ. Salzburg, Draft version, April 1991.Google Scholar
  10. [10]
    WOLFRAM S.: Cryptography with cellular automata, Proceeding of Crypto 85 (1985), 429–432Google Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • S. Harari
    • 1
  • P. Liardet
    • 2
  1. 1.Groupe d’Étude du Codage de ToulonUniversité de Toulon et du VarLa GardeFrance
  2. 2.URA CNRS No225 Equipe DSA, case 96Université de ProvenceMarseille cedex 3France

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