The extension of collision and avalanche effect to k ary sequences
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Viktória Tóth Eötvös Loránd University, Budapest Department of Algebra and Number Theory, Department of Computer Algebra 9-12th June, 2010, Bedlewo. THE EXTENSION OF COLLISION AND AVALANCHE EFFECT TO k -ARY SEQUENCES. Pseudorandom sequences. They have many applications Cryptography:

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The extension of collision and avalanche effect to k ary sequences

Viktória Tóth

Eötvös Loránd University, Budapest

Department of Algebra and Number Theory,

Department of Computer Algebra

9-12th June, 2010, Bedlewo

THE EXTENSION OF COLLISIONAND AVALANCHE EFFECT TO k-ARY SEQUENCES


Pseudorandom sequences
Pseudorandom sequences

  • They have many applications

    Cryptography:

    keystream in the Vernam cipher

  • The notion of pseudorandomness can be defined in different ways


Motivation
Motivation

  • The standard approach:

    • based on computational complexity

    • limitations and difficulties

  • New, constructive approach:

    Mauduit, Sárközy

  • about 50 papers in the last 10-15 years


The standard approach
The standard approach

Notions:

  • PRBG seed, PR sequence

  • next bit test unpredictable

  • cryptographically secure PRBG


Problems
Problems

  • „probability significantly greater than ½”

  • The non-existence of a polynomial time

    algorithm has not been shown unconditionally

    yet

    • There is no PRBG whose

      cryptographycal sequrity has been

      proved unconditionally.

  • These definitions measure only the quality of

    PRBG’s, not the output sequences


  • Goal s
    Goals

    • More constructive

    • We do not want to use unproved hypothesis

    • We describe the single sequences

    • Apriori testing

    • Characterizing with real-valued function

      • comparable


    Historical background
    Historical background

    • Infinity sequences:

      normality (Borel)

    • Finite sequences:

      • Golomb

      • Knuth

      • Kolmogorov

      • Linear complexity


    Advantages
    Advantages

    • Normality

    • Well-distribution

    • Low correlation of low order

    • characterizing with real-valued function

      comperable




    Previous results
    Previous results

    • „good” sequence:

      If both and (at least for

      small k) are „small” in terms of N

    • This terminology is justified:

      Theorem: for truly random sequences


    Further properties
    Further properties

    • collision free: two different choice of the parameters should not lead to the same sequence;

    • avalanche effect: changing only one bit on the input leads to the change about half of the bits on the output.



    1 construction generalized legendre symbol
    1.construction: sequences with strong pseudorandom properties.Generalized Legendre symbol


    2 construction
    2. construction: sequences with strong pseudorandom properties.


    My results
    My results sequences with strong pseudorandom properties.

    • These constructions are ideal of this point of view as well:

      • both possess the strong avalanche effect

        AND

      • they are collision free


    Extension to k symbol
    Extension to sequences with strong pseudorandom properties.k symbol

    • Mauduit and Sárközy studied k-ary sequences instead of binary ones

    • They extended the notion of

      well-distribution measure and correlation measure


    The construction
    The construction sequences with strong pseudorandom properties.

    • They generated the sequences with a character of order k:

    • Mauduit and Sárközy proved that both the correlation measure and the

      well-distribution measure are „small”

    • So we can say that this is a good construction of pseudorandom k-ary sequences


    A good family of pseudorandom sequences of k symbols
    A good family of pseudorandom sequences of sequences with strong pseudorandom properties.k symbols

    • Ahlswede, Mauduit and Sárközy extended:

    • They proved that both measures are small


    New results
    New results sequences with strong pseudorandom properties.

    • I extended the notion of collisions and avalanche effect to k symbol

    • I studied the previous family of k-ary sequences with strong pseudorandom properties.


    • Let sequences with strong pseudorandom properties.Hd be the set of polynomials of degree d which do not have multiple zeroes

    • Theorem: If f is an element of Hd , then the family of k-ary sequences constructed above is collision free and it also possesses the avalanche effect.


    Conclusion
    Conclusion sequences with strong pseudorandom properties.

    • If we have a large family of sequences

      with strong pseudorandom properties,

      then it worth studying it from other point of view

      • In this way we can get further beneficial properties, which can be profitable, especially in applications


    Thank you for your attention! sequences with strong pseudorandom properties.


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