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THE EXTENSION OF COLLISION AND AVALANCHE EFFECT TO k -ARY SEQUENCESPowerPoint Presentation

THE EXTENSION OF COLLISION AND AVALANCHE EFFECT TO k -ARY SEQUENCES

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THE EXTENSION OF COLLISION AND AVALANCHE EFFECT TO k -ARY SEQUENCES

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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 SEQUENCESPseudorandom sequences

- They have many applications
Cryptography:

keystream in the Vernam cipher

- The notion of pseudorandomness can be defined in different ways

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

Notions:

- PRBG seed, PR sequence
- next bit test unpredictable
- cryptographically secure PRBG

Problems These definitions measure only the quality of

- „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.

- There is no PRBG whose

PRBG’s, not the output sequences

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

- Infinity sequences:
normality (Borel)

- Finite sequences:
- Golomb
- Knuth
- Kolmogorov
- Linear complexity

Advantages

- Normality
- Well-distribution
- Low correlation of low order
- characterizing with real-valued function
comperable

Measures

- mmm

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

- 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.

- In the applications one usually needs LARGE FAMILIES of sequences with strong pseudorandom properties.
- I have tested two of the most important constructions:

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

2. construction: sequences with strong pseudorandom properties.

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

- both possess the strong avalanche effect

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 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 sequences with strong pseudorandom properties.k symbols

- Ahlswede, Mauduit and Sárközy extended:
- They proved that both measures are small

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 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.