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

Pseudorandom sequences They have many applications Cryptography: keystream in the Vernam cipher The notion of pseudorandomness can be defined in different ways

3 Motivation The standard approach: – based on computational complexity – limitations and difficulties New, constructive approach: Mauduit, Sárközy about 50 papers in the last years

The standard approach Notions: PRBG seed, PR sequence next bit test unpredictable cryptographically secure PRBG

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

6 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

9 Measures mmm

10 Measures

11 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

12 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.

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

14 1.construction: Generalized Legendre symbol

2. construction:

16 My results These constructions are ideal of this point of view as well: – both possess the strong avalanche effect AND –they are collision free

17 Extension to k symbol Mauduit and Sárközy studied k-ary sequences instead of binary ones They extended the notion of well-distribution measure and correlation measure

18 The construction 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 Ahlswede, Mauduit and Sárközy extended: They proved that both measures are small

New results 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 H d be the set of polynomials of degree d which do not have multiple zeroes Theorem: If f is an element of H d, then the family of k-ary sequences constructed above is collision free and it also possesses the avalanche effect.

22 Conclusion 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

23 Thank you for your attention!