A note on the distinctness of some Kloosterman sums
- PDF / 229,344 Bytes
- 6 Pages / 439.642 x 666.49 pts Page_size
- 95 Downloads / 442 Views
A note on the distinctness of some Kloosterman sums Yuri Borissov1
· Lyubomir Borissov1
Received: 18 September 2019 / Accepted: 4 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Fischer result about distinctness of the Kloosterman sums on F ∗ p is extended for the finite fields of degrees of extension that are powers of 2. To obtain the desired outcome, we give an elementary proof of the fact that there does not exist a pair of Kloosterman sums on same odd characteristic fields which are opposite to each other. Keywords Kloosterman sum · Kloosterman zero · Distinctness of Kloosterman sums. Mathematics Subject Classification (2010) 11L05 · 11T71
1 Introduction The Kloosterman sums on finite fields play an important role in areas such as cryptology and coding theory being connected by definition with the inverse function over finite field. They are relevant, for instance, to design and cryptanalysis of some block ciphers; as a special kind of exponential sums they are related to some notable coding-theoretical and combinatorial objects like Kloosterman and Melas codes, (hyper)bent functions, etc.. For more details about these links the reader is referred to the surveys [1] and [2]. The issue of distinctness of Kloosterman sums was studied for the first time by B. Fischer in his paper [3]. Namely: – –
This author has proved distinctness of the simplest Kloosterman sums, i.e., those for the non-zero elements of the prime field Fp ; In the same work it was pointed out that Kloosterman sums tend to be distinct if the field characteristic p is sufficiently larger than the degree of the field extension m of
This article belongs to the Topical Collection: Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth Dedicated to Prof. Claude Carlet’s 70th Birthday Yuri Borissov
[email protected] Lyubomir Borissov [email protected] 1
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G. Bontchev Str., 1113 Sofia, Bulgaria
Cryptography and Communications
the finite field Fq , q = p m . For example, it was proved that when p > (2.4m + 1)2 the q−1 Kloosterman sums on F∗q = Fq \{0} are distinct (except for Frobenius conjugation, of course). Indeed, the referee of Fischer’s work had conjectured that a much stronger bound p ≥ 2m should hold, and a weaker version of this conjecture for p obeying certain additional constraints, was proved in [4]. However, there are not definitive results concerning the distinctness of Kloosterman sums when p is small compared with m (see, e.g., [5]). In this regard, the present article makes a partial progress focusing on a certain set of Kloosterman sums for fixed p when m = 2n , n ≥ 1, and can be considered in some sense as an extension of the first result of Fischer. This paper is organized as follows. In the next section, we recall the background needed to present our results. Then in Section 3, we exhibit the results and their proofs. Finally, some conclusions are drawn in the las
Data Loading...
