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Türkiye'deki Matematiksel Etkinlikler


04 Mayıs 2017, 15:00


Mimar Sinan Güzel Sanatlar Üniversitesi Matematik Bölümü Seminerleri

On a conjecture of Morgan and Mullen

Giorgos Kapetanakis
Sabancı University, Türkiye

Let $F_q$ be the finite field of cardinality $q$ and $F_{q^n}$ its extension of degree $n$, where $q$ is a prime power and $n$ is a positive integer. A generator of the multiplicative group $F_{q^n}^*$ is called primitive. Besides their theoretical interest, primitive elements of finite fields are widely used in various applications, including cryptographic schemes, such as the Diffieellman key exchange. An $F_{q}$-normal basis of $F_{q^n}$ is an $F_{q}$-basis of $F_{q^n}$ of the form $\{x, x^q, . . . , x^{q^{n−1}}\}$ and the element $x\in F_{q^n}$ is called normal over $F_{q}$. These bases bear computational advantages for finite field arithmetic, so they have numerous applications, mostly found in coding theory and cryptography. An element of $F_{q^n}$ that is simultaneously normal over $F_{q^l}$ for all $l|n$ is called completely normal over $F_{q}$. It is well-known that primitive and normal elements exist for every $q$ and $n$. The existence of elements that are simultaneously primitive and normal is also well-known for every $q$ and $n$. Further, it is also known that for all q and n there exist completely normal elements of $F_{q^n}$ over $F_{q}$. Morgan and Mullen [Util. Math., 49:21–43, 1996], took the next step and conjectured that for any $q$ and $n$, there exists a primitive completely normal element of $F_{q^n}$ over $F_{q}$. In order to support their claim, they provided examples for such elements for all pairs $(q, n)$ with $q\leq 97$ and $q^n < 10^{50}$. This conjecture is yet to be established for arbitrary $q$ and $n$, but instead we have partial results, covering special types of extensions. Recently, Hachenberger [Des. Codes Cryptogr., 80(3):577– 586, 2016] using elementary methods, proved the validity of the Morgan-Mullen conjecture for $q\leq n^3$ and $n\geq 37$. In this work, we use character sum techniques and prove the validity of the Morgan-Mullen conjecture for all $q$ and $n$, provided that $q > n4. In the talk, the previous results will briefly be presented, our proof will be outlined and possible improvements will be discussed.
Cebir İngilizce
Seminar room, Bomonti Campus, MSGSÜ

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