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Pseudorandomness at prime times and digits of Mersenne numbers
Virtualconference
Authors : Shparlinski, Igor (Author of the conference)
CIRM (Publisher )
CIRM (Publisher )
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Abstract :
We consider two common pseudorandom number generators constructed from iterations of linear and Möbius maps
$x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these orbits at prime times, that is, we study elements $u_{p}$, $p = 2, 3, . . .$, where $p$ runs over primes.
We give a short survey of previous results on the distribution of $u_{p}$ for the above maps and then:
- Explain how B. Kerr, L. Mérai and I. E. Shparlinski (2019) have used a method of N. M. Korobov (1972) to study the congruential generator on primes modulo a large power of a fixed prime, e.g. $q=3^{\gamma }$ with a large $\gamma$. We also give applications of this result to digits of Mersenne numbers $2^{p}-1$.
- Present a result of L. Mérai and I. E. Shparlinski (2020) on the distribution of the inversive generator on primes modulo a large prime, q. The proof takes advantage of the flexibility of Heath-Brown's identity, while Vaughan's identity does not seem to be enough for our purpose. We also pose several open questions and discuss links to Sarnak's conjecture on pseudorandomness of the Möbius function.
MSC Codes : $x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these orbits at prime times, that is, we study elements $u_{p}$, $p = 2, 3, . . .$, where $p$ runs over primes.
We give a short survey of previous results on the distribution of $u_{p}$ for the above maps and then:
- Explain how B. Kerr, L. Mérai and I. E. Shparlinski (2019) have used a method of N. M. Korobov (1972) to study the congruential generator on primes modulo a large power of a fixed prime, e.g. $q=3^{\gamma }$ with a large $\gamma$. We also give applications of this result to digits of Mersenne numbers $2^{p}-1$.
- Present a result of L. Mérai and I. E. Shparlinski (2020) on the distribution of the inversive generator on primes modulo a large prime, q. The proof takes advantage of the flexibility of Heath-Brown's identity, while Vaughan's identity does not seem to be enough for our purpose. We also pose several open questions and discuss links to Sarnak's conjecture on pseudorandomness of the Möbius function.
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 01/12/2020 Conference Date : 25/11/2020 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:51:44 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-11-25_Shparlinski.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoireEvent Organizers : Rivat, Joël ; Tichy, Robert Dates : 23/11/2020 - 27/11/2020 Event Year : 2020 Event URL : https://www.chairejeanmorlet.com/2256.html
Citation Data
DOI : 10.24350/CIRM.V.19689303Cite this video as: Shparlinski, Igor (2020). Pseudorandomness at prime times and digits of Mersenne numbers. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19689303 URI : http://dx.doi.org/10.24350/CIRM.V.19689303 |
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