# Abstracts for Joint MPIM and B-IT workshop on Number Theory and Cryptography

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## On the counting function of the range of the Carmichael $\lambda$-function

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Speaker:
Florian Luca
Date:
Wed, 2013-11-20 10:00 - 11:00
Location:
MPIM Lecture Hall

The Carmichael $\lambda$-function associates to $n$ the exponent $\lambda(n)$ of the multiplicative group modulo $n$. In my talk, I will describe the main ideas behind the proof that the counting function $\#\{\lambda(n)\le x\}$ of the range of the Carmichael function $\lambda(n)$ below $x$ is $x/(\log x)^{\eta+o(1)}$ as $x\to\infty$, where $\eta=1-(1+\log\log 2)/\log 2=0.08607\ldots$ is the Erdős-Tenebaum-Ford constant. The proof uses sieve methods. This is joint work with Kevin Ford and Carl Pomerance.

## On the period of some pseudo-random number generators and "number-theoretical turbulence"

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Speaker:
Par Kurlberg
Date:
Wed, 2013-11-20 11:15 - 12:15
Location:
MPIM Lecture Hall

Given coprime integers b and n, let ord(b,n) be the
multiplicative order of b modulo n.  The length of the periods of some
popular pseudorandom number generators (e.g., the linear congruential
generator, and the Blum-Blum-Shub generator) turns out to be related
to ord(b,n) for appropriately chosen b and n.  We will investigate
some conclusions by V.I. Arnold (based on numerics by F. Aicardy as
well as analogies with the physical principle of turbulence) on the
average of ord(b,n), as n ranges over integers.  We will also give

## A refinement of the abc conjecture

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Speaker:
Cameron Stewart
Date:
Wed, 2013-11-20 14:45 - 15:45
Location:
MPIM Lecture Hall

We shall discuss joint work with Robert and Tenenbaum on a
proposed refinement of the well known abc conjecture.

## Complex multiplication in cryptography

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Speaker:
Peter Stevenhagen
Date:
Wed, 2013-11-20 16:30 - 17:30
Location:
MPIM Lecture Hall

Algorithms going under the name `complex multiplication' typically have a run time that is exponential in the size of the input data. We will show that nevertheless such algorithms may sometimes be profitably used in cryptographic settings.

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