2026-04-22_Saad-Eddin.txt

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Computer Algebra and Applicaitons Seminar

Title: Relaxed Prime Structures in the Study of RSA Integers

Speaker: Priv.-Doz. Dr. Sumaia Saad Eddin (RICAM)

Time and Location: Wednesday, April 22, 2026, 2 p.m

Abstract:

In this talk, I study integers represented as the product of two
primes satisfying additional constraints, with a particular focus on
biases in their distribution in arithmetic progressions. We present
recent results describing these distributional phenomena and discuss
their relevance to cryptographic settings, in particular the RSA
cryptosystem.

In the second part, we turn to the study of safe primes and related
structures. A prime p is called safe if (p−1)/2 is also prime, in
which case (p−1)/2 is a Sophie Germain prime. Such primes play an
important role in cryptography, notably in finite-field Diffie–Hellman
protocols and, increasingly, in certain approaches to RSA key
generation.

Despite their importance, the distribution of safe primes remains
poorly understood from an analytic perspective. In particular, no
asymptotic formula is known, even under strong hypotheses, and very
little is established for RSA-type integers composed of two safe
primes.

Motivated by these difficulties, we propose a new framework based on
relaxed prime structures, aimed at preserving key arithmetic features
of safe primes while providing a more tractable setting for analytic
investigation.