## Public defence in Mathematics, M.Sc. (Tech) Pihla Karanko Sept. 19, 2024

9. September 2024

**Doctoral student:** Pihla Karanko

**Opponent:** Assistant Professor Pavel Hubáček, Czech Academy of Sciences, Czech Republic

**Custos:** Associate Professor Chris Brzuska, Aalto University School of Science, Department of Mathematics and Systems Analysis

Cryptography uses mathematical models to simulate real-world scenarios involving secrecy. Since we cannot know what an adversary might do (e.g. use supercomputers to break encryption), researchers try to model worst-case scenarios. Security is defined through "games" where a powerful unspecified adversarial algorithm attempts to break the system under controlled conditions. For example, in studying pseudorandom functions (PRFs), the adversary tries to distinguish between true PRF outputs and random bitstrings. If the adversary cannot guess correctly more than 50 % of the time, the PRF in question is considered secure.

Currently, no algorithm is proven to satisfy the rigorous security definitions for PRFs or other cryptographic tools. Instead, real-life systems rely on plausible candidates that have withstood extensive scrutiny. This reliance on unproven assumptions motivates efforts to reduce and better understand them. E.g. a PRF can be built from a one-way function (OWF), a tool with a simpler security definition.

Main Results:

- The thesis studies ways to get a OWF from a weaker (i.e. easier to break) OWF. We show that the existing methods are likely optimal in efficiency, highlighting the importance of the input distribution in such security amplification techniques. - We transform a weak PRF into a strong one using a special technique. This approach has practical applications in secure password handling, allowing more efficient authentication, where the user does not need to reveal the password to the server it is authenticating to.

- We propose a modification to a popular public key encryption mechanism, Fujisaki-Okamoto (FO) Transform, used in post-quantum secure encryption schemes. Our modification provides a more robust security proof.

- We propose a new security definition for garbling which is a method that allows outsourcing computations to untrusted servers without revealing the function or data. Our definition ensures strong security and efficient input encoding, overcoming current limitations when garbling cryptographic functions, useful for maintaining security when multiple servers encrypt a message jointly and one server becomes corrupted.

Key words: theoretical cryptography, one-way function, pseudorandom function

Thesis available for public display 10 days prior to the defence at: https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Contact information:

Email pihla.karanko@aalto.fi

Doctoral theses at the School of Science: https://aaltodoc.aalto.fi/handle/123456789/52

## Armaan Hooda spent one day as a math professor in Aalto

30. August 2024

Here you can read the whole article with Armaan's interview on Aalto's website.

## Public defence in Mathematics, M.Sc. (Tech) Kim Myyryläinen June 6, 2024

28. May 2024

Functions of bounded mean oscillation (BMO) and Muckenhoupt weights are essential concepts in modern harmonic analysis. They are used to measure the oscillation of a function at respective scales and to distribute mass to the underlying space. The theory of BMO and Muckenhoupt weights is comprehensive and has many applications also in other fields of mathematics such as partial differential equations.

This thesis studies parabolic bounded mean oscillation and Muckenhoupt weights that are generalizations of the classical time-independent concepts. The objective of the study is to extend the classical theory of BMO and Muckenhoupt weights to the time-dependent parabolic setting. Parabolic BMO and Muckenhoupt weights arise from parabolic partial differential equations, in particular, a doubly nonlinear equation that models nonlinear diffusion. The doubly nonlinear equation is a generalization of the classical heat equation modelling heat conduction.

The results obtained in the thesis include various decay estimates for the oscillation of a function. We show weighted norm inequalities for a parabolic maximal function and obtain a complete theory for the limiting parabolic Muckenhoupt class including characterization and factorization results. The theory for the other endpoint class is also developed and several new characterizations and self-improving phenomena are discovered. The proofs of our results apply covering, chaining and decomposition techniques adapted to the parabolic geometry. In addition to the parabolic framework, other forms of oscillation are considered in the general context of metric measure spaces.

Thesis available for public display 10 days prior to the defence at: https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Contact information: kim.myyrylainen@aalto.fi

## Camilla Hollanti elected as a member of Finnish Academy of Science and Letters

26. April 2024

The Finnish Academy of Science and Letters invites academics and scientists who have distinguished themselves in their own fields to become members. Membership of the Finnish Academy of Science and Letters, in common with membership of any other academy of science, is looked on as a considerable achievement in a person’s academic career.

One of the new members is professor of mathematics Camilla Hollanti. Hollanti leads a prominent research group on applications of algebra and number theory. The central topic of her research is to apply the methods of algebra and number theory to problems encountered in communication systems. Applications include wireless security and secure distributed computation.

The academy was founded in 1908 with the aims of promoting scientific research and acting as a bond between those engaged in advanced research.

It arranges meetings, discussions and educational events, in addition to producing scientific publications and issuing comments on topical issues pertaining research and researchers. The Academy of Science and Letters also distributes grants chiefly to young researchers.

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