Royal Holloway, University of London

The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. Decomposing a plethysm product into Schur functions, or equivalently, writing the corresponding composition of Schur functors as a direct sum of Schur functors, is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between representation theory and combinatorics. I will end with new results obtained in joint work with Rowena Paget (University of Kent) on stability on plethysm coefficients. No specialist background knowledge will be assumed.

The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.

In this talk, I'll discuss some personal experiences - good, bad, and
ugly - of disclosing vulnerabilities in a range of different cryptographic
standards and implementations. I'll try to draw some general lessons about
what works well and what does not.

The Learning with Errors (LWE) problem has become a central building block of modern cryptographic constructions. We will discuss hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and propose a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the Learning with Errors problem.

Quantum computers derive their computational power from the ability to manipulate superposition states of quantum registers. The generic quantum attack against a symmetric encryption scheme with key length n using Grover's algorithm has O(2^(n/2)) time complexity. For this kind of attack, an adversary only needs classical access to an encryption oracle. In this talk I discuss adversaries with quantum superposition access to encryption and decryption oracles. First I review and extend work by Kuwakado and Morii showing that a quantum computer with superposition access to an encryption oracle can break the Even-Mansour block cipher with key length n using only O(n) queries. Then, improving on recent work by Boneh and Zhandry, I discuss indistinguishability notions in chosen plaintext and chosen ciphertext attacks by a quantum adversary with superposition oracle access and give constructions that achieve these security notions.