Homomorphic encryption and some black box attacks

This paper is a compressed summary of some principal definitions and concepts in the approach to the black box algebra being developed by the authors. We suggest that black box algebra could be useful in cryptanalysis of homomorphic encryption schemes, and that homomorphic encryption is an area of research where cryptography and black box algebra may benefit from exchange of ideas

Representation Theory of Finite Semigroups over Semirings

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory

Tribimaximal Mixing From Small Groups

Current experimental data on the neutrino parameters is in good agreement with tribimaximal mixing and may indicate the presence of an underlying family symmetry. For 76 flavor groups, we perform a systematic scan for models: The particle content is that of the Standard Model plus up to three flavon fields, and the effective Lagrangian contains all terms of mass dimension <=6. We find that 44 groups can accommodate models that are consistent with experiment at 3 sigma, and 38 groups can have models that are tribimaximal. For one particular group, we look at correlations between the mixing angles and make a prediction for theta13 that will be testable in the near future. We present the details of a model with theta12=33.9, theta23=40.9, theta13=5.1 to show that the recent tentative hints of a non-zero theta13 can easily be accommodated. The smallest group for which we find tribimaximal mixing is T7. We argue that T7 and T13 are as suited to produce tribimaximal mixing as A4 and should therefore be considered on equal footing. In the appendices, we present some new mathematical methods and results that may prove useful for future model building efforts.Comment: 44 pages, 7 figures. Typos corrected, references added, figures update

Black Box White Arrow

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting ${\rm (P)SL}_n(\mathbb{F}_q)$. One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting ${\rm SL}_2(2^n)$. Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory. The paper is further development of our text "Fifty shades of black" [arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248

Extending tensors on polar manifolds

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $\Sigma\subset M$ and generalized Weyl group $W$. We show that restriction to $\Sigma$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $\Sigma$. Moreover, we show that every smooth $W$-invariant Riemannian metric on $\Sigma$ can be extended to a smooth $G$-invariant Riemannian metric on $M$ with respect to which the $G$-action remains polar with the same section $\Sigma$.Comment: arXiv admin note: text overlap with arXiv:1205.476