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

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

F-Theory on Spin(7) Manifolds: Weak-Coupling Limit

F-theory on appropriately fibered Spin(7) holonomy manifolds is defined to arise as the dual of M-theory on the same space in the limit of a shrinking fiber. A class of Spin(7) orbifolds can be constructed as quotients of elliptically fibered Calabi-Yau fourfolds by an anti-holomorphic involution. The F-theory dual then exhibits one macroscopic dimension that has the topology of an interval. In this work we study the weak-coupling limit of a subclass of such constructions and identify the objects that arise in this limit. On the Type IIB side we find space-time filling O7-planes as well as O5-planes and orbifold five-planes with a (-1)^{F_L} factor localised on the interval boundaries. These orbifold planes are referred to as X5-planes and are S-dual to a D5-O5 system. For other involutions exotic O3-planes and X3-planes on top of a six-dimensional orbifold singularity can appear. We show that the objects present preserve a mutual supersymmetry of four supercharges in the bulk of the interval and two supercharges on the boundary. It follows that in the infinite-interval and weak-coupling limit full four-dimensional N=1 supersymmetry is restored, which on the Type IIA side corresponds to an enhancement of supersymmetry by winding modes in the vanishing interval limit.Comment: 23 page