Conditionals in Homomorphic Encryption and Machine Learning Applications

Homomorphic encryption aims at allowing computations on encrypted data without decryption other than that of the final result. This could provide an elegant solution to the issue of privacy preservation in data-based applications, such as those using machine learning, but several open issues hamper this plan. In this work we assess the possibility for homomorphic encryption to fully implement its program without relying on other techniques, such as multiparty computation (SMPC), which may be impossible in many use cases (for instance due to the high level of communication required). We proceed in two steps: i) on the basis of the structured program theorem (Bohm-Jacopini theorem) we identify the relevant minimal set of operations homomorphic encryption must be able to perform to implement any algorithm; and ii) we analyse the possibility to solve -- and propose an implementation for -- the most fundamentally relevant issue as it emerges from our analysis, that is, the implementation of conditionals (requiring comparison and selection/jump operations). We show how this issue clashes with the fundamental requirements of homomorphic encryption and could represent a drawback for its use as a complete solution for privacy preservation in data-based applications, in particular machine learning ones. Our approach for comparisons is novel and entirely embedded in homomorphic encryption, while previous studies relied on other techniques, such as SMPC, demanding high level of communication among parties, and decryption of intermediate results from data-owners. Our protocol is also provably safe (sharing the same safety as the homomorphic encryption schemes), differently from other techniques such as Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical section on polynomial approximatio

Search for the most stable massive state in superstring theory

In ten dimensional type II superstring, all perturbative massive states are unstable, typically with a short lifetime compared to the string scale. We find that the lifetime of the average string state of mass M has the asymptotic form T < const.1/(g^2 M). The most stable string state seems to be a certain state with high angular momentum which can be classically viewed as a circular string rotating in several planes ("the rotating ring"), predominantly decaying by radiating soft massless NS-NS particles, with a lifetime T = c_0 M^5/g^2. Remarkably, the dominant channel is the decay into a similar rotating ring state of smaller mass. The total lifetime to shrink to zero size is ~ M^7. In the presence of D branes, decay channels involving open strings in the final state are exponentially suppressed, so the lifetime is still proportional to M^5, except for a D brane at a special angle or flux. For large mass, the spectrum for massless emission exhibits qualitative features typical of a thermal spectrum, such as a maximum and an exponential tail. We also discuss the decay properties of rotating rings in the case of compact dimensions.Comment: 24 pages, 1 figure. Correction on lifetime of average stat

Cross sections for production of closed superstrings at high energy colliders in brane world models

In brane world string models with large extra dimensions, there are processes where fermion and antifermion (or two gluons) can annihilate producing a light particle (e.g. gluon) carrying transverse momentum and a Kaluza-Klein graviton or an excited closed string that propagates in the extra dimensions. In high energy colliders, this process gives a missing momentum signature. We compute the total cross section for this process within the context of type II superstring theory in the presence of a D brane. This includes all missing energy sources for this string theory model up to s=8M_s^2, and it can be used to put new limits on the string scale M_s.Comment: 18 pages, 3 figures. Reference adde

Self-interacting fundamental strings and black holes

We study the size distribution of very massive close string states and the typical string configuration as one slowly increase the string coupling, both in the case of zero and of non-zero Neveu-Schwarz charges. The computations are performed rigorously in string theory, starting from quantities that are well-defined in the theory and therefore clarify previous works on the subject which were based on various approximation techniques. We find that, starting from a value of the coupling in agreement with the one predicted by the black hole correspondence principle, the string ensemble is dominated in any dimensions by compact states whose size is within the correspondent black hole horizon radius, which is of the order of the string scale at the black hole/string transition point.Comment: 31 pages plus references, final version accepted for publicatio