Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry

We generalize the supersymmetry method in Random Matrix Theory to arbitrary rotation invariant ensembles. Our exact approach further extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent Random Matrix Ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A projector is identified that provides the mapping of the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the non-linear sigma model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green functions.Comment: 36 page

Family of solvable generalized random-matrix ensembles with unitary symmetry

We construct a very general family of characteristic functions describing Random Matrix Ensembles (RME) having a global unitary invariance, and containing an arbitrary, one-variable probability measure which we characterize by a `spread function'. Various choices of the spread function lead to a variety of possible generalized RMEs, which show deviations from the well-known Gaussian RME originally proposed by Wigner. We obtain the correlation functions of such generalized ensembles exactly, and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality.Comment: 4 pages, to be published in Phys. Rev. E, Rapid Com

How to Construct Cryptosystems and Hash Functions in Weakened Random Oracle Models

In this paper, we discuss how to construct secure cryptosystems and secure hash functions in weakened random oracle models. ~~~~The weakened random oracle model (\wrom), which was introduced by Numayama et al. at PKC 2008, is a random oracle with several weaknesses. Though the security of cryptosystems in the random oracle model, \rom, has been discussed sufficiently, the same is not true for \wrom. A few cryptosystems have been proven secure in \wrom. In this paper, we will propose a new conversion that can convert \emph{any} cryptosystem secure in \rom to a new cryptosystem that is secure in the first preimage tractable random oracle model \fptrom \emph{without re-proof}. \fptrom is \rom without preimage resistance and so is the weakest of the \wrom models. Since there are many secure cryptosystems in \rom, our conversion can yield many cryptosystems secure in \fptrom. ~~~~The fixed input length weakened random oracle model, \filwrom, introduced by Liskov at SAC 2006, reflects the known weakness of compression functions. We will propose new hash functions that are indifferentiable from \ro when the underlying compression function is modeled by a two-way partially-specified preimage-tractable fixed input length random oracle model (\wfilrom). \wfilrom is \filrom without two types of preimage resistance and is the weakest of the \filwrom models. The proposed hash functions are more efficient than the existing hash functions which are indifferentiable from \ro when the underlying compression function is modeled by \wfilrom

Polymer Statistics and Fermionic Vector Models

We consider a variation of $O(N)$-symmetric vector models in which the vector components are Grassmann numbers. We show that these theories generate the same sort of random polymer models as the $O(N)$ vector models and that they lie in the same universality class in the large-$N$ limit. We explicitly construct the double-scaling limit of the theory and show that the genus expansion is an alternating Borel summable series that otherwise coincides with the topological expansion of the bosonic models. We also show how the fermionic nature of these models leads to an explicit solution even at finite-$N$ for the generating functions of the number of random polymer configurations.Comment: 13 pages LaTeX, run twice. Minor technical details corrected (mainly in combinatorics for Feynman graphs) and clarifying comments added; additional reference include

A Hash Table Without Hash Functions, and How to Get the Most Out of Your Random Bits

This paper considers the basic question of how strong of a probabilistic guarantee can a hash table, storing $n$ $(1 + \Theta(1)) \log n$-bit key/value pairs, offer? Past work on this question has been bottlenecked by limitations of the known families of hash functions: The only hash tables to achieve failure probabilities less than 1 / 2^{\polylog n} require access to fully-random hash functions -- if the same hash tables are implemented using the known explicit families of hash functions, their failure probabilities become 1 / \poly(n). To get around these obstacles, we show how to construct a randomized data structure that has the same guarantees as a hash table, but that \emph{avoids the direct use of hash functions}. Building on this, we are able to construct a hash table using $O(n)$ random bits that achieves failure probability $1 / n^{n^{1 - \epsilon}}$ for an arbitrary positive constant $\epsilon$. In fact, we show that this guarantee can even be achieved by a \emph{succinct dictionary}, that is, by a dictionary that uses space within a $1 + o(1)$ factor of the information-theoretic optimum. Finally we also construct a succinct hash table whose probabilistic guarantees fall on a different extreme, offering a failure probability of 1 / \poly(n) while using only $\tilde{O}(\log n)$ random bits. This latter result matches (up to low-order terms) a guarantee previously achieved by Dietzfelbinger et al., but with increased space efficiency and with several surprising technical components