On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG

On the Uniformity of Distribution of the Naor–Reingold Pseudo-Random Function

AbstractWe show that the new pseudo-random number function, introduced recently by M. Naor and O. Reingold, possesses one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions

Nonlinear complexity of the Naor-Reingold pseudo-random function

This is a preprint of a book chapter published in Lecture Notes in Computer Science,1787, Springer-Verlag, Berlin (2000). The original publication is available at www.springerlink.com.We obtain an exponential lower bound on the non-linear complexity of the new pseudo-random function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower bound on the linear complexity of this function that has been obtained by F. Griffin and I. E. Shparlinski

Distribution and Polynomial Interpolation of the Dodis-Yampolskiy Pseudo-Random Function

International audienceWe give some theoretical support to the security of the cryptographic pseudo-random function proposed by Dodis and Yampolskiy in 2005. We study the distribution of the function values over general finite fields and over elliptic curves defined over prime finite fields. We also prove lower bounds on the degree of polynomials interpolating the values of these functions in these two settings

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

Algebraic Frameworks for Cryptographic Primitives

A fundamental goal in theoretical cryptography is to identify the conceptually simplest abstractions that generically imply a collection of other cryptographic primitives. For symmetric-key primitives, this goal has been accomplished by showing that one-way functions are necessary and sufficient to realize primitives ranging from symmetric-key encryption to digital signatures. By contrast, for asymmetric primitives, we have no (known) unifying simple abstraction even for a few of its most basic objects. Moreover, even for public-key encryption (PKE) alone, we have no unifying abstraction that all known constructions follow. The fact that almost all known PKE constructions exploit some algebraic structure suggests considering abstractions that have some basic algebraic properties, irrespective of their concrete instantiation. We make progress on the aforementioned fundamental goal by identifying simple and useful cryptographic abstractions and showing that they imply a variety of asymmetric primitives. Our general approach is to augment symmetric abstractions with algebraic structure that turns out to be sufficient for PKE and much more, thus yielding a “bridge” between symmetric and asymmetric primitives. We introduce two algebraic frameworks that capture almost all concrete instantiations of (asymmetric) cryptographic primitives, and we also demonstrate their applicability by showing their cryptographic implications. Therefore, rather than manually building different cryptosystems from a new assumption, one only needs to build one (or more) of our simple structured primitives, and a whole host of cryptosystems immediately follows.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/166137/1/alamati_1.pd

Statically Aggregate Verifiable Random Functions and Application to E-Lottery

Cohen, Goldwasser, and Vaikuntanathan (TCC\u2715) introduced the concept of aggregate pseudo-random functions (PRFs), which allow efficiently computing the aggregate of PRF values over exponential-sized sets. In this paper, we explore the aggregation augmentation on verifiable random function (VRFs), introduced by Micali, Rabin and Vadhan (FOCS\u2799), as well as its application to e-lottery schemes. We introduce the notion of static aggregate verifiable random functions (Agg-VRFs), which perform aggregation for VRFs in a static setting. Our contributions can be summarized as follows: (1) we define static aggregate VRFs, which allow the efficient aggregation of VRF values and the corresponding proofs over super-polynomially large sets; (2) we present a static Agg-VRF construction over bit-fixing sets with respect to product aggregation based on the q-decisional Diffie-Hellman exponent assumption; (3) we test the performance of our static Agg-VRFs instantiation in comparison to a standard (non-aggregate) VRF in terms of costing time for the aggregation and verification processes, which shows that Agg-VRFs lower considerably the timing of verification of big sets; and (4) by employing Agg-VRFs, we propose an improved e-lottery scheme based on the framework of Chow et al.\u27s VRF-based e-lottery proposal (ICCSA\u2705). We evaluate the performance of Chow et al.\u27s e-lottery scheme and our improved scheme, and the latter shows a significant improvement in the efficiency of generating the winning number and the player verification

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes