Abstract.We provide the first construction of a hash function into ordinary elliptic curves that is indifferentiable from a random oracle, based on Icart's deterministic encoding from Crypto 2009. While almost as efficient as Icart's encoding, this hash function can be plugged into any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle model.We also describe a more general (but less efficient) construction that works for a large class of encodings into elliptic curves, for example the Shallue-Woestijne-Ulas (SWU) algorithm. Finally we describe the first deterministic encoding algorithm into elliptic curves in characteristic 3.
Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.
It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. A lot of attacks has led to design criteria for these functions; mainly: balancedness, a high algebraic degree, a high nonlinearity, a good behavior against Fast Algebraic Attacks and also a high algebraic immunity (which is now an absolutely necessary criterion (but not sufficient) for cryptographic Boolean functions).Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many good cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true:Conjecture 0.1. Let S t,k be the following set:Then:The main contribution of the present paper is the reformulation of the problem in terms of carries which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of |S t,k | for numbers whose binary expansion is made of one block (see theorem 3.8), a proof that the conjecture is asymptotically true (see theorem 3.12) and a proof that a family of numbers (whose binary expansion has a high number of 1s and isolated 0s) reaches the bound of the conjecture (see theorem 3.17). We also conjecture that the numbers in that family are the only ones reaching the bound (see conjecture 3.20).
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