16,500 research outputs found
Counting Prime Numbers with Short Binary Signed Representation
Modular arithmetic with prime moduli has been crucial in
present day cryptography. The primes of Mersenne, Solinas,
Crandall and the so called IKE-MODP have been extensively used
in efficient implementations.
In this paper we study the density of primes with binary
signed representation involving a small number of non-zero -digits
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Affine Determinant Programs: A Framework for Obfuscation and Witness Encryption
An affine determinant program ADP: {0,1}^n → {0,1} is specified by a tuple (A,B_1,...,B_n) of square matrices over F_q and a function Eval: F_q → {0,1}, and evaluated on x \in {0,1}^n by computing Eval(det(A + sum_{i \in [n]} x_i B_i)).
In this work, we suggest ADPs as a new framework for building general-purpose obfuscation and witness encryption. We provide evidence to suggest that constructions following our ADP-based framework may one day yield secure, practically feasible obfuscation.
As a proof-of-concept, we give a candidate ADP-based construction of indistinguishability obfuscation (iO) for all circuits along with a simple witness encryption candidate. We provide cryptanalysis demonstrating that our schemes resist several potential attacks, and leave further cryptanalysis to future work. Lastly, we explore practically feasible applications of our witness encryption candidate, such as public-key encryption with near-optimal key generation
Hard Instances of the Constrained Discrete Logarithm Problem
The discrete logarithm problem (DLP) generalizes to the constrained DLP,
where the secret exponent belongs to a set known to the attacker. The
complexity of generic algorithms for solving the constrained DLP depends on the
choice of the set. Motivated by cryptographic applications, we study sets with
succinct representation for which the constrained DLP is hard. We draw on
earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such
as generalized Menelaus' theorem for proving lower bounds on the complexity of
the constrained DLP, and construct sets with succinct representation with
provable non-trivial lower bounds
Hard isogeny problems over RSA moduli and groups with infeasible inversion
We initiate the study of computational problems on elliptic curve isogeny
graphs defined over RSA moduli. We conjecture that several variants of the
neighbor-search problem over these graphs are hard, and provide a comprehensive
list of cryptanalytic attempts on these problems. Moreover, based on the
hardness of these problems, we provide a construction of groups with infeasible
inversion, where the underlying groups are the ideal class groups of imaginary
quadratic orders.
Recall that in a group with infeasible inversion, computing the inverse of a
group element is required to be hard, while performing the group operation is
easy. Motivated by the potential cryptographic application of building a
directed transitive signature scheme, the search for a group with infeasible
inversion was initiated in the theses of Hohenberger and Molnar (2003). Later
it was also shown to provide a broadcast encryption scheme by Irrer et al.
(2004). However, to date the only case of a group with infeasible inversion is
implied by the much stronger primitive of self-bilinear map constructed by
Yamakawa et al. (2014) based on the hardness of factoring and
indistinguishability obfuscation (iO). Our construction gives a candidate
without using iO.Comment: Significant revision of the article previously titled "A Candidate
Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the
constructions by giving toy examples, added "The Parallelogram Attack" (Sec
5.3.2). 54 pages, 8 figure
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