Blum Integers Research Articles

Improvements on Non-Interactive Zero-Knowledge Proof Systems Related to Quadratic Residuosity Languages

Non-interactive zero-knowledge (NIZK) proof systems are very useful for statement verification without interaction in cryptography; they entail just one message, called the proof, that convinces the verifier of the truth of the statement without leaking any extra information. Many NIZK proof systems are related to quadratic residuosity languages and follow three main steps. First, the prover and the verifier sample a common reference string (CRS) in some particular form. Second, the prover proves that n is a Blum integer (BL,n=p1t1·p2t2, where p1 and p2 are different primes both congruent to 3 modulo 4, and t1 and t2 are odd). Third, various statements (e.g., NQRn,ORn,ANDn,T(k,t), and 3SAT) can be proven by the prover based on the properties of BL. In this study, we improve the NIZK proof system for n∈BL based on the special distribution of the square roots modulo n and embed this proof in the third step. Moreover, we show that the CRS can be sampled more efficiently using the improved process.

Double-authentication-preventing signatures

Digital signatures are often used by trusted authorities to make unique bindings between a subject and a digital object; for example, certificate authorities certify a public key belongs to a domain name, and time-stamping authorities certify that a certain piece of information existed at a certain time. Traditional digital signature schemes however impose no uniqueness conditions, so a trusted authority could make multiple certifications for the same subject but different objects, be it intentionally, by accident, or following a (legal or illegal) coercion. We propose the notion of a double-authentication-preventing signature, in which a value to be signed is split into two parts: a subject and a message. If a signer ever signs two different messages for the same subject, enough information is revealed to allow anyone to compute valid signatures on behalf of the signer. This double-signature forgeability property discourages signers from misbehaving--a form of self-enforcement--and would give binding authorities like CAs some cryptographic arguments to resist legal coercion. We give a generic construction using a new type of trapdoor functions with extractability properties, which we show can be instantiated using the group of sign-agnostic quadratic residues modulo a Blum integer; we show an additional application of these new extractable trapdoor functions to standard digital signatures.

Cca-Secure Key Encapsulation Mechanism Based on Factoring Assumption

ABSTRACT In this article a key encapsulation mechanism is presented which is based on squaring function, where the input element is from QR+N, where QR+N denotes the signed quadratic residue group, and N is a Blum integer. The article presents the soundness, the efficiency and the proof of CCA security of the proposed mechanism

Non-Automatizability of Bounded-Depth Frege Proofs

In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that bounded-depth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the Diffie-Hellman function is sufficiently hard. It follows as a corollary that bounded-depth Frege is not automatizable; in other words, there is no deterministic polynomial-time algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.

Number-theoretic constructions of efficient pseudo-random functions

We describe efficient constructions for various cryptographic primitives in private-key as well as public-key cryptography. Our main results are two new constructions of pseudo-random functions. We prove the pseudo-randomness of one construction under the assumption that factoring (Blum integers) is hard while the other construction is pseudo-random if the decisional version of the Diffie--Hellman assumption holds. Computing the value of our functions at any given point involves two subset products. This is much more efficient than previous proposals. Furthermore, these functions have the advantage of being in TC 0 (the class of functions computable by constant depth circuits consisting of a polynomial number of threshold gates). This fact has several interesting applications. The simple algebraic structure of the functions implies additional features such as a zero-knowledge proof for statements of the form " y = f s ( x )" and " y ≠ f s ( x )" given a commitment to a key s of a pseudo-random function f s .

Equivalences and Separations Between Quantum and Classical Learnability

We consider quantum versions of two well-studied models of learning Boolean functions: Angluin's model of exact learning from membership queries and Valiant's probably approximately correct (PAC) model of learning from random examples. For each of these two learning models we establish a polynomial relationship between the number of quantum or classical queries required for learning. These results contrast known results that show that testing black-box functions for various properties, as opposed to learning, can require exponentially more classical queries than quantum queries. We also show that, under a widely held computational hardness assumption (the intractability of factoring Blum integers), there is a class of Boolean functions which is polynomial-time learnable in the quantum version but not the classical version of each learning model. For the model of exact learning from membership queries, we establish a stronger separation by showing that if any one-way function exists, then there is a class of functions which is polynomial-time learnable in the quantum setting but not in the classical setting. Thus, while quantum and classical learning are equally powerful from an information theory perspective, the models are different when viewed from a computational complexity perspective.

On the linear complexity profile of the power generator

We obtain a lower bound on the linear complexity profile of the power generator of pseudo-random numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the Blum-Blum-Shub (1986) generator. In particular, these results imply that lattice reduction attacks on such generators are not feasible.

On Interpolation and Automatization for Frege Systems

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC0-Frege. As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

STRONG THEORETICAL STEGNOGRAPHY

Hiding the existence of a message can be an important technique in this era of terabit networks. One technique for practicing this obfuscation, Mimic Functions, is derived from Context-Free Grammars and can be as secure as inverting RSA or factoring Blum integers. This paper discusses the implications of the result and presents a practical solution for securely hiding information from inspection.

When Won′t Membership Queries Help?

We investigate cryptographic limitations on the power of membership queries to help with concept learning. We extend the notion of prediction-preserving reductions to prediction with membership queries. We exhibit a number of reductions and show several prediction problems to be complete for different complexity classes. We show that assuming the intractability of (1) quadratic residues module a composite, (2) inverting RSA encryption, or (3) factoring Blum integers, there is no polynomial time prediction algorithm with membership queries for Boolean formulas, constant depth threshold circuits, 3 μ-Boolean formulas, finite unions or intersections of DFAs, 2-way DFAs, NFAs, or CFGs. Also, we show that if there exist one-way functions that cannot be inverted by polynomial-sized circuits, then CNF or DNF formulas and convex polytopes intersected with the Boolean hypercube are either polynomial time predictable without membership queries, or they are not polynomial time predictable even with membership queries; so, in effect, membership queries will not help with predicting CNF or DNF formulas.

Cryptographic limitations on learning Boolean formulae and finite automata

In this paper, we prove the intractability of learning several classes of Boolean functions in the distribution-free model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent , in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of well-known public-key cryptosystems to the learning problems. We prove that a polynomial-time learning algorithm for Boolean formulae, deterministic finite automata or constant-depth threshold circuits would have dramatic consequences for cryptography and number theory. In particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.

The discrete logarithm modulo a composite hides 0(n) Bits

In this paper we consider the one-way function f g, N ( X) = g X (mod N), where N is a Blum integer. We prove that under the commonly assumed intractability of factoring Blum integers, all its bits are individually hard, and the lower as well as upper halves of them are simultaneously hard. As a result, f g, N can be used in efficient pseudo-random bit generators and multi-bit commitment schemes, where messages can be drawn according to arbitrary probability distributions.