Number-theoretic Problems Research Articles

Identity-based threshold group signature scheme based on multiple hard number theoretic problems

We introduce in this paper a new identity-based threshold signature (IBTHS) technique, which is based on a pair of intractable problems, residuosity and discrete logarithm. This technique relies on two difficult problems and offers an improved level of security relative to an individual hard problem. The majority of the denoted IBTHS techniques are established on an individual difficult problem. Despite the fact that these methods are secure, however, a prospective solution of this sole problem by an adversary will enable him/her to recover the entire private data together with secret keys and configuration values of the associated scheme. Our technique is immune to the four most familiar attack types in relation to the signature schemes. Enhanced performance of our proposed technique is verified in terms of minimum cost of computations required by both of the signing algorithm and the verifying algorithm in addition to immunity to attacks.

Four rational squares in arithmetic progressions and a\xa0family of elliptic curves with positive Mordell-Weil rank

We study the question at which relative distances four squares of rational numbers can occur as terms in an arithmetic progression. This number-theoretical problem is seen to be equivalent to finding rational points on certain elliptic curves. Both number-theoretical results and results concerning the associated elliptic curves are derived; i.e., the correspondence between rational squares in arithmetic progressions and elliptic curves is exploited both ways.

Design of Identity-Based Blind Signature Scheme Upon Chaotic Maps

Cryptosystems relying on chaotic maps have been presented lately. As a result of inferred and convenient connections amongst the attributes of conventional cryptosystems and chaotic frameworks, the concept of chaotic systems with applications to cryptography has earned much consideration from scientists working in the various domains. Hence, we suggest a novel IDentity-based Blind Signature (ID-BS) based technique in this paper that relies on a pair of hard number theoretic problems, namely, the Chaotic Maps (CM) and FACtoring (FAC) problems. The technique is immune to attacks, in addition to its efficiency in application. Relative to other related schemes, it requires fewer module operations. In summary, our proposed technique is superior to similar schemes within the cryptosystems domain.

A Headache-Causing Problem

After disproving the celebrated Conway–Paterson–Moscow theorem [1], we prove that theorem and make an application to a well-known number-theoretic problem.

Hash-enhanced elliptic curve bit-string generator for medical image encryption

Bit-string generator (BSG) is based on the hardness of known number theoretical problems, such as the discrete logarithm problem with the elliptic curve (ECDLP). Such type of generators will have good randomness and unpredictability properties as it is challenged to find a solution regarding this mathematical dilemma. Hash functions in turn play a remarkable role in many cryptographic tasks to accomplish different security levels. Hash-enhanced elliptic curve bit-string generator (HEECBSG) mechanism is proposed in this study based on the ECDLP and secure hash function. The cryptographic hash function is used to achieve integrity and security of the obtained bit-strings for highly sensitive plain data. The main contribution of the proposed HEECBSG is transforming the x-coordinate of the elliptic curve points using a hash function H to generate bit-strings of any desirable length. The obtained pseudo-random bits are tested by the NIST test suite to analyze and verify its statistical and randomness properties. The resulted bit-string is utilized here for encrypting various medical images of the vital organs, i.e. the brain, bone, fetuses, and lungs. Then, extensive evaluation metrics have been applied to analyze the successful performance of the cipherimage, including key-space analysis, histogram analysis, correlation analysis, entropy analysis and sensitivity analysis. The results demonstrated that our proposed HEECBSG mechanism is feasible for achieving security and privacy purposes of the medical image transmission over unsecure communication networks.

Public key encryption with equality test in the standard model

Public key encryption with equality test (PKEET) is a cryptosystem that allows a tester who has trapdoors issued by one or more users Ui to perform equality tests on ciphertexts encrypted using public key(s) of Ui. Since this feature has a lot of practical applications including search on encrypted data, several PKEET schemes have been proposed so far. However, to the best of our knowledge, all the existing proposals are proven secure only under the hardness of number-theoretic problems and/or the random oracle heuristics.In this paper, we show that this primitive can be achieved not only generically from well-established other primitives but also even without relying on the random oracle heuristics. More precisely, our generic construction for PKEET employs a two-level hierarchical identity-based encryption scheme, which is selectively secure against chosen plaintext attacks, a strongly unforgeable one-time signature scheme and a cryptographic hash function. Our generic approach toward PKEET has several advantages over all the previous works; it directly leads the first standard model construction and also directly implies the first lattice-based construction. Finally, we show how to extend our approach to the identity-based setting.

A Survey on Fixed Divisors

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of $\Z,$ progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of $n \times n$ matrices over $\Z$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.

Idealisers in skew group rings

Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B=C#G be the skew group ring. For a prime ideal I◁C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. We obtain necessary and sufficient conditions for when this idealiser subring is left and right noetherian. We also give an example of these conditions in practice which translates to an interesting number theoretic problem.

A Hybrid-Based Verifiable Secret Sharing Scheme Using Chinese Remainder Theorem

It is not always in the best interests to rely on an individual to have control of entire sensitive information. This has led to the need for secret sharing schemes, which divide secret (key) among many participants or shareholders. To avoid any cheating by any of the shareholders, the need for verifiable secret sharing (VSS) has emerged. In this context, a hybrid approach for VSS scheme is suggested in this paper. The proposed algorithm shares multiple secrets among shareholders, where shareholders are also divided/classified into different levels. Hence, it includes multiple as well as multilevel secret sharing. Secrets can be recovered at intra- or inter-level, where shareholders of higher level can contribute their shares to lower levels. To reduce the complexity, the one-way hash function is used instead of the hard number-theoretic problems. The proposed scheme stands against the dishonest dealer and shareholders. To rule out a typical dishonest strategy of leaking secret information in the valid shares, the concept of dealer leakage resilience is used by reducing the dealer’s powers of selecting random values on his own. The execution is also done using cryptographic libraries. Finally, it is demonstrated that the scheme satisfies the security requirements of VSS.

Efficient identity-based multi-bit proxy re-encryption over lattice in the standard model

Proxy re-encryption (PRE) can share ciphertext data without leaking the decryption key of the data owner in cloud computing. Because of the special property, PRE achieves good reliability and secrecy. Nevertheless, most of the proposed proxy re-encryption schemes are based on the number theoretic problem and cannot resist quantum attacks. In addition, most existing researches over lattice focus on the single bit encryption, which leads to low efficiency. Therefore, according to these problems, an efficient identity-based multi-bit proxy re-encryption over lattice in the standard model is constructed, which enlarges the plaintext space and improves the efficiency. What's more, the proposed multi-use and bidirectional scheme is proved to be CPA secure in the standard model.

Quantum path computing: computing architecture with propagation paths in multiple plane diffraction of classical sources of fermion and boson particles

Quantum computing (QC) architectures utilizing classical or coherent resources with Gaussian transformations are classically simulable as an indicator of the lack of QC power. Simple optical set-ups utilizing wave-particle duality and interferometers achieve QC speed-up with the cost of exponential complexity of resources in time, space or energy. However, linear optical networks composed of single photon inputs and photon number measurements such as boson sampling achieve solving problems which are not efficiently solvable by classical computers while emphasizing the power of linear optics. In this article, quantum path computing (QPC) set-up is introduced as the simplest optical QC satisfying five fundamental properties all-in-one: exploiting only the coherent sources being either fermion or boson, i.e., Gaussian wave packet of standard laser, simple set-up of multiple plane diffraction (MPD) with multiple slits by creating distinct propagation paths, standard intensity measurement on the detector, energy efficient design and practical problem solving capability. MPD is unique with non-Gaussian transformations by realizing an exponentially increasing number of highly interfering propagation paths while making classical simulation significantly hard. It does not require single photon resources or number resolving detection mechanisms making the experimental implementation of QC significantly low complexity. QPC set-up is utilized for the solutions of specific instances of two practical and hard number theoretical problems: partial sum of Riemann theta function and period finding to solve Diophantine approximation. Quantumness of MPD with negative volume of Wigner function is numerically analyzed and open issues for the best utilization of QPC are discussed.

Threshold-directed signature scheme based on hybrid number theoretic problems

Directed signature is a type of function-based signature with the property that the signature only can be verified by a designated verifier and at certain times, the verifier should be able to convince anyone about the validity of the signature without revealing any secret information about the signature to the public. Taking into consideration the involvement of group decision making, some threshold directed signature schemes based on single number theoretic problems, such as integer factorization, discrete logarithm problem, and elliptic curve discrete logarithm problem, have been developed by cryptographers. Although the single-problem-based schemes are still invincible because there is still no cryptanalyst to find the solution to the problems, in the future, if the enemy or attacker manages to get the polynomial algorithm to solve the single problems, the schemes will no longer be practiced and applied. For such reason, in this paper, we propose a new threshold-directed signature scheme based on integer factorization and discrete logarithm problems. The advantage of our scheme is based on the assumption that it is very unlikely to solve two hard number theoretic problems simultaneously. We also show that our scheme is secured against some cryptographic attacks and also significantly efficient compared with threshold signature scheme based on single problem.

PKCHD: Towards a Probabilistic Knapsack Public-Key Cryptosystem with High Density

By introducing an easy knapsack-type problem, a probabilistic knapsack-type public key cryptosystem (PKCHD) is proposed. It uses a Chinese remainder theorem to disguise the easy knapsack sequence. Thence, to recover the trapdoor information, the implicit attacker has to solve at least two hard number-theoretic problems, namely integer factorization and simultaneous Diophantine approximation problems. In PKCHD, the encryption function is nonlinear about the message vector. Under the re-linearization attack model, PKCHD obtains a high density and is secure against the low-density subset sum attacks, and the success probability for an attacker to recover the message vector with a single call to a lattice oracle is negligible. The infeasibilities of other attacks on the proposed PKCHD are also investigated. Meanwhile, it can use the hardest knapsack vector as the public key if its density evaluates the hardness of a knapsack instance. Furthermore, PKCHD only performs quadratic bit operations which confirms the efficiency of encrypting a message and deciphering a given cipher-text.

New efficient identity based encryption without pairings

Identity based encryption (IBE) schemes were first constructed with, and often have been since, bilinear mappings (a.k.a. pairings) on elliptic curves. But the multiply and exponent operation using pairings is slowly and inefficiency in implementation. There were, however, some successful attempts to construct IBE schemes based on more traditional number theoretic problems. Unfortunately, most of the proposed schemes are impractical as a result of bandwidth utilization or the time complexity of performance. By this work, we present a new efficient IBE scheme without pairings, which is inspired from the trapdoor technique rooted in composite residuosity class problem. Firstly, our converted basic IBE scheme is proven, in the random oracle model, secure against chosen-plaintext attacks (CPA) under the assumptions that the decision composite residuosity and decision partial discrete logarithm problems are intractable. Moreover, we employ the technique of Fujisaki–Okamoto to transform the basic scheme into enhanced one for resisting chosen-ciphertext attacks (CCA).

Graph coloring and Graham’s greatest common divisor problem

In this paper we introduce and study two graph coloring problems and relate them to some deep number-theoretic problems. For a fixed positive integer k consider a graph Bk whose vertex set is the set of all positive integers with two vertices a,b joined by an edge whenever the two numbers a∕gcd(a,b) and b∕gcd(a,b) are both at most k. We conjecture that the chromatic number of every such graph Bk is equal to k. This would generalize the greatest common divisor problem of Graham from 1970; in graph-theoretic terminology it states that the clique number of Bk is equal to k. Our conjecture is connected to integer lattice tilings and partial Latin squares completions.

Counting isospectral manifolds

Given a simple Lie group H of real rank at least 2 we show that the maximum cardinality of a set of isospectral non-isometric H-locally symmetric spaces of volume at most x grows at least as fast as xclog⁡x/(log⁡log⁡x)2 where c=c(H) is a positive constant. In contrast with the real rank 1 case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [2]. Our proof uses Sunada's method, results of [2], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.

Quantum security analysis of a lattice-based oblivious transfer protocol

Because of the concise functionality of oblivious transfer (OT) protocols, they have been widely used as building blocks in secure multiparty computation and high-level protocols. The security of OT protocols built upon classical number theoretic problems, such as the discrete logarithm and factoring, however, is threatened as a result of the huge progress in quantum computing. Therefore, post-quantum cryptography is needed for protocols based on classical problems, and several proposals for post-quantum OT protocols exist. However, most post-quantum cryptosystems present their security proof only in the context of classical adversaries, not in the quantum setting. In this paper, we close this gap and prove the security of the lattice-based OT protocol proposed by Peikert et al. (CRYPTO, 2008), which is universally composably secure under the assumption of learning with errors hardness, in the quantum setting. We apply three general quantum security analysis frameworks. First, we apply the quantum lifting theorem proposed by Unruh (EUROCRYPT, 2010) to prove that the security of the lattice-based OT protocol can be lifted into the quantum world. Then, we apply two more security analysis frameworks specified for post-quantum cryptographic primitives, i.e., simple hybrid arguments (CRYPTO, 2011) and game-preserving reduction (PQCrypto, 2014).

Build your own clarithmetic I: Setup and completeness

Clarithmetics are number theories based on computability logic (see http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized/schematic version CLA11(P1,P2,P3,P4). By tuning the three parameters P1,P2,P3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a wide range of target tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2) and so called amplitude (set by P1) complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a solution from the given tricomplexity class is a theorem of the system. This article is published in two parts. The present Part I introduces the system and proves its completeness, while Part II is devoted to proving soundness.

Build your own clarithmetic II: Soundness

Clarithmetics are number theories based on computability logic (see http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized/schematic version CLA11(P1,P2,P3,P4). By tuning the three parameters P1,P2,P3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a wide range of target tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2) and so called amplitude (set by P1) complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a solution from the given tricomplexity class is a theorem of the system. This article is published in two parts. The previous Part I has introduced the system and proved its completeness, while the present Part II is devoted to proving soundness.

Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques

This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}\{z\}$, and an algebraic number $\alpha$, $0<\vert \alpha\vert <1$, that is not a pole for $f$, we show that one can always determined whether the number $f(\alpha)$ is transcendental or not. In the latter case, we obtain that $f(\alpha)$ belong to the number fields ${\bf k}(\alpha)$. We also consider some consequences of such results to a classical number theoretical problem: the study of sequences of digits of algebraic numbers in an integer (or, more generally, algebraic) base. Our results are based on a theorem of Philippon [31] that we refine. We also simplify his proof.