Intractability Assumption Research Articles

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

The security of several fully homomorphic encryption (FHE) schemes depends on the intractability assumption of the approximate common divisor (ACD) problem over integers. Subsequent efforts to solve the ACD problem as well as its variants were also developed during the past decade. In this paper, an improved orthogonal lattice (OL)-based algorithm, AIOL, is proposed to solve the general approximate common divisor (GACD) problem. The conditions for ensuring the feasibility of AIOL are also presented. Compared to the Ding–Tao OL algorithm, the well-known LLL reduction method is used only once in AIOL, and when the error vector r is recovered in AIOL, the possible difference between the restored and the true value of p is given. Experimental comparisons between the Ding-Tao algorithm and ours are also provided to validate our improvements.

Identity-based ring signature scheme with multi-designated verifiers

As far as we know, there is only one ring signature scheme with multi-designated verifiers in the literature and the scheme is based on the bilinear pairings which need to consume a lot of computation and under the traditional public key infrastructure (PKI). In order to improve computational efficiency and solve the problem of certificate management in PKI, in this paper, we propose the first identity-based ring signature scheme with multi-designated verifiers without pairings. We prove that our novel scheme's unforgeability, anonymity in the random oracle model and indistinguishability in the standard model under the intractability assumption of discrete logarithm problem and decisional Diffie-Hellman problem, and we prove that our novel scheme's non-transferability in the standard model. We compare the new scheme with a previous scheme in terms of computation and communication at last. The results show that our new scheme is more efficient than the previous scheme and is more suitable for the multi-user setting.

Flaw and amendment of a two-party authenticated key agreement protocol for post-quantum environments

Recent advances in quantum computers challenge the security of key agreement protocol that are designed with the intractability assumptions of discrete logarithmic problem and integer factorization problem. Hence, motivated to develop the key agreement protocol that is secure in post-quantum era, recently Islam proposed a provably secure two-party authenticated key agreement (2PAKA) protocol. The protocol is based on the intractability assumption of the famous Ring Learning With Error (RLWE) problem. In this work, we show that Islam’s two-party key agreement is vulnerable to the modified version of the signal leakage attack (SLA) which is termed as improved-signal leakage attack (i-SLA) in this article. Using i-SLA, the attacker can successfully recover the long term private key of the honest user by instantiating utmost q number of key exchange sessions with the honest user using q number of malformed public keys. To overcome the attack, we provide a countermeasure without changing the original design of the protocol.

Post-Quantum Cryptosystems for Internet-of-Things: A Survey on Lattice-Based Algorithms

The latest quantum computers have the ability to solve incredibly complex classical cryptography equations particularly to decode the secret encrypted keys and making the network vulnerable to hacking. They can solve complex mathematical problems almost instantaneously compared to the billions of years of computation needed by traditional computing machines. Researchers advocate the development of novel strategies to include data encryption in the post-quantum era. Lattices have been widely used in cryptography, somewhat peculiarly, and these algorithms have been used in both; (a) cryptoanalysis by using lattice approximation to break cryptosystems; and (b) cryptography by using computationally hard lattice problems (non-deterministic polynomial time hardness) to construct stable cryptographic functions. Most of the dominant features of lattice-based cryptography (LBC), which holds it ahead in the post-quantum league, include resistance to quantum attack vectors, high concurrent performance, parallelism, security under worst-case intractability assumptions, and solutions to long-standing open problems in cryptography. While these methods offer possible security for classical cryptosytems in theory and experimentation, their implementation in energy-restricted Internet-of-Things (IoT) devices requires careful study of regular lattice-based implantation and its simplification in lightweight lattice-based cryptography (LW-LBC). This streamlined post-quantum algorithm is ideal for levelled IoT device security. The key aim of this survey was to provide the scientific community with comprehensive information on elementary mathematical facts, as well as to address real-time implementation, hardware architecture, open problems, attack vectors, and the significance for the IoT networks.

An Efficient Access Control Scheme With Outsourcing and Attribute Revocation for Fog-Enabled E-Health

Fog computing is increasingly popular partly due to its capability to minimize data transfer and latency requirements, for example by moving some of the computational operations away from the cloud servers and closer to the users. To achieve fine-grained access control in fog-enabled application scenarios to guarantee data security and user privacy, one could use ciphertext-policy attribute-based encryption (CP-ABE). However, the lack of an effective mechanism to carry out access right revocation in conventional CP-ABE schemes limits the deployment of such schemes in practice. Thus, we propose an efficient CP-ABE scheme with attribute revocation capability, designed to construct a fine-grained access control system in fog-enabled E-health (referred to as AC-FEH). In our AC-FEH system, fog nodes undertake data encryption and decryption operations; thus, computational costs for data owners and users are minimized. In comparison to several other competing access control schemes based on CP-ABE, our AC-FEH system reduces the computational costs associated with encryption and decryption. We also prove the selective security of the underlying CP-ABE scheme under the intractability assumption of the $q$ -parallel BDHE problem.

An Efficient Hybrid Signcryption Scheme With Conditional Privacy-Preservation for Heterogeneous Vehicular Communication in VANETs

Vehicular ad hoc networks (VANETs) ensure improvement in road safety and traffic management by allowing the vehicles and infrastructure that are connected to them to exchange safety messages. Due to the open wireless communication channels, security and privacy issues are a major concern in VANETs. A typical attack consists of a malicious third party intercepting, modifying and retransmitting messages. Heterogeneous vehicular communication in VANETs occurs when vehicles (only) or vehicles and other infrastructure communicate using different cryptographic techniques. To address the security and privacy issues in heterogeneous vehicular communication, some heterogeneous signcryption schemes have been proposed. These schemes simultaneously satisfy the confidentiality, authentication, integrity and non-repudiation security requirements. They however fail to properly address the efficiency with respect to the computational cost involved in unsigncrypting ciphertexts, which is often affected by the speeds at which vehicles travel in VANETs. In this paper, we propose an efficient conditional privacy-preserving hybrid signcryption (CPP-HSC) scheme that uses bilinear pairing to satisfy the security requirements of heterogeneous vehicular communication in a single logical step. Our scheme ensures the transmission of a message from a vehicle with a background of an identity-based cryptosystem (IBC) to a receiver with a background of a public-key infrastructure (PKI). Furthermore, it supports a batch unsigncryption method, which allows the receiver to speed up the process by processing multiple messages simultaneously. The security of our CPP-HSC scheme ensures the indistinguishability against adaptive chosen ciphertext attack (IND-CCA2) under the intractability assumption of q -bilinear Diffie-Hellman inversion ( q -BDHI) problem and the existential unforgeability against adaptive chosen message attack (EUF-CMA) under the intractability assumption of q -strong Diffie-Hellman ( q -SDH) problem in the random oracle model (ROM). The performance analysis indicates that our scheme has an improvement over the existing related schemes with respect to the computational cost without an increase in the communication cost.

Provably secure two-party authenticated key agreement protocol for post-quantum environments

A two-party authenticated key agreement (2PAKA) protocol is a cryptographic tool employed widely to allow two users to generate a shared and fresh session key between them in each session over an insecure network. The authenticated version of a two-party key agreement protocol is popular because it can easily withstand the impersonation of the user. In the literature, many 2PAKA protocols have been put forward with the intractability assumptions of the discrete logarithm (DLP) problem and integer factorization problem (IFP). Some recent studies showed that the 2PAKA protocols based on these assumptions are insecure in post-quantum environments. To resolve this issue, we have designed a lattice-based 2PAKA (LB-2PAKA) protocol with the intractability of the ring-learning-with-errors (RLWE) problem. The proposed LB-2PAKA protocol is also analyzed in the random oracle model to measure provable security and to estimate the breaching time. To evaluate the performance, we used the LatticeCrypto Library and estimated the running time of our LB-2PAKA protocol. Besides, we analyzed the communication cost requirement of our LB-2PAKA protocol.

Provably Secure Encryption Schemes With Zero Setup and Linear Speed by Using Rubik’s Cubes

Recently, new paradigms for designing modern cryptographic schemes were proposed based on Rubik's rotations. However, most of them lack rigorous provable security reductions. Enlightened by this interesting progress, we present a novel method for designing encryption schemes by using Rubik's groups. Different from most naive designs of permutation ciphers based on Rubik's cubes, our proposals are probabilistic encryption schemes that combine some of the newest cryptographic primitives with modern coding theory. More specifically, under the intractability assumption of the conjugacy decision problem over Rubik's groups, the proposed schemes have provable security reductions (in the random oracle model). Furthermore, the proposed schemes have two remarkable performance advantages: zero setup and linear encryption/decryption speed. In addition, the processes of encoding/encryption and decryption/decoding are demonstrated graphically.

A Miniature CCA Public Key Encryption Scheme Based on Non-abelian Factorization Problem in Finite Groups of Lie Type

Abstract With the development of Lie theory, Lie groups have attained profound significance in several branches of Mathematics and Physics. In Lie theory, the matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as the finite analogue of Lie groups, finite groups of Lie type have potential applications in cryptography due to their unique mathematical structures. In this paper, we first put forward a novel idea of designing cryptosystems based on Lie theory. First of all, combing with discrete logarithm problem and group factorization problem, we proposed several new intractable assumptions based on the matrix exponential in finite groups of Lie type. Subsequently, in analog with Boyen’s scheme (Asiacrypt 2007), we designed a public-key encryption scheme based on the non-abelian factorization problem in finite groups of Lie type. Finally, our proposal was proved to be indistinguishable against adaptively chosen-ciphertext attack in the random oracle model. It is encouraging that our scheme also has the potential to resist against Shor’s quantum algorithm attack.

Constructing a pseudo-free family of finite computational groups under the general integer factoring intractability assumption

Abstract We provide a correct version of Remark 3.5 of the paper mentioned in the title. Also, we fix a typo in Remark 4.4 of that paper.

A certain family of subgroups of ℤ𝑛⋆ is weakly pseudo-free under the general integer factoring intractability assumption

AbstractLet {\mathbb{G}_{n}} be the subgroup of elements of odd order in the group {\mathbb{Z}^{\star}_{n}}, and let {\mathcal{U}(\mathbb{G}_{n})} be the uniform probability distribution on {\mathbb{G}_{n}}. In this paper, we establish a probabilistic polynomial-time reduction from finding a nontrivial divisor of a composite number n to finding a nontrivial relation between l elements chosen independently and uniformly at random from {\mathbb{G}_{n}}, where {l\geq 1} is given in unary as a part of the input. Assume that finding a nontrivial divisor of a random number in some set N of composite numbers (for a given security parameter) is a computationally hard problem. Then, using the above-mentioned reduction, we prove that the family {((\mathbb{G}_{n},\mathcal{U}(\mathbb{G}_{n}))\mid n\in N)} of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble {(\mathcal{U}(\mathbb{G}_{n})\mid n\in N)} is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function {\nu\colon D\to N} (where {D\subseteq\{0,1\}^{*}}) and a polynomial-time samplable probability ensemble {(\mathcal{G}_{d}\mid d\in D)} (where {\mathcal{G}_{d}} is a distribution on {\mathbb{G}_{\nu(d)}} for each {d\in D}) such that the family {((\mathbb{G}_{\nu(d)},\mathcal{G}_{d})\mid d\in D)} of computational abelian groups is weakly pseudo-free.

D-NTRU: More efficient and average-case IND-CPA secure NTRU variant

NTRU is a fast public key cryptosystem remaining unbroken. However, there is no known worst-to-average reduction for the original NTRU cryptosystem. Several provably secure NTRU modifications such as NAEP, pNE and NTRUCCA were proposed in the literature at the cost of inefficiency in encryption/decryption and enlarged ciphertext expansion. NAEP completes the IND-CCA security of the original NTRU under the average-case NTRU one-wayness intractability assumption. Both pNE and NTRUCCA obtain provable security goals under worst-case lattice assumptions. In this paper, a general framework for NTRU is considered, and a new PKC called D-NTRU is proposed. It is shown that the D-NTRU cryptosystem reduces the ciphertext expansion of the NTRU algorithm, and the encryption and decryption algorithms of D-NTRU perform even asymptotically faster than the NTRU algorithm only at the cost of slightly enlarged secret and public keys. The security of D-NTRU is proven in the standard model and under the average-case NTRU one-wayness assumption. The proof of the IND-CPA security of D-NTRU is completed by introducing another NTRU variant called C-NTRU as a bridge, defining some problems, and then proving the equivalence of these problems. So the proposed D-NTRU algorithm is more advantageous than the original NTRU algorithm, and much more efficient than all the provably secure variants of NTRU.

Arithmetic Cryptography

We study the possibility of computing cryptographic primitives in a fully black-box arithmetic model over a finite field F . In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits that make only a black-box use of the underlying field, and the adversary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions. We prove several positive and negative results in this model for various cryptographic tasks. On the positive side, we show that, under coding-related intractability assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, additively homomorphic encryption, and secure computation with low online complexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard Boolean model and the arithmetic model, and explain, in retrospect, some of the limitations of previous constructions.

Design of a provably secure identity-based digital multi-signature scheme using biometrics and fuzzy extractor

A novel biometric identity-based digital multi-signature BIO-IDMS scheme is put forwarded in this paper. The proposed scheme is constructed with the help of fuzzy extractor and elliptic curve bilinear pairings. Furthermore, we designed the formal model and the security model of the proposed BIO-IDMS scheme. The formal security analysis demonstrates that the forgery of the proposed scheme is infeasible in the random oracle model based on the intractability assumption of the computational Diffie-Hellman CDH problem. The proposed scheme outperforms in terms of computational cost compared with other related existing multi-signature schemes. Copyright © 2016 John Wiley & Sons, Ltd.

Unprovable Security of Perfect NIZK and Non-interactive Non-malleable Commitments

We present barriers to provable security of two important cryptographic primitives, perfect non-interactive zero knowledge (NIZK) and non-interactive non-alleable commitments: źBlack-box reductions cannot be used to demonstrate adaptive soundness (i.e., that soundness holds even if the statement to be proven is chosen as a function of the common reference string) of any statistical NIZK for NP based on any standard intractability assumptions. źBlack-box reductions cannot be used to demonstrate non-malleability of non-interactive, or even 2-message, commitment schemes based on any standard intractability assumptions. We emphasize that the above separations apply even if the construction of the considered primitives makes a non-black-box use of the underlying assumption. As an independent contribution, we suggest a taxonomy of game-based intractability assumptions.

Secure delegation of services through proxy signature with revocation

The trusted delegation of services requires an effective synchronised and verifiable signature scheme to convince the authorisation of involving entities in the delegation processes. The proxy signature scheme enables an original signer to delegate her signing capability to a proxy signer in a secure manner. In this paper, we present a controlled delegation of generating a proxy signature using a control entity (mediator). The mediator controls the delegation capability of the proxy signer for making the signature even before/after the delegation period. We propose a new construction of proxy signature with revocation by using a mediator that controls the delegation capability and authorisation of the proxy signer. The security hardness of the proposed scheme is based on the computational bilinear Diffie-Hellman (CBDH) intractability assumption in the random oracle model. Furthermore, we verify the revocation process of our proposed scheme using model checking.

Secure delegation of services through proxy signature with revocation

The trusted delegation of services requires an effective synchronised and verifiable signature scheme to convince the authorisation of involving entities in the delegation processes. The proxy signature scheme enables an original signer to delegate her signing capability to a proxy signer in a secure manner. In this paper, we present a controlled delegation of generating a proxy signature using a control entity (mediator). The mediator controls the delegation capability of the proxy signer for making the signature even before/after the delegation period. We propose a new construction of proxy signature with revocation by using a mediator that controls the delegation capability and authorisation of the proxy signer. The security hardness of the proposed scheme is based on the computational bilinear Diffie-Hellman (CBDH) intractability assumption in the random oracle model. Furthermore, we verify the revocation process of our proposed scheme using model checking.

Identity‐based signcryption from lattices

AbstractSigncryption as a cryptographic primitive can carry out signature and encryption simultaneously at a remarkably reduced cost. Identity‐based cryptography is more convenient than public key infrastructure‐based cryptography in certificate management. As a result, identity‐based signcryption has been studied extensively, and many efficient and provably secure constructions have been proposed. However, most of these schemes are based on intractability assumptions from number theory, and these assumptions have been threatened by the booming quantum computation. Therefore, a recent trend in cryptography is to construct cryptosystems that are based on lattice‐based intractability assumptions because of their plausible features of quantum attack resistance. In this paper, several identity‐based signcryption schemes from lattice hardness assumptions are proposed. In the standard model, these schemes are indistinguishable against inner adaptively chosen ciphertext attacks (IND‐CCA2) and strongly unforgeable against inner chosen message attacks. In our construction, it does not matter whether the original encryption scheme used to construct signcryption is deterministic or probabilistic; the resulted signcryption schemes can reach IND‐CCA2 security. To achieve this, we carefully combine three techniques—the identity‐based encryption from lattice due to Agrawal–Boneh–Boyen (EUROCRYPT 2010), the framework of lattice‐based short signature due to Boyen (Public Key Cryptography 2010), and the Canetti–Halevi–Katz (abbr. CHK) technique, with necessary and tailored optimization—for transforming an (ℓ + 1)‐level indistinguishable under chosen plaintext attack secure hierarchical identity‐based encryption (HIBE) into an ℓ level IND‐CCA2 secure HIBE scheme. In addition, our security proof also contains a more efficient simulation tool that might have separate interest in cryptographic applications. Copyright © 2015 John Wiley & Sons, Ltd.

Provably secure one-round certificateless authenticated group key agreement protocol for secure communications

A novel one-round authenticated group key agreement (CL-AGKA) protocol is devised in this paper on certificateless public key cryptography. Our CL-AGKA protocol abolished the certificate management burden existing in certificate authority-based public key cryptography and the private-key escrow problem occurring in identity-based cryptography. Our CL-AGKA protocol is designed to establish an authenticated group session key between a group participants by ensuring that the session key is not revealed to any outsiders. The proposed CL-AGKA protocol designed a security model in the computational model, called the random oracle model. It is formally examined that our CL-AGKA protocol achieved the strong authenticated key exchange security and thus provably secured in the random oracle model. The security of the designed protocol is achieved against the intractability assumptions of the computational Diffie---Hellman problem and bilinear Diffie---Hellman problem. In addition, the proposed CL-AGKA protocol offered low message exchange cost and computational cost against the related group key agreement protocols.

Provably Secure and Pairing-Based Strong Designated Verifier Signature Scheme with Message Recovery

In this paper, an efficient and secure strong designated verifier signature with message recovery scheme is presented using elliptic curve and bilinear pairing. In our scheme, the signer implants a message on the signature and sends it without message to the verifier, who then extracts the original message and validates the message-signature pair. However, an outsider is unable to verify the message-signature pair since the verifier’s private key is strictly required for verification. Our scheme has been designed to achieve confidentiality, integrity, authentication and non-repudiation of message transmitted through hostile networks. Our scheme is secure against adaptive chosen message attack in the random oracle model under the intractability assumption of Co-Bilinear Diffie–Hellman problem. Besides, our scheme is computation and communication efficient than other schemes, and hence, it may be useful in many small message applications and also for the resource-constrained environments.