Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem

Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC, and ElGamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found that are able to solve certain mathematical problems on non-commutative algebraic structures. Against this background, some novel public key cryptography based on Polynomial symmetrical decomposition (PSD) problem have been proposed. We find that these schemes are not secure. We present that they are vulnerable to structural attack, linearization equations attack, overdefined systems of multivariate polynomial equations attack and that, they only require polynomial time complexity to retrieve the same secret key for some given public keys respectively. We also propose an improvement to enhance public key cryptography based on PSD problem. In addition, we discuss possible lines of future work.

This paper introduces a multicast method for renewing secrets which are shared by a set of hosts. The method is centralized, secure, efficient, scalable to a reasonable size and compatible with any multicast topology configuration underneath. It can be used to achieve privacy in a centralized multicast overlay. Additionally, the method can be used to renew an asymmetric key pair when a cryptosystem based on discrete logarithm is used. Knowledge of the public key is then restricted to the group it is communicated to. Security and scalability are discussed, and a comparison with other well-known alternatives is shown.

Problem statement: A cryptosystem is a way for a sender and a receiver to communicate digitally by which the sender can send receiver any confidential or private message by first encrypting it using the receiver’s public key. Upon receiving the encrypted message, the receiver can confirm the originality of the message’s contents using his own secret key. Up to now, most of the existing cryptosystems were developed based on a single cryptographic assumption like factoring, discrete logarithms, quadratic residue or elliptic curve discrete logarithm. Although these schemes remain secure today, one day in a near future they may be broken if one finds a polynomial algorithm that can efficiently solve the underlying cryptographic assumption. Approach: By this motivation, we designed a new cryptosystem based on two cryptographic assumptions; quadratic residue and discrete logarithms. We integrated these two assumptions in our encrypting and decrypting equations so that the former depends on one public key whereas the latter depends on one corresponding secret key and two secret numbers. Each of public and secret keys in our scheme determines the assumptions we use. Results: The newly developed cryptosystem is shown secure against the three common considering algebraic attacks using a heuristic security technique. The efficiency performance of our scheme requires 2Texp+2Tmul +Thash time complexity for encryption and Texp+2Tmul +Tsrt time complexity for decryption and this magnitude of complexity is considered minimal for multiple cryptographic assumptions-like cryptosystems. Conclusion: The new cryptosystem based on multiple cryptographic assumptions offers a greater security level than that schemes based on a single cryptographic assumption. The adversary has to solve the two assumptions simultaneously to recover the original message from the received corresponding encrypted message but this is very unlikely to happen.

Multisignatures allow n signers to produce a short joint signature on a single message. Multisignatures were achieved in the plain model with a non-interactive protocol in groups with bilinear maps, by Boneh et al, and by a three-round protocol under the Discrete Logarithm (DL) assumption, by Bellare and Neven, with multisignature verification cost of, respectively, O(n) pairings or exponentiations. In addition, multisignatures with O(1) verification were shown in so-called Key Verification (KV) model, where each public key is accompanied by a short proof of well-formedness, again either with a non-interactive protocol using bilinear maps, by Ristenpart and Yilek, or with a three-round protocol under the Diffie-Hellman assumption, by Bagherzandi and Jarecki.

In 1984, Shamir proposed the concept of the ID-based cryptosystem (IBC). Instead of generating and publishing a public key for each user, the ID-based scheme permits each user to choose his name or network address as his public key. This is advantageous to public-key cryptosystems because the public-key verification is so easy and direct. In such a way, a large public key file is not required. Since new cryptographic schemes always face security challenges and many integer factorization and discrete logarithm based cryptographic systems have been deployed, therefore, the purpose of this paper is to design a transformation process that can transfer the entire integer factorization problem and discrete logarithm problem based cryptosystems into the ID-based systems rather than re-invent a new system. We consider the security against a conspiracy of some entities in the proposed system and show the possibility of establishing a more secure system. Rapid advances in computer technology and the development of the Internet are changing the way, we conduct our daily and business lives. Secrecy is an important issue with respect to sensitive data transferred over insecure public channels. In an open network environment, secret session key needs to be shared between two users before it establishes a secret communication. While the number of users in the network is increasing, key distribution will become a serious problem. In 1976, Diffie and Hellman (4) introduced the concept of the public key distribution system (PKDS). In the PKDS, each user needs to select a secret key and compute a corresponding public key and store in the public directory. The common secrete session key, which will be shared between two users can then be determined by either user, based on his own secret key and the partner's public key. Although the PKDS provides an elegant way to solve the key distribution problem, the major concern is the authentication of the public keys used in the cryptographic algorithm. Many attempts have been made to deal with the public key authentication issue. Kohnfelder (5) used the RSA digital signature scheme to provide public key certification. His system involves two kinds of public key cryptography: one is in modulo p, where p is a large prime number; the other is in modulo N, where N = pq, and p and q are large primes. Blom (7) proposed a symmetric key generation system (SKGS) based on secret sharing schemes. The problems of SKGS however, are the difficulty of choosing a suitable threshold value and the requirement of large memory space for storing the secret shadow of each user. In 1984, Shamir (1) introduced the concept of an identity. In this system, each user needs to visit key authentication center (KAC) and identify himself before joining the network. Once a user's identity is accepted, the KAC will provide him with a secret key.

To ensure secure communication between two parties (Alice and Bob), Public Key Agreement (PKA) algorithms play a crucial role. The Diffie-Hellman (D-H) algorithm, introduced by W. Diffie and M. Hellman in 1976, is the most well-known PKA algorithm, based on exponentiation over a finite field. Over time, various PKA algorithms have been developed based on this original idea—these are referred to as D-H realizations or D-H algorithms. A common issue with D-H realizations is their computational inefficiency, particularly in environments with resource-constrained devices. This inefficiency arises from the high computational complexity of generating public and secret shared keys (SSKs), often resulting in communication delays. In this paper, we propose a modified D-H protocol designed to reduce the computational time required by one party, thereby decreasing the total time needed to establish the SSK in asymmetric environments where Alice and Bob have different computational capabilities. The core of this approach leverages the group homomorphism property of the underlying function to enable parallelization and reduce the complexity of Alice’s computations, together with an asymmetric key agreement structure in which both parties may follow distinct computational rules for public key and SSK generation. We demonstrate that the proposed protocol retains the security properties of the original D-H realizations, preserving the hardness of the discrete logarithm (DL), computational Diffie-Hellman (CDH), and decisional Diffie-Hellman (DDH) problems under an asymmetric setting. We also discuss its applicability to other cryptographic protocols. Experimental results show that the modified protocol significantly improves computational efficiency, particularly by reducing Alice's computational burden through parallel processing.

Privacy preservation is very essential in various real life applications such as medical science and financial analysis. This paper focuses on implementation of an asymmetric secure multi-party computation protocol using anonymization and public-key encryption where all parties have access to trusted third party (TTP) who (1) doesn't add any contribution to computation (2) doesn't know who is the owner of the input received (3) has large number of resources (4) decryption key is known to trusted third party (TTP) to get the actual input for computation of final result. In this environment, concern is to design a protocol which deploys TTP for computation. It is proposed that the protocol is very proficient (in terms of secure computation and individual privacy) for the parties than the other available protocols. The solution incorporates protocol using asymmetric encryption scheme where any party can encrypt a message with the public key but decryption can be done by only the possessor of the decryption key (private key). As the protocol works on asymmetric encryption and packetization it ensures following: (1) Confidentiality (Anonymity) (2) Security (3) Privacy (Data).

Network security is an important aspect of information sharing. Attempts have been made to remove various insecurities over internet. For this, many technological implementations and security policies have been developed. The amount of data, transferred, is not a factor. The basic factor is, how much security, the channel provides while transmitting data. Cryptography is one such technique, which allows secure data transmission without losing its confidentiality and integrity. Based on the key distribution, cryptography is further classified into two major types-Symmetric Key Cryptography and Asymmetric Key Cryptography. In this paper, we have surveyed the traditional algorithms, along with the proposed algorithms based on their pros and cons, related to Symmetric and Asymmetric Key Cryptography. We have also compared the importance of both these cryptographic techniques. The proposed algorithms proved to be highly efficient in their respective grounds but there are certain areas that remained open, related to these algorithms, and have not yet been thoroughly discussed. This paper also presents an appropriate future scope related to these open fields.

Public-key encryption is extensively used to provide digital data confidentiality and deliver the security features, such as nonrepudiation (digital signature) and secure key exchange. Conventional public-key schemes are based on mathematical problems with inflexible constraints, and the security of digital contents relies on computational complexity. In the era of emerging technologies, most public-key image encryption schemes are susceptible to various threats. We propose a novel public-key encryption in this article with near-ring criteria and provide confidentiality to private data with the microstates of the Einstein crystal model. The virtual oscillator generated by microstates of initial oscillators for the common secrets with the public-key scheme produces unique states to encrypt digital data. The privacy-preserved structure, that mimics the data stream of digital content with the behavior of the improved Einstein crystal model, describes a system in terms of microstates to generate diffusion in the plain data with unique states of a virtual oscillator. The performance and digital forensic evaluations, such as randomness, histogram uniformity, pixels’ correlation, pixels’ similarity, visual strength, pixels’ incongruity, key sensitivity, linear and differential attacks, noise, and occlusion attacks analyses, certify the resistivity of the proposed algorithm against potential threats and provide superior capacity in comparison to existing methodologies to hostile certain attacks.

Certificateless public key cryptography (CL-PKC) can solve the problems of certificate management in a public key infrastructure (PKI) and of key escrows in identity-based public key cryptography (ID-PKC). In CL-PKC, the key generation center (KGC) does not know the private keys of all users, and their public keys need not be certificated by certification authority (CA). At present, however, most certificateless encryption schemes are based on large integer factorization and discrete logarithms that are not secure in a quantum environment and the computation complexity is high. To solve these problems, we propose a new certificate-less encryption scheme based on lattices, more precisely, using the hardness of the learning with errors (LWE) problem. Compared with schemes based on large integer factorization and discrete logarithms, the most operations are matrixvector multiplication and inner products in our scheme, our approach has lower computation complexity. Our scheme can be proven to be indistinguishability chosen ciphertext attacks (IND-CPA) secure in the random oracle model.

Anonymous channels or similar techniques that achieve sender's anonymity play important roles in many applications, e.g. electronic voting. However, they will be meaningless if cryptographic primitives containing sender's identity are carelessly used during the transmission. In computationally secure settings, this problem may be easily overcome by using public key encryption and group signatures. However, in an unconditionally secure setting, in which no computational difficulty is assumed, this is not an easy case as such. As the increasing computational power approaches the point where security policy can no longer assume the difficulty of solving factoring or discrete logarithm problems, it must shift its focus to assuring the solvency of unconditionally secure schemes that provide long-term security. The main contribution of this paper is to study the security primitives for the above problem. In this paper, we first define the unconditionally secure asymmetric encryption scheme, which is an encryption scheme with unconditional security and where it is impossible for a receiver to deduce the identity of a sender from the encrypted message. We also investigate tight lower bounds on required memory sizes from an information theoretic viewpoint and show an optimal construction based on polynomials. It is remarkable to see that these bounds are considerably different from those in Shannon's model of the conventional unconditionally secure symmetric encryption. Other than the polynomial-based scheme, we also show a construction based on combinatorial theory, a non-malleable scheme and a multi-receiver scheme. Then, we define and formalize the group authentication code (GA-code), which is an unconditionally secure authentication code with anonymity like group signatures. In this scheme, any authenticated user will be able to generate and send an authenticated message while the receiver can verify the legitimacy of the message---that it has been sent from a legitimate user but at the same time retains his anonymity. However, by cooperating with the group authority, such as in the case of disputes, the receiver is able to obtain information of the user's identity. For GA-code, we show two concrete constructions.

Matrix equations involving a family of unknown matrices $$X_1, \ldots , X_k$$ can be written in a general form $$f(X_1, \ldots , X_k) = 0$$ . One of the fundamental problems in matrix equation theory is to establish identifying conditions for the matrix equation to always hold for all matrices $$X_1, \ldots , X_k$$ . In this article, we select some nonlinear matrix equations with two, three, and four unknown matrices as illustrative examples, and show how to establish a family of simple and well-organized necessary and sufficient conditions for these matrix equations to always hold with respect to all unknown matrices in the equations. The methods, results, and facts in this article can be used to efficiently establish various nonlinear identities with variables in different noncommutative algebraic structures.

In this paper we survey the recent work on Auto-Recoverable Auto-Certifiable Cryptosystems. This notion has been put forth to solve the “software key escrow” problem in an efficient manner within the context of a Pubic Key Infrastructure (PKI). This survey presents the exact specification of the problem which is based on what software key escrow can hope to achieve. The specification attempts to separate the truly difficult technical issues in the area from the ones that are only seemingly difficult. We then review the work in Eurocrypt ’98 and PKC ’99, which gives an efficient reduction to a software key escrow system from a certified public key system (PKI). Namely, we show how to construct an escrowed PKI for essentially the same cost and effort required for a regular PKI. More specifically, the schemes presented are as efficient for users to use as a PKI, do not require tamper-resistant hardware (i.e., they can be distributed in software to users), and the schemes are shadow public key resistant as defined in Crypto ’95 by Kilian and Leighton (namely, they do not allow the users to publish public keys other then the ones certified). The schemes enable the efficient verification of the fact that a given user’s private key is escrowed properly. They allow the safe and efficient recovery of keys (and plaintext messages) which is typical in emergency situations such as in the medical area, in secure file systems, and in criminal investigations. We comment that we do not advocate nor deal with the policy issues regarding the need of governments to control access to messages; our motivation is highly technical: in cases that escrow is required or needed we would like to minimize its effect on the overall PKI deployment. We then briefly mention forthcoming developments in the area which include further flexibility/compatibility requirements for auto-recoverable cryptosystems, as well as design of such systems which are based on traditional public key methods (RSA and discrete logs).KeywordsRandom OraclePrimitive RootNIZK ProofRump SessionEscrow EncryptionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Quantum cryptography and quantum search algorithm are considered as two important research topics in quantum information science. An asymmetrical quantum encryption protocol based on the properties of quantum one-way function and quantum search algorithm is proposed. Depending on the no-cloning theorem and trapdoor one-way functions of the public-key, the eavesdropper cannot extract any private-information from the public-keys and the ciphertext. Introducing key-generation randomized logarithm to improve security of our proposed protocol, i.e., one private-key corresponds to an exponential number of public-keys. Using unitary operations and the single photon measurement, secret messages can be directly sent from the sender to the receiver. The security of the proposed protocol is proved that it is information-theoretically secure. Furthermore, compared the symmetrical Quantum key distribution, the proposed protocol is not only efficient to reduce additional communication, but also easier to carry out in practice, because no entangled photons and complex operations are required.

A public-key encryption algorithm and a practical hybrid scheme are proposed in this paper. This algorithm uses the advanced transform techniques, i.e. the Number Theoretic Transforms (NTTs) to diffuse data, and uses a non-linear technique based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) to confuse data. The main advantage of the proposed algorithm is its high efficiency and capability. The plaintext can be encrypted in blocks in parallel, thus saves computational time. The proposed algorithm also can be easily applied in a hybrid scheme for practical applications.