A factoring and discrete logarithm based cryptosystem

The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

In 1976 Diffie and Hellman [152] described the framework for public-key cryptography. It was not until 1978 that three designs for public-key cryptosystems were published. Rivest, Shamir, and Adleman [431] showed how the discrete logarithm and factorization problems could be used to construct a public-key cryptosystem. This is the well-known RSA cryptosystem. Merkle and Hellman [339] used the knapsack problem in their construction. McEliece [329] built a system based on error correcting codes. Later in 1985 ElGamal [163] designed a public-key cryptosystem using the discrete logarithm problem. Koblitz [283] and Miller [346] suggested the use of elliptic curves in the design of public-key cryptosystems. Nowadays, there are quite a few more suggestions as to how to design public-key cryptosystems, but none so popular as the RSA and ElGamal cryptosystems.

The concept of public key cryptography was first introduced by Diffie and Hellman in 1976 using discrete logarithm problem as base of difficulty. In 1985, T. ElGamal proposed public key cryptosystem scheme based on discrete logarithm problem. Elliptic curve cryptosystems were first proposed in 1985 independently by Neil Koblitz and Victor Miller. Elliptic curve cryptosystems are unique in using elliptic curve groups for arithmetic. This cryptosystem is based on discrete logarithm problem in the group of points of an elliptic curve defined over a finite field. The discrete logarithm problem in an elliptic curve group appears to be much harder than the discrete logarithm problem in other groups. Hence elliptic curves cryptosystem can match the security of other cryptosystems while using smaller key. In this paper we will discuss a VLSl implementation of elliptic curves cryptosystem for ElGamal encryption scheme.

Research in the field of public key cryptography in general and digital signatures in particular is often considered and evaluated at two levels: The first level is considering the mathematical basis for constructing cryptographic and digital signature algorithms, especially these difficult problems: factorizing a large integer into prime factors, finding root problem, discrete logarithm problem, discrete logarithm problem on elliptic curve... The second level is constructing cryptographic and digital signatures algorithms on these hard problems including RSA, DSA, Schnorr, GOST R34.10-94. At the first level, research focuses mainly on improving algorithms for finding large primes, structures for strong primes and algorithms to attack these problems efficiently. At the second level, the research concentrates on improving the existing algorithms to enhance the safety or effective performance of the algorithm. In this article, the authors propose a solution to improve the security of digital signature schemes, implemented in both levels of digital signatures. At the first level, the authors propose a new hard problem - different from the hard problems used before and hasn’t been solved by anyone so far (except by "brute force attack" method). At the second level, the authors propose a method to construct new digital signature algorithms that can help create not only one but also a family of new digital signature algorithms with high security level for practical applications.

Cryptography defines different methods and technologies used in ensuring that communication between two parties over any communication medium is secure, especially in presence of a third part. This is achieved through the use of several methods, such as encryption, decryption, signing, generating of pseudo-random numbers, among many others. Cryptography uses a key or some sort of a password to either encrypt or decrypt a message that needs to be kept secret. This is made possible using two classes of key-based encryption and decryption algorithms, namely symmetric and asymmetric algorithms. The best known and the most widely used public key system is ElGamal. This algorithm comprises of three phases, which are the key generation phase, encryption phase, and the decryption phase. Owing to the advancement in computing technology, ElGamal is prone to some security risks, which makes it less secure. The following paper previews combination of three paring function used to enhance the ElGamal algorithm and increase its security. The results showed that the modified algorithm gives 93% accuracy.KeywordsCryptographyElGamal algorithmEncryptionDecryptionCryptosystemSecurityPublic keyPrivate keyParing functions

We describe several cryptographic schemes in quadratic function fields of odd characteristic. In both the real and the imaginary representation of such a field, we present a Diffie-Hellman-like key exchange protocol as well as a public-key cryptosystem and a signature scheme of ElGamal type. Several of these schemes are improvements of systems previously found in the literature, while others are new. All systems are based on an appropriate discrete logarithm problem. In the imaginary setting, this is the discrete logarithm problem in the ideal class group of the field, or equivalently, in the Jacobian of the curve defining the function field. In the real case, the problem in question is the task of computing distances in the set of reduced principal ideals, which is a monoid under a suitable operation. Currently, the best general algorithms for solving both discrete logarithm problems are exponential (subexponential only in fields of high genus), resulting in a possibly higher level of security than that of conventional discrete logarithm based schemes.

The discrete logarithm problem is one of the backbones in public key cryptography. In this paper we study the discrete logarithm problem in the group of circulant matrices over a finite field.

In the early years after the invention of public key cryptography by Diffie and Hellman in 1976, the design and evaluation of public key cryptosystems has been done merely in ad-hoc manner based on trial and error. The public key cryptosystem said to be secure as long as there is no successful cryptanalytic attack on it. But due to various successful attacks on the cryptosystems after development, the cryptographic community understood that this ad-hoc approach might not be good enough. The paradigm of provable security is an attempt to get rid of ad hoc design. The goals of provable security are to define appropriate models of security on the one hand, and to develop cryptographic designs that can be proven to be secure within the defined models on the other. There are two general approaches for structuring the security proof. One is reductionist approach and other is game-based approach. In these approaches, the security proofs reduce a well known problem (such as discrete logarithm, RSA) to an attack against a proposed cryptosystem. With this approach, the security of public key cryptosystem can be proved formally under the various models viz. random oracle model, generic group model and standard model. In this chapter, we will briefly explain these approaches along with the security proofs of well known public key cryptosystems under the appropriate model.

We are going to observe special algebraic Turing machines designed for different assignments of cryptography such as classical symmetric encryption, public key algorithms, problems of secure key exchange, development of hash functions. The security level of related algorithms is based on the discrete logarithm problem (DLP) in Cremona group of free module over finite commutative ring. In the case of symbolic computations with “sufficiently large number of variables” the order of generator (base of DLP) is impossible to evaluate and we have “hidden discrete logarithm problem”. In the case of subgroups of Cremona group DLP is closely connected with the following classical difficult mathematical problems:

Identity-based signatures (IBS) can be verified using the signer identity information as the public key, and hence, there is no need for certificate management that proves the corresponding public key ownership. Unfortunately, none of the existing IBS schemes has security proven as tight as the discrete logarithm (DL) problem, the hardest problem in the cyclic group setting, under the standard EUF-CMA security model. Recently, the introduction of proving security in the algebraic group model (AGM), where the adversary’s computation is algebraic, enables some ordinary signature schemes to be proven tightly reducible under DL assumption and EUF-CMA. To date, however, it remains unknown whether IBS schemes can also be proven as secure as the DL problem in the AGM. Achieving tight security in IBS schemes under standard EUF-CMA is challenging, due to the need to take extra precautions against adaptive queries on user private keys by the adversary. In this work, we show, for the first time, an IBS scheme with tight security under DL assumption and EUF-CMA in the AGM. The scheme features a minimal signature size of two group elements, with a reduction loss factor of two.

Modern ransomware often generate and manage cryptographic keys on the victim's machine, giving defenders an opportunity to capture exposed keys and recover encrypted data without paying the ransom. However, recent work has raised the possibility of future enclave-enhanced malware that could avoid such mitigations using emerging support for hardware-enforced secure enclaves in commodity CPUs. Nonetheless, the practicality of such enclave-enhanced malware and its potential impact on all phases of the ransomware lifecyle remain unclear. Given the demonstrated capacity of ransomware authors to innovate in order to better extort their victims (e.g. through the adoption of untraceable virtual currencies and anonymity networks), it is important to better understand the risks involved and identify potential mitigations. As a basis for comprehensive security and performance analysis of enclave-enhanced ransomware, we present RansomClave, a family of ransomware that securely manage their cryptographic keys using an enclave. We use RansomClave to explore the implications of enclave-enhanced ransomware for the key generation, encryption and key release phases of the ransomware lifecycle, and to identify potential limitations and mitigations. We propose two plausible victim models and analyse, from an attacker's perspective, how RansomClave can protect cryptographic keys from each type of victim. We find that some existing mitigations are likely to be effective during the key generation and encryption phases, but that RansomClave enables new trustless key release schemes that could potentially improve attacker's profitability and, by extension, make enclaves an attractive target for future attackers.

For a finite group G to be used in the MOR public key cryptosystem, it is necessary that the discrete logarithm problem(DLP) over the inner automorphism group Inn (G) of G must be computationally hard to solve. In this paper, under the assumption that the special conjugacy problem of G is easy, we show that the complexity of the MOR system over G is about log |G| times larger than that of DLP over G in a generic sense. We also introduce a group-theoretic method, called the group extension, to analyze the MOR cryptosystem. When G is considered as a group extension of H by a simple abelian group, we show that DLP over Inn (G) can be ‘reduced’ to DLP over Inn (H). On the other hand, we show that the reduction from DLP over Inn (G) to DLP over G is also possible for some groups. For example, when G is a nilpotent group, we obtain such a reduction by the central commutator attack.

Encryption scheme using heavy computations and large amounts of computing resources is not applicable for lightweight devices, due to the limited storage and power. The technique of offline/online is a possible way to solve this dilemma. Based on bilinear pairing on elliptic curves, a novel identity-based offline/online encryption scheme for lightweight devices is proposed. In the offline encryption phase, point multiplication and other heavy computations are preprocessed by powerful devices. Only light computations are needed in the online encryption phase, which is suitable for lightweight devices and accelerates the encryption speed. Formal security analysis indicates that the proposed scheme is IND-ID-CCA2 secure under the k-BCAA2 assumption. Compared to other related schemes, our scheme has less offline storage and shorter ciphertext, and achieves better performance in both encryption and decryption phase.

ElGamal system was introduced by Taher ElGamal in his landmark paper in 1985 [1]. ElGamal is an important asymmetric cryptosystem that is based on the discrete logarithm problem. In brief, this cryptosystem relies on the difficulty of computing discrete logarithms over finite fields [1]. Towards solving discrete logarithms, Shor demonstrated the existence of a polynomial algorithm that can solve these logarithms on quantum machines [2]. In this paper we propose the Convolution Power Cryptosystem (CPC-H2); an extension to the ElGamal cryptosystem that is based on a novel hard problem. CPC-H2 is designed to defend against a Chosen Ciphertext Attack. We show that CPC-H2 is less complicated and has a higher security level than the Number Theory Research Unit (NTRU).

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.