Mathematical Hard Problem Research Articles

A Framework for Filtering Step of Number Field Sieve and Function Field Sieve

ABSTRACT Public key cryptography is based on two popular mathematical hard problems namely the integer factorization and discrete logarithm problem. Number Field Sieve (NFS) and Function Field Sieve (FFS) are the most efficient and widely used algorithms to solve integer factorization and discrete logarithm problems. The main phases of these algorithms are polynomial selection, relation collection and linear algebra. Among these steps, relation collection and linear algebra are the most computationally costly. The filtering step is used in between relation collection and linear algebra step, which employs several strategies to reduce the coefficient matrix size and makes the matrix suitable for linear algebra, so that linear algebra takes comparatively lesser time. This paper presents a complete framework for the filtering step in the NFS class of algorithms. In this paper, the filtering step is considered as two main sub-phases. The two phases are namely matrix construction and matrix reduction. In the first sub-phase, the relations collected from the relation collection step are structured to represent in a matrix form. The main goal of this phase is to construct a relation from the smooth elements obtained in the relation collection phase along with the other necessary inputs with respect to NFS class of algorithms. The second sub-phase deals with matrix reduction in which various strategies such as duplicate removal, singleton removal, clique removal and merge are applied to reduce the overall matrix size preserving the sparsity at the same time. Algorithms for the first and second phases are designed and analyzed in the paper. The experiments are conducted on the relations collected from standard tool CADO-NFS and the results are reported. From the results, it is shown that the filter map structure presented in the current work for matrix representation helps to improve the overall performance of filtering module. Specifically, the method to handle duplicates in the matrix instead of the traditional method improves the duplicate removal module of filtering. Additionally, an optimized clique removal method of filtering which is introduced in the present work further improves the performance of filtering.

Certificateless Signature Scheme Based on Rabin Algorithm and Discrete Logarithm

Certificateless signature can effectively immue the key escrow problem in the identity-based signature scheme. But the security of the most certificateless signatures usually depends on only one mathematical hard problem, which makes the signature vulnerable when the underlying hard problem has been broken. In order to strengthen the security, in this paper, a certificateless signature whose security depends on two mathematical hard problems, discrete logarithm and factoring problems, is proposed. Then, the proposed certificateless signature can be proved secure in the random oracle, and only both of the two mathematical hard problems are solved, can the proposed signature be broken. As a consequence, the proposed certificateless signature is more secure than the previous signatures. On the other hand, with the pre-computation of the exponential modular computation, it will save more time in the signature signing phase. And compared with the other schemes of this kind, the proposed scheme is more efficient.

SON CONSTRUCTION OF CRYPTOGRAPHIC SYSTEMS OVER UNITS OF GROUP RINGS

Encryption schemes can be derived from the units which are known as invertible elements in a group ring. Besides there are many studies on units in group rings in the literature, we can also see some studies of units in terms of applicability to cryptography and coding theory. In this work, we shall establish the relations between units and RSA problem. Our motivation is to construct a much stronger public key cryptosystem. It is clear that we must consider a mathematical hard problem to do this. Thus, we investigate some interesting properties of units which are no deterministic algorithm in general. Our notations follow (8).

Towards ID-Based Authenticated Group Key Exchange Protocol with Identifying Malicious Participants

An authenticated group key exchange (AGKE) protocol allows participants to construct a common key and provide secure group communications in cooperative and distributed applications. Recently, Choi et al. proposed an identity (ID)-based authenticated group key exchange (IDAGKE) protocol from bilinear pairings. However, their protocol suffered from an insider colluding attack because it didn't realize the security issue of withstanding insider attacks. Withstanding insider attacks mean that it can detect whether malicious participants exist in the group key exchange protocol. Nevertheless, an AGKE protocol resistant to insider attacks is still unable to find “who are malicious participants”. In this paper, we propose an ID-based AGKE protocol with identifying malicious participants. In our protocol, we use a confirmed computation property to achieve identifying malicious participants. Certainly, it is also secure against insider attacks. In the random oracle model and under related mathematical hard problems, we prove that the proposed protocol a secure AGKE protocol with identifying malicious participants.

Ephemeral key recovery using index calculus method

The present study investigates the problem of retrieving the ephemeral keys, which are used in the Discrete Logarithm Problem (DLP) based public key cryptosystems. The ephemeral key can be retrieved by solving the mathematical hard problem, namely DLP. The DLP defined over a prime field is considered in the present study. An efficient way of computing the DLP for retrieving the ephemeral key by using a new variant of Index Calculus Method (ICM) when the factors of p − 1 are known and small is proposed. The Pohlig-Hellman is the best known method to solve the DLP on the prime field with factors of p − 1 are small, while the ICM is an efficient method for a general DLP. The ICM has two steps, such as a pre-computation and an individual logarithm computation. In the pre-computation step, the logarithms of elements of a subset of a group, which is known as a factor base is computed and in the individual logarithm step the DLP is computed with the help of pre-computed logarithms of factor base. Since the ephemeral keys are dynamic and changes for every session, once the logarithms of a subset of a group is known, the DLP for the ephemeral key can be obtained by using the individual logarithm step. Therefore, an efficient way of solving the individual logarithm step is presented based on the newly proposed pre-computation method and the performance is analyzed on a comprehensive set of experiments. From the experimental results, it is observed that the individual logarithm (computation) step outperforms the Pohlig-Hellman method on some special cases. The property of generators of prime field is the main motivation for the current study.

Improved random method using smooth numbers over Zp

The Index Calculus Method(ICM) is the most effective attack on the mathematical hard problem, namely Discrete Logarithm Problem (DLP). The ICM has two steps: A pre-computation step and an individual logarithm computation step. In the pre-computation step, the logarithms of elements of a subset of a group known as factor base is computed and in the individual logarithm step, the DLP is computed with the help of pre computed logarithms of factor base. The pre-computation of ICM is based on the probability of field elements getting smoothening i.e., the factors of field elements are less than a prescribed bound. In the present work the ICM is improved through the property of smoothness concept over the the prime field $Z_p^{*}$ for certain instances of primes. This new smoothness concept is introduced and found to exhibit different characteristics on some types of primes. The property based on the characteristics of smooth numbers over $Z_p^{*}$ allows to map the field elements from one subset to another. Based on this concept an improved ICM by increasing the probability of numbers getting smoothening is devised. The efficiency of the improved ICM is analyzed experimentally. Through the experimental results it is observed that the performance of ICM is enhanced by $\approx 50\%$.