Public-Key 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.

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.

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:

The new signature scheme presented by the authors in [13] is the first signature scheme based on the discrete logarithm problem that gives message recovery. The purpose of this paper is to show that the message recovery feature is independent of the choice of the signature equation and that all ElGamal-type schemes have variants giving message recovery. For each of the six basic ElGamal-type signature equations five variants are presented with different properties regarding message recovery, length of commitment and strong equivalence. Moreover, the six basic signature schemes have different properties regarding security and implementation. It turns out that the scheme proposed in [13] is the only inversionless scheme whereas the message recovery variant of the DSA requires computing of inverses in both generation and verification of signatures. In general, message recovery variants can be given for ElGamal-type signature schemes over any group with large cyclic subgroup as the multiplicative group of GF(2n) or elliptic curve over a finite field. The present paper also shows how to integrate the DLP-based message recovery schemes with secret session key establishment and ElGamal encryption. In particular, it is shown that with DLP-based schemes the same functionality as with RSA can be obtained. However, the schemes are not as elegant as RSA in the sense that the signature (verification) function cannot at the same time be used as the decipherment (encipherment) function.

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.

A number of public key cryptosystems such as El Gamal [2] are based on the difficulty of solving the discrete logarithm problem in certain groups, that is, solving g x = h for x, where g and h are given group elements. The computational difficulty of the discrete logarithm depends on the representation of the group. The additive version in ℤ m is essentially trivial, involving only the solution of a linear congruence. However, in \({\Bbb F}_{{p^d}}^*\), the multiplicative group of the finite field with p d elements, the problem is intractable if p d is large enough, even though it is isomorphic to the cyclic group \({{\Bbb Z}_{{p^d} - 1}}\). The computation appears even somewhat more difficult in groups based on elliptic and hyperelliptic curves, with some exceptions such as supersingular curves.

The ElGamal cryptosystem was originally proposed by Taher ElGamal in 1985, in which its security level is based on the Discrete Logarithm Problem (DLP). ElGamal cryptosystem is relatively an expensive algorithm. For security guarantees, ElGamal cryptosystem requires modulo operation of large prime integer whose size range approximately from 1,024 to 4,096 bits. As a consequence of such requirement, the application of ElGamal cryptosystem is limited for securing only small messages such as secret keys. This paper aims to propose an efficient variant of ElGamal cryptosystem. The proposed scheme is designed based on quotient ring of polynomial, \( Z_{2} [x]/{ } \), where \( f\left( x \right) \) is an irreducible polynomial. The decryption algorithm was further optimized with the use of the multiplicative inverse of the generator g(x), which only generated once during the key generation algorithm, thus leading to a simpler and faster decryption process. The proposed scheme is as secure as the original ElGamal scheme, since both schemes are based on the DLP. The preliminary result shows that the proposed scheme minimizes complex arithmetic operations and achieves very practical performance compared to the classic ElGamal algorithm and its variants. The proposed \( F_{2}^{n} \) based ElGamal scheme outperforms the \( F_{p} \) based scheme by significantly reducing 69.74% of the numbers of required logic gates in the case study of VLSI implementation.

This paper introduces a new public key cryptosystem based on two hard problems : the cube root extraction modulo a composite moduli (which is equivalent to the factorisation of the moduli) and the discrete logarithm problem. These two hard problems are combined during the key generation, encryption and decryption phases. By combining the IFP and the DLP we introduce a secure and efficient public key cryptosystem. To break the scheme, an adversary may solve the IFP and the DLP separately which is computationally infeasible. The key generation is a simple operation based on the discrete logarithm modulo a composite moduli. The encryption phase is based both on the cube root computation and the DLP. These operations are computationally efficient.

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.

Elliptic curves cryptography was introduced independently by Victor Miller (Miller, 1986) and Neal Koblitz (Koblitz, 1987(Koblitz, ) in 1985. .At that time elliptic curve cryptography was not actually seen as a promising cryptographic technique.As time progress and further research and intensive development done especially on the implementation side, elliptic curve cryptography is now being implemented widely.Elliptic curves cryptography offers smaller key size, bandwidth savings and faster in implementations when compared to the RSA (Rivest-Shamir-Adleman) cryptography which based its security on the integer factorization problem.The most interesting feature of the elliptic curves is the group structure of the points generated by the curves, where points on the elliptic curves form a group.The security of elliptic curves cryptography relies on the elliptic curves discrete logarithm problem.The elliptic curve discrete logarithm problem is analogous to the ordinary algebraic discrete logarithm problem, l = g x , where given the l and g, it is infeasible to compute the x.Elliptic curve discrete logarithm problem deals with solving for n the relation P = nG.Given the point P and the point G, then it is very hard to find the integer n.To implement the discrete logarithm problem in elliptic curve cryptography, the main task is to compute the order of group of the curves or in other words the number of points on the curve.Computation to find the number of points on a curve, has given rise to several point counting algorithms.The Schoof and the SEA (Schoof-Elkies-Atkin) point counting algorithms will be part of the discussion in this chapter.This chapter is organized as follows: Section 2, gives some preliminaries on elliptic curves, and in section 3, elliptic curve discrete logarithm problem is discussed.Some relevant issues on elliptic curve cryptography is discussed in section 4, in which the Diffie-Hellman key exchange scheme, ElGamal elliptic curve cryptosystem and elliptic curve digital signature scheme are discussed here accompanied with some examples.Section 5 discussed the two point counting algorithms, Schoof algorithm and the SEA (Schoof-Elkies-Atkin) algorithm.Following the discussion in section 5, section 6 summaries some similarities and the differences between these two algorithms.Section 7 gives some brief literature on these two point counting algorithms.Finally, section 8 is the concluding remarks for this chapter. Elliptic curvesElliptic curves obtained their name from their relation to elliptic integrals that arise from the computation of the arc length of ellipses (Lawrence & Wade, 2006).Elliptic curves are www.intechopen.com Cryptography and Security in Computing1 1 2 3 4 2 2 4 Fig. 1.1. 2(1 ) (1 ) yx x x   www.intechopen.com

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.

The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$G=?P? of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$P,?P,?2P,?,?dP?G, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? in the case of $$d|(p\pm 1)$$d|(p±1) with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$O((p±1)/d+di) group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$O(max{(p±1)/d,d}) storage ($$i=\frac{1}{2}$$i=12 or 1 for $$d|(p-1)$$d|(p-1) case or $$d|(p+1)$$d|(p+1) case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$P,?P,?kP,?k2P,?k3P,?,?k?(d)-1P?G and multiplicative cyclic group $$K=\langle k \rangle $$K=?k? are given, where d is a prime, $$\varphi (d)$$?(d) is the order of K and $$\varphi $$? is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$O((p-1)/d+d) group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$O((p-1)/d) storage.

One of the earliest examples of public key cryptography is the Diffie-Hellman key exchange protocol, which is based on the discrete logarithm problem. In 2000, Ko, Lee et al. modified the Diffie-Hellman protocol by using braid groups as platform where the security of the protocol lies on the conjugacy search problem. In this paper, we propose two key exchange protocols that use two well known classes of nonassociative Moufang loops as potential platforms.

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.