An Enhanced Cryptographic Method for Securing and Accelerating Digital Signatures

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.

RSA [1] is one of the mostly used cryptosystem in securing data and information. Though, it has been recently discovered that RSA has some weaknesses and in advance technology, RSA is believed to be inefficient especially when it comes to decryption. Thus, a new algorithm called Multi prime RSA, an extended version of the standard RSA is studied. Then, a modification is made to the Multi prime RSA where another keys is shared secretly between the receiver and the sender to increase the securerity. As in RSA, the methodology used for modified Multi-prime RSA also consists of three phases; 1. Key Generation in which the secret and public keys are generated and published. In this phase, the secrecy is improved by adding more prime numbers and addition of secret keys. 2. Encryption of the message using the public and secret keys given. 3. Decryption of the secret message using the secret key generated. For the decryption phase, a method called Chinese Remainder Theorem is used which helps to fasten the computation. Since Multi prime RSA use more than two prime numbers, the algorithm is more efficient and secure when compared to the standard RSA. Furthermore, in modified Multi prime RSA another secret key is introduced to increase the obstacle to the attacker. Therefore, it is strongly believed that this new algorithm is better and can be an alternative to the RSA.

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.

Implementing an improved ID‐based cryptographic mechanism is the main objective of the proposed work. In this article, an ID‐based cryptographic (IBC) technique using generalized discrete logarithm problem (GDLP) and the integer factorization problem (IFP) presented by Meshram et al have improved. Meshram et al have given IBC technique without using the bilinear pair and also, reveal that their technique can attain data protection and security objectives. Besides, their technique will deter the adversary from eavesdropping the encrypted information or the secret key of the user. However, it has found that their system carries a deadlock problem. Encryption process, as expected by the user, is not guaranteed to be secure. It is because the user may require private information about key authentication center (KAC), which has kept secret from users. Pang et al have proposed an improved technique that overcomes the deadlock problem. It has found that Pang et al have not discussed the analysis and proofs of security. In this article, generalized discrete logarithms in multiplicative group over finite fields and IFP have used to improve the technique and also a key distribution system has discussed. It has analyzed that the proposed strategy is safer than the technique described by Pang et al. Also, it has found that the proposed technique addresses the deadlock problem.

This paper presents a strong designated verifier group signature scheme, in which signature is generated by any member of the group and it is verified by only a specific receiver, called a designated verifier using the public parameters of the group and his own secret key. The verifier assumes that the signature indeed is from the group itself, not from any individual group member. In this scheme, anonymity of the signer is completely preserved. The verifier cannot link a signature to the signer. The scheme is secure as an adversary cannot verify the signature even if the secret key of the signer is compromised or leaked. Also, neither the group manager nor any member of the group can produce a valid signature on behalf of another member. Only in case of legal disputes, the group manager can disclose the identity of a group member who has signed a document. The major benefit of this scheme is that the length of the group signature is independent of number of group members and it remains unaffected if some members left the group or new members join the group. Security of the proposed scheme lies in the complexity of solving three computationally hard problems, namely, Computational Diffiee-Helmann (CDH) problem, Discrete Logarithm Problem (DLP) and Integer Factorization Problem (IFP). This scheme can be useful in organizations where there is a need to send confidential documents to a specific recipient. This scheme can also be applicable to real life scenarios, such as, e-commerce applications, e-banking and e-voting.

This paper presents a strong verifiable signcryption scheme, in which a message is signcrypted by a signcryptor and only a specific receiver, who called a designated verifier, verifies it using his own secret key. The scheme is secure, as an adversary can not verify the signature even if the secret key of the signer is compromised or leaked. The security of the proposed scheme lies in the complexity of solving two computationally hard problems, namely, the Discrete Logarithm Problem (DLP) and the Integer Factorization Problem (IFP). The security analysis of the scheme has been done and it is proved that, the proposed scheme can withstand an adaptive chosen ciphertext attack. This scheme can be very useful in organizations where there is a need to send confidential documents to a specific recipient. This scheme can also be applicable to real life scenarios, such as, e-commerce applications, e-banking and e-voting.

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

This paper introduces two new public key encryption techniques namely the "logarithmic-factorized (LF)" and the "double-exponentiation (DE)" cryptosystems. The main one-way trapdoor functions of these algorithms are the general form of the generalized discrete logarithm problem (GDLP), the integer factorization problem (IFP) and the double IFP (DIFP). The LF algorithm depends simultaneously on both the GDLP and the IFP, while the DE algorithm depends on the DIFP. The validity of these algorithms is proven by applying them to messages and returning the original messages in various numerical examples. The proposed cryptosystems have many advantages that are explained in detail.

The Internet of Things (IoT) revolutionizes how we interact with smart devices. However, ensuring security in IoT systems presents complex challenges due to their centralized architecture and inherent limitations. The increasing attention towards blockchain technology stems from its decentralized structure, transparency, immutable records, and cryptographic hash functions, particularly when combined with the Internet of Things (IoT). Cryptographic hash algorithms are essential to blockchain technology, ensuring secure data transmission by linking private and public keys through mathematical functions. Existing cryptographic algorithms typically produce variable hash values accessible to a single node and support input key sizes up to 128 bytes. However, larger key sizes can reduce transaction accuracy, lower key entropy, decrease throughput, and increase transaction complexity. Conversely, smaller key sizes increase vulnerability to attacks. To overcome these challenges, several advancements have been proposed. A modified key signature scheme extends the key size to 256 bytes, achieving 82.6% accuracy, a time complexity of 1.68 nanoseconds, and a throughput of 15.24 kilobytes per 200 nanoseconds. However, this scheme operates on a single-node basis, limiting accessibility. These advancements aim to bolster security in IoT data transmission by integrating encryption techniques with blockchain technology. The performance of the modified signature scheme is analyzed through transaction parameters such as accuracy, time complexity, throughput, and hash output with cryptographic hash functions like SHA224, SHA256, SHA384, SHA512, MD5, SHA3-224, SHA3-256, SHA3-384, SHA3-512, and ECDSA. From this analysis, the elliptic curve cryptography algorithm gives higher accuracy and throughput with less time complexity.

NTRUSign is a lattice-based digital signature scheme proposed by Hoffstein et al. NTRUSign is quite different from many other signature schemes in a sense that its security depends on neither the integer factorization problem nor the discrete logarithm problem but on a geometric problem called the close vector searching problem. However, it is known that there is some vulnerability in NTRUSign, namely there is an attack called the transcript attack. In this paper, we propose a countermeasure for protecting NTRUSign against the transcript attack, and give an improved NTRUSign algorithm.

Quantum computing represents a transformative advancement in computational power, with profound implications for cryptographic systems. This paper explores the intersection of quantum computing and cryptography, examining the potential for quantum computers to both break traditional cryptographic algorithms and enable new forms of secure communication. Classical cryptographic systems, such as RSA and ECC, rely on the computational difficulty of problems like integer factorization and discrete logarithms. However, quantum algorithms, notably Shor's algorithm, can solve these problems exponentially faster than the best-known classical algorithms, posing a significant threat to the security of current cryptographic practices. This vulnerability necessitates the development and implementation of quantum-resistant algorithms to ensure data security in a post-quantum world. The paper reviews the state-of-the-art in quantum-resistant cryptographic algorithms, including lattice-based, hash-based, code-based, and multivariate polynomial cryptography. These algorithms are analyzed for their security properties, efficiency, and practicality for real-world deployment. Additionally, the paper discusses the ongoing standardization efforts led by organizations such as NIST, which are crucial for establishing widely accepted quantum-safe cryptographic protocols. Beyond the threat posed by quantum computing, this paper also highlights the potential benefits quantum computing can bring to cryptographic systems. Quantum Key Distribution (QKD) is one such advancement, offering theoretically unbreakable encryption by leveraging the principles of quantum mechanics. QKD enables secure communication channels that are immune to any computational advances, including those of quantum computers, by ensuring that any attempt at eavesdropping can be detected. The practical implementation challenges of QKD and other quantum cryptographic protocols are also addressed, including issues related to transmission distance, error rates, and integration with existing communication infrastructures. Experimental results from recent QKD trials are presented, showcasing significant progress in extending the viability of quantum-secure communication over long distances. Furthermore, the paper delves into the hybrid approaches combining classical and quantum cryptographic techniques. These approaches aim to leverage the strengths of both paradigms to enhance security and performance. For instance, hybrid protocols might use classical encryption for bulk data transfer, supplemented by quantum techniques for key exchange, thereby ensuring robust security even against future quantum adversaries. In conclusion, while quantum computing poses substantial risks to existing cryptographic systems, it also offers novel opportunities for creating highly secure communication protocols. The transition to quantum-safe cryptography is imperative to protect sensitive information in the quantum era. This paper underscores the urgency of advancing quantum-resistant cryptographic research, promoting international collaboration for standardization, and exploring innovative applications of quantum technologies in cryptography. Continued research and development in these areas are essential to stay ahead of the evolving threat landscape and harness the full potential of quantum computing for secure communications.

A post-quantum secure PUF based cross-domain authentication mechanism for Internet of drones

Due to the scalability of the stateless and compact JSON Web Tokens (JWT), they are increasingly used in securing modern applications to support single-sign-on context and establish trust in microservices or connected-devices architectures. However, currently JWT relies on traditional encryption algorithms that are based on integer factorization or discrete logarithm problems; this exposes JWT signatures to vulnerability when quantum computers become available. Recently, NIST has announced new standards which are quantum safe. In this paper, we investigate the usage of two quantum-resistant algorithms, which are NIST candidates of round 2 standardization process in 2019, to create and verify JWT signatures. Namely, we consider lattice-based schemes (DILITHIUM and qTESLA) and compare their performance against RSA for digital signature for different security levels defined by NIST. The results show that lattice-based digital signature schemes have better performance than RSA in terms of request per seconds and average response time. DILITHIUM has demonstrated the best performance while also having smaller performance loss when scaling the security level.

This paper presents the generation of digital signature along with message recovery based on Elliptic Curve Cryptography (ECC) and knapsack algorithm. In digital signature along with message recovery scheme, signature alone is sent and message is recovered from the signature (r, s). ECC provides greater security with less key size, when compared to integer factorization and discrete logarithm system. As the strength of knapsack algorithm depends on the selection of the series, the proposed algorithm uses modified Stern series which not only reduces the time complexity but also provides better security.

The need for storage and movement of data in today's ever increasing data usage cannot be overemphasized. As a result, storage and movement of data within the cloud becomes a viable option, even though this is a third party and some of the data may be sensitive. Therefore, the security methods in place have to continually be reviewed so as to get the best method or a combination of methods in this ever changing and demanding field. In this paper, the authors have done a formal assessment and examination of cloud systems key cryptographic algorithms and given a recommendation of the best combination of algorithms as a result of our different parameters of considerations. These parameters include whether they are a stream or block cipher and their key size or the hash value. The authors found out that the most viable cryptographic methods for securing the cloud systems are AES, Blowfish, Diffie-Hellman, RSA and SHA-1 which are from symmetric, asymmetric and hashing cryptographic algorithm. They further noted that a hybrid of the three cryptographic algorithms is more effective than an exclusive use of one cryptographic algorithm. They propose a combination of AES, Diffie-Hellman and SHA-1 algorithm for optimum cloud security.