An authenticated group key transfer protocol using elliptic curve cryptography

With the growing practical implementation of smart home, the attacks on smart homes are proportionally increasing. Residents can only be benefited from smart home technology if they and their home-assets are secured against cyber-attacks. A number of PKI-based communication security models have been proposed for data authentication and confidentiality in smart homes. However, it is not convenient for a home device with the limited capacity to store, verify, and manage public keys (certificates) of all other devices. Identity-based cryptography (IBC) is one of the asymmetric cryptographic solutions that does not require certificates. However, due to the central storage of the secret at the key generation center (KGC), the security fully relies on the KGC in IBC environment. Thus, to resolve these issues while providing the security to smart homes, in this paper, we proposed a straightforward and light-weight security model based on IBC, wheel pairing, and elliptic curves. The proposed model performs distributed key generation where the main secret is generated by all participating home devices, instead of a central KGC. We designed a complete protocol, which illuminates the fundamental steps of new device enrollment, distributed key generation, device to device encryption, data integrity, and entity authentication. Moreover, the commitment procedure is introduced that ensures no party can change its partial-secret after he has committed to it. The elliptic curve cryptography (ECC) based Diffie-Hellman (DH) model is deployed for session key generation for device to device data encryption, whereas IBC-based private key is used for signatures. Finally, the feasibility of the model is evaluated by implementing the system on various numbers of IoT machines, while considering them as home devices. Also, the security of the proposed model is verified technically and formally by a software verification tool called Automated Validation of Internet Security Protocols and Applications (AVISPA) against popular known attacks.

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

Elliptic curve cryptography is a public key cryptosystem based on the elliptic curve discrete logarithm problem. The reason for the attractiveness of ECC is that there is no sub-exponential algorithm known to solve the elliptic curve discrete logarithm problem. This leads to lower memory requirements, lower computation requirements and higher security than other public key cryptosystems. This project describes an implementation of a micro coded elliptic curve processor using FPGA technology. This processor implements optimal normal basis field operations. The design is generated by a parameterized module generator which can accommodate arbitrary n and also produce field multipliers with different speed/area tradeoffs. The control part of the processor is micro coded, enabling curve operations to be incorporated into the processor hence reducing the chip's I/O requirements. The micro coded approach also facilitates rapid development and algorithmic optimization. for example, projective and affine coordinates were supported using different sets of micro codes. The design was successfully tested on a Xilinx Spartan -II kit device. Elliptic curve processor is particularly suitable for recourse limited devices because of its lower requirements of memory and computation. It is applicable to be implanted into smart cards, cellular phones and other hand-held devices so that secure communication can be provided. For based on implement the FPGA Using Verilog HDL.

Special designs are needed for cryptographic schemes in wireless sensor networks (WSNs). This is because sensor nodes are limited in memory storage and computational power. The existing group key transfer protocols for WSNs using classical secret sharing require that a t-degree interpolating polynomial be computed in order to encrypt and decrypt the secret group key. This approach is too computationally intensive. In this paper, we propose a new group key transfer protocol using a linear secret sharing scheme and factoring assumption. The proposed protocol can resist potential attacks and also significantly reduce the computation complexity of the system while maintaining low communication cost. Such a scheme is desirable for secure group communications in WSNs, where portable devices or sensors need to reduce their computation as much as possible due to battery power limitations.

Key establishment protocols are fundamental for establishing secure communication channels over public insecure networks. Security must be given the topmost priority in the design of a key establishment protocol. In this work, we provide a security analysis on two recent key establishment protocols: Harn and Lin’s group key transfer protocol and Dutta and Barua’s group key agreement protocol. Our analysis shows that both the Harn-Lin protocol and the Dutta-Barua protocol have a flaw in their design and can be easily attacked. The attack we mount on the Harn-Lin protocol is a replay attack whereby a malicious user can obtain the long-term secrets of any other users. The Dutta-Barua protocol is vulnerable to an unknown key-share attack. For each of the two protocols, we present how to eliminate their security vulnerabilities. We also improve Dutta and Barua’s proof of security to make it valid against unknown key share attacks.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

This paper examines the cryptographic security of fixed versus random elliptic curves over the field GF(p). Its basic assumption is that a large precomputation to aid in breaking the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve. We take this into account when examining curve security as well as considering a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. We present a lower bound on the expected time to solve such ECDLPs using this method, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. We conclude that adding 5 bits to the size of a fixed curve to avoid general software attacks and an extra 6 bits to avoid attacks on special moduli and a parameters provides an equivalent level of security.

Cryptographic security is essential for ensuring data integrity and secure communication. Traditional encryption methods such as RSA and elliptic curve cryptography (ECC), while widely used, suffer from high computational complexity, large key sizes, and vulnerability to evolving attack strategies. This paper introduces geometric mean labeling(GML), a novel graph-based cryptographic technique that enhances encryption efficiency, key management, and security. Unlike RSA, which relies on prime factorization, or ECC, which depends on elliptic curve logarithms, GML applies geometric mean-based transformations to encrypt data using structured graph relationships, making it computationally efficient. Specifically, GML achieves encryption in just 0.085 ms with a 512-bit key, compared to 600 ms (RSA, 2,048-bit) and 30 ms (ECC, 256-bit), leading to a 99.99% speed improvement over RSA and 99.7% over ECC. Theoretical analysis confirms that GML operates with a computational complexity of O(V3), making it scalable for large-scale encryption while maintaining strong security against brute-force and chosen-plaintext attacks. Experimental validation was conducted on an Intel i7-based system, confirming the efficiency and reliability of GML for real-world applications such as IoT security and cloud encryption. By integrating GML into modern cryptographic frameworks, this study provides a scalable, efficient, and future-ready encryption solution, with potential applications in post-quantum cryptography.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

Privacy Preservation in Data Mining (PPDM) including for Privacy Preserving Association Rule Mining (PPARM) has attracted lots of attention in recent research and practice. However, the current method or approach still have drawbacks in the sense that there are trade-offs between efficiency and privacy preservation. This paper describes our work towards providing a new efficient PPARM protocol. We reviewed current literature on PPARM and mapped the methods or approaches involved. As previous research showed that Elliptic Curve Cryptography (ECC) perform better than the other Public Key systems such as RSA and Diffie-Hellman, we will utilize ECC for reducing the computational cost of the new PPARM protocol. In choosing good elliptic curves for ECC, we measured the running time of the key generation for various group of recommended elliptic curves i.e. Brainpool curves (by Brainpool), Prime, C2pnb, C2tnb curves (by ANSI X9.62), Secp curves (by SECG), and PrimeCurve curves (by CDC Group). As the result, Secp curves outperformed all of the other curves on overall average ratio of running time and key size of key generation by 4.4% up to 357.6%.

The Residue Number System (RNS) is a numeral representation enabling for more efficient addition and multiplication implementations. However, due its non-positional nature, modular reductions, required for example by Elliptic Curve (EC) Cryptography (ECC), become costlier. Traditional approaches to RNS modular reduction resort to the Montgomery algorithm, underpinned by large basis extensions. Recently, Hybrid-Positional Residue Number Systems (HPRs) have been proposed, providing a trade-off between the efficiency of RNS and the flexibility of positional number representations. Numbers are represented in a positional representation with the coefficients represented in RNS. By crafting primes of a special form, the complexity of reductions modulo those primes is mitigated, relying on extensions of smaller bases. Due to the need of crafting special primes, this approach is not directly extensible to group operations over currently standardised elliptic curves. In this paper, the Hybrid-Polynomial Residue Number System (HyPoRes) is proposed, enabling for improved modular reductions for any prime. Experimental results show that the modular reduction of HyPoRes, although at most 1.4 times slower than HPR for HPR-crafted primes, is up to 1.4 times faster than a generic RNS approach for primes of ECC standards.

Known group key transfer protocols in group communications using classical secret sharing require that a t-degree interpolating polynomial be computed in order to encrypt and decrypt the secret group key. Secret sharing plays an important role in ensuring the group communications security. A verifiable multi-secret sharing (VMSS) scheme is a multi-secret sharing scheme with the verifiable property. Recently, Zhao et al. and Dehkordi et al. successively proposed two threshold VMSS schemes. Shortly, using the same verification mechanism, Dehkordi et al. presented another two VMSS schemes. In these schemes, authors claimed that the dealer was absolutely impossible to become a cheater. In this paper, we show that in both Zhao scheme and Dehkordi scheme, a dishonest dealer may distribute a fake share to a certain participant, and then that participant would subsequently never obtain the true secret. Indeed, verification mechanism should be improved in these schemes; and furthermore our results highlight that extra cautions still be exercised when constructing schemes in this direction.

Anonymous channel tickets have been proposed as a way to provide user anonymity and to reduce the overhead of re-authentication for authentication in wireless environments. Chen et al. proposed a secure and efficient protocol, based on a protocol proposed by Yang et al., which is resistant to guessing attacks on networks from which users’ secret keys are easy to obtain. However, their scheme is time-consuming in the phases of ticket issuing and authentication. Furthermore, a malicious attacker can utilize the expired time, Texp, to launch a denial of authentication (DoA) attack, which is a type of denial of service attack. Because Texp is exposed to any user, it would be easy to launch a DoA attack that could make the scheme impractical. To resist against DoAs that the scheme of Chen et al. might suffer, we propose an improved scheme based on elliptic curve cryptography in this paper. Our scheme not only reduces time cost but also enhances security. The basis of the proposed scheme is the elliptic curve discrete logarithm problem. The operations of points of an elliptic curve are faster and use fewer bits to achieve the same level of security. Therefore, our scheme is more suitable for mobile devices, which have limited computing power and storage. Copyright © 2012 John Wiley & Sons, Ltd.

A pairing over an elliptic curve E(F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> ) to an extension field of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pmk</sub> has begun to be attractive in cryptosystems, where k is called the embedding degree. The cryptosystems using a pairing are called the pairing-based cryptosystems. The embedding degree k is also an indicator of the relationship between the elliptic curve discrete logarithm problem (ECDLP) and the discrete logarithm problem (DLP), where ECDLP over E(F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> ) is reduced to DLP over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pmk</sub> . An elliptic curve is determined by j-invarient or order, however the explicit condition between these parameters and an embedding degree has been described only in some degrees. In this paper, we investigate a new condition of the existence of elliptic curves with pre-determined embedding degrees, and present some examples of the elliptic curves over 160-bit, 192-bit and 224-bit F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> .