Attacking elliptic curve cryptosystems with special-purpose hardware

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.

The acceleration of cryptographic applications on embedded devices is a topic of increasingly importance, due to their massive use. In this paper, the efficiency of these devices when operating as cryptographic accelerators is assessed, using a two-tiered parallelism model, where not only multi-core, but also Single Instruction Multiple Data (SIMD) parallelism is exploited to increase the throughput of modular multiplications. In addition, the applicability of the developed parallel modular multiplications are tested for the Rivest-Shavir-Adleman (RSA) and Elliptic Curve (EC) cryptosystems. The relative efficiency of the two-tiered model for implementing cryptosystems on general purpose processors and embedded devices is analysed and experimentally evaluated. It can be concluded that it is possible to achieve the same levels of execution enhancement on both platforms. Moreover, speedups of up to 7.3 and 3.9 are obtained for the RSA and EC cryptosystems on the ARM A15 quad-core processor, respectively.

Elliptic curve cryptosystem (ECC) was proposed by Miller [10] and Koblitz [9] which relies on the difficulty of elliptic curve discrete logarithmic problem (ECDLP). It is gaining wide acceptance as an alternative to the conventional public key cryptosystem such as RSA [24], DSA [25], DH [26]. Also it is noted that the wireless sensor networks (WSN) based on the rapid progress of wireless communications and embedded micro electro mechanical systems technologies are becoming important part in our daily life. The security of the WSN becomes one of the major concerns in its applications. Even ECC prominently offers great potential benefits for WSN security there is still a lot of work needs to be done due to WSN has very restraint running conditions such as limited energy source, capability of computing, etc. It is well known that scalar multiplication operation in ECC accounts for about 80% of key calculation time on wireless sensor network motes. In this paper we present an optimized dynamic window based on our previous research works. The whole quality of service (QoS) has been improved under this algorism in particularly the power consuming is more efficiently. The simulation results showed that the average calculation time, due to fuzzy conditions decreased from previous 26 to current 9 as a whole the calculation time, decreased by approximately 18% in comparison to our previous algorithms in an ECC wireless sensor network [23].

In this digital world, communication systems have witnessed abundant usage of media over the platforms. Among these, providing security in transmission of images is highly important, and attained a lot of research interest due to its high consideration in both the industry and the academic community. This paper proposes a hybrid asymmetric image encryption algorithm using Elliptic curve cryptosystem (ECC), Hadamard transform and Hill cipher algorithms. Based on the Diffie–Hellman public key exchange method a point on the elliptic curve is selected and agreed between both the sender and receiver. The key relies upon the ECC and it is difficult to resolve the ECDLP to get it. The proposed algorithm involves two stages of encryption, primarily, the XOR function is applied on the Elliptic curve Diffie-Hellman (ECDH) shared secret key and the hadamard image. In the subsequent stage, ECC is combined with the hill cipher algorithm. Encryption and decryption uses self-invertible key matrix, hence the process of finding inverse key becomes redundant during decryption which improves the speed of execution. It also enhances the security and efficiency compared to original hill cipher method. The results are compared with other ECC methods proves that the current cryptosystem attains large key space, highly key sensitive, low correlation and can resist against differential and statistical attacks.

An ID-based remote mutual authentication with key agreement scheme for mobile devices on elliptic curve cryptosystem

Elliptic curve cryptosystem — The answer to strong, fast public-key cryptography for securing constrained environments

In 2009, Yang and Chang proposed an ID-based remote mutual authentication with key agreement scheme on elliptic curve cryptosystem (ECC). Based upon ID-based concept, Yang and Chang scheme (YC scheme) does not require additional computations for certificate and is not constructed by bilinear-pairings, which is an expensive operation on elliptic curve. In addition, YC scheme not only provides mutual authentication but also supports a session key agreement between the user and the server. Therefore, YC scheme is more efficient and practical than the related works. However, we find YC scheme is vulnerable to an impersonation attack and does not provide perfect forwardsecrecy in spite of efforts to perform mutual authentication and session key agreement between the user and the remote server and reduce the computational costs than the related works. Therefore, this paper proposes an improved ID-based remote mutual authentication with key agreement scheme for mobile devices on ECC. Compared with YC scheme, the proposed scheme is more secure, efficient, and practical for mobile devices because the proposed scheme not only eliminates the security flaws of YC scheme but also reduces the computational costs between the user and the server.

Elliptic curve cryptosystems (ECCs) are becoming more popular because of the reduced number of key bits required in comparison to other cryptosystems (for example, a 160 bit ecc has roughly the same security as 1024 bit RSA). ECCs are especially suited to smart cards because of the limited memory and computational power available on these devices. This paper discusses an optimized implementation of the elliptic curve Digital Signature Algorithm implemented over the field $GF(p)$ on a Motorola smart card. Algorithms for point addition, point doubling and scalar multiplication are compared according to their timings. The effects of different memory usage, code size and speed tradeoffs which were considered during the implementation are discussed. Also, optimized point addition and doubling algorithms are presented.

Elliptic curve cryptosystems (ECCs) are becoming more popular because of the reduced number of key bits required in comparison to other cryptosystems (e.g. a 160 bit ECC has roughly the same security as 1024 bit RSA). ECCs are especially suited to smart cards because of the limited memory and computational power available on these devices. However, the side-channel attacks which have recently been proposed can obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time. This information may be used to break implementations that have not incorporated defences against these attacks. This paper presents a new defence against Simple Power Analysis (SPA). This new defence is based on the NAF (non-adjacent form) representation of a scalar and requires 44% fewer additions and 11% extra doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm.

Elliptic curve cryptosystems, proposed by Koblitz ([12]) and Miller ([16]), can be constructed over a smaller field of definition than the ElGamal cryptosystems ([6]) or the RSA cryptosystems ([20]). This is why elliptic curve cryptosystems have begun to attract notice. In this paper, we investigate efficient elliptic curve exponentiation. We propose a new coordinate system and a new mixed coordinates strategy, which significantly improves on the number of basic operations needed for elliptic curve exponentiation.

An efficient key-management scheme for hierarchical access control based on elliptic curve cryptosystem

One of the main barriers to the development of multi-application smart card is the limitation of the memory size and processing power. This paper gives a solution to it by using an elliptic curve cryptosystem (ECC) as the public key scheme. The basic mathematics principles of ECC are discussed. Based on the ECC, we also put a digital certificate inside the smart card. This paper presents a prototype of a multi-application smart card system using the ECC certificate. Several security issues are also discussed.

The cryptosystem of elliptic curve has gained more and more researchers' attention from home and abroad in recent years as it owns a short key and a fine security. This paper elaborates the definition of elliptic curve which is used to establish cryptosystem and also makes some explanation to the realization of the Add Point Operation on the elliptic curve together with the fast Add Point Operation of its corresponding coordinate system. In addition, the security of the elliptic curve cryptosystem is also analyzed starting from the theory of the elliptic curve in this paper. And the issue of production of the safe elliptic curve is discussed. Finally, the elliptic curve cryptosystem has aroused great attention of the researchers as the public key system of the next generation. The production arithmetic of the safe elliptic curve, the security analysis of the elliptic curve cryptosystem together with the fast arithmetic on the elliptic curve will be the hotspot and the issues that are remained to be solved in its future development.

Recently the study and implementation of elliptic curve cryptosystems (ECC) have developed rapidly and its achievements have become a center of attraction. ECC has the advantage of high-speed processing in software even on restricted environments such as smart cards. In this paper, we concentrate on implementation of ECC over a field of prime characteristic on a 16-bit microcomputer M16C (10MHz). We report a practical software implementation of a cryptographic library which supports 160-bit elliptic curve DSA (ECDSA) signature generation, verification and SHA-1 on the processor. This library also includes general integer arithmetic routines for applicability to other cryptographic algorithms. We successfully implemented the library in 4Kbyte code/data size including SHA-1, and confirmed a speed of 150msec for generating an ECDSA signature and 630msec for verifying an ECDSA signature on M16C.

Elliptic curve based hardware architecture using cellular automata