Security Of Elliptic Curve Cryptography Research Articles

Implementation and evaluation of EMAES – A hybrid encryption algorithm for sharing multimedia files with more security and speed

In this era of smartphones, a huge amount of multimedia files like audio, video, images, animation, and plain text are shared. And with this comes the threat of data being stolen and misused. Most people don’t think about the security of data before uploading it to any platform. Most apps used on smartphones upload our data to their server. Not only this, but other third-party apps can also read that data while it is being transmitted. One solution to this problem is encrypting the data before sharing it and decrypting it back at the other end so that even if it is intercepted in between the transmission, it would be impossible to decrypt it. In this paper, a newly designed hybrid encryption algorithm EMAES that includes the efficiency of MAES (Modified Advanced Encryption Standard) and security of ECC (Elliptic Curve Cryptography) was implemented in MATLAB as well as in android studio 4.0. using a mobile messaging application. Also, it was tested for different speeds and security parameters. Further, it was compared with standard algorithms like the RC4, RC6 and Blowfish as well as with other hybrid algorithms like RC4+ECC, RC6+ECC and Blowfish+ECC. The EMAES was found 30% more efficient in terms of encryption and decryption time. The security of EMAES also showed improvement when compared with other hybrid algorithms for parameters like SSIM (structural similarity index measure), SNR (Signal to Noise Ratio), PSNR(Peak Signal to Noise Ratio), MSE (Mean Squared Error) and RMSE (Root Mean Squared Error). And finally, no significant improvement was found in the CPU and RAM usage.

ELLIPTIC CURVE CRYPTOGRAPHY AND ITS PRACTICAL APPLICATION

Elliptic curves are one of the most promising tools for constructing modern cryptographic algorithms. The security of elliptic curve cryptography is based on the complexity of solving the discrete logarithm problem in the group of points of the elliptic curve over a finite field. Elliptic curve cryptography enables two parties communicating over public channel using elliptic curve encryption and signing algorithms. Elliptic curves allow to achieve the same level of security with small key sizes than other asymmetric cryptographic algorithms. The article describes the mathematical apparatus of elliptic curves used for cryptographic purposes, the basic operations in the group of points of elliptic curves, such as addition of points, doubling of a point, and scalar multiplication of a point by a number are given. The steps and principles of the Diffie-Hellman key exchange algorithm (ECDH) and the digital signature scheme (ECDSA) on elliptic curves are considered. An overview of standards establishing recommendations and requirements for the use of elliptic curves in cryptographic systems is provided. The advantages of elliptic curve cryptography compared to traditional asymmetric algorithms, such as smaller key sizes, computational speed, and efficient use of resources, are analyzed. Potential threats and vulnerabilities of cryptographic algorithms based on elliptic curves are discussed. The main practical application areas of cryptographic algorithms on elliptic curves, including network security, cryptocurrency operations, message exchange, the Internet of Things, and government institutions are investigated. Examples of popular standardized curves (Curve25519, Curve448, secp256k1) that have been tested and recommended by specialized organizations such as NIST are given.

An Elliptic Curve Signcryption Scheme and Its Application

Two basic security requirements in communication are confidentiality and authentication. Signcryption is an ideal technique to transmit encrypted and authenticated data. In view of the shortcomings of existing signcryption schemes and the high security of elliptic curve cryptography (ECC), we design a ECC-based signcryption scheme and evaluate it in terms of security, computational overhead, and communication overhead. Finally, we consider the application of our secure and efficient signcryption scheme in the smart lock key management system and analyze the bit-oriented performance of the designed key management scheme.

Implementation of Pollard Rho over binary fields using Brent Cycle Detection Algorithm

The security of Elliptic Curve Cryptography depends on how to solve the Elliptic Curve Cryptography Discrete Logarithm Problem (ECDLP). In this paper we propose the use of modified Pollard Rho Algorithm by using Brent Cycle Detection Algorithm to solve the ECDLP. We give performance comparison on time and the number of iterations between Pollard Rho with Brent Cycle Detection and Pollard Rho with Negation map. In particular, for Koblitz curve, we also give comparison between Pollard Rho with Brent Cycle Detection and Pollard Rho with Negation and Frobenius maps.

Web-based Volunteer Computing for Solving the Elliptic Curve Discrete Logarithm Problem

Elliptic curve discrete logarithm problem (ECDLP) is the basis of security of elliptic curve cryptography (ECC). The security evaluation of ECC has been studied by solving an ECDLP. We need a large amount of computational resources for the evaluation. This paper proposes a new system collecting computational resources with Web-based volunteer computing (Web-based VC). In the system, web applications are allocated to volunteer participants (workers) as jobs. Web applications are built by utilizing technologies called Native Client (NaCl) and Portable NaCl (PNaCl). This paper evaluates the performance of the system and solves 70-bit ECDLP with Web-based VC. The performance of the web application utilizing NaCl is approximately 6.4 times higher than that of the web application written in JavaScript. Also, the performance of the web application utilizing PNaCl is approximately 4.2 times higher. In the case of NaCl, 70-bit ECDLP is solved in only 1389 seconds.If we collect 100,000 PCs by Web-based VC, 114-bit ECDLP will be solved in only approximately 1 day.

Improvement of FPPR method to solve ECDLP

Solving the elliptic curve discrete logarithm problem (ECDLP) by using Grobner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugere, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough.

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Elliptic curve cryptography (ECC) is an approach to public key cryptography (PKC) that is based on algebraic operations with elliptic curves defined over finite fields. Security of elliptic curve cryptography is based on the hardness of the elliptic curve discrete logarithm problem (ECDLP). Although there is no theoretical proof that ECDLP is intractable, no general-purpose sub-exponential running time algorithm has been found for solving the ECDLP if the elliptic curve parameters are chosen properly. In this study, we develop a new security attack based on the binary division of elliptic curve points over prime fields that may be used to solve the ECDLP when the order q of elliptic curve satisfies the congruence q = 2 (mod 4). To perform the binary division, we devise a novel algorithm of point halving on elliptic curves defined over prime fields that applies to the cases when q = 1 (mod 2) and q = 2 (mod 4). The binary division attack has exponential worst-case asymptotic time complexity but in certain practical cases can be used to solve the ECDLP in a relatively efficient way. We therefore make a recommendation to avoid the case of q = 2 (mod 4) in elliptic curve cryptosystems.

A Hardware-Accelerated ECDLP with High-Performance Modular Multiplication

Elliptic curve cryptography (ECC) has become a popular public key cryptography standard. The security of ECC is due to the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). In this paper, we demonstrate a successful attack on ECC over prime field using the Pollard rho algorithm implemented on a hardware-software cointegrated platform. We propose a high-performance architecture for multiplication over prime field using specialized DSP blocks in the FPGA. We characterize this architecture by exploring the design space to determine the optimal integer basis for polynomial representation and we demonstrate an efficient mapping of this design to multiple standard prime field elliptic curves. We use the resulting modular multiplier to demonstrate low-latency multiplications for curves secp112r1 and P-192. We apply our modular multiplier to implement a complete attack on secp112r1 using a Nallatech FSB-Compute platform with Virtex-5 FPGA. The measured performance of the resulting design is 114 cycles per Pollard rho step at 100 MHz, which gives 878 K iterations per second per ECC core. We extend this design to a multicore ECDLP implementation that achieves 14.05 M iterations per second with 16 parallel point addition cores.

Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a ‘shared secret’ related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.