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.

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A High-Speed Elliptic Curve Cryptography Processor for Teleoperated Systems Security

Mathematical Problems in Engineering ◽

10.1155/2021/6633925 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Yong Xiao ◽

Weibin Lin ◽

Yun Zhao ◽

Chao Cui ◽

Ziwen Cai

Keyword(s):

Elliptic Curve ◽

Elliptic Curve Cryptography ◽

High Speed ◽

High Performance ◽

Prime Field ◽

Practical Applications ◽

Cell Library ◽

Cryptographic Algorithm ◽

Human Operators ◽

Key Length

Teleoperated robotic systems are those in which human operators control remote robots through a communication network. The deployment and integration of teleoperated robot’s systems in the medical operation have been hampered by many issues, such as safety concerns. Elliptic curve cryptography (ECC), an asymmetric cryptographic algorithm, is widely applied to practical applications because its far significantly reduced key length has the same level of security as RSA. The efficiency of ECC on GF (p) is dictated by two critical factors, namely, modular multiplication (MM) and point multiplication (PM) scheduling. In this paper, the high-performance ECC architecture of SM2 is presented. MM is composed of multiplication and modular reduction (MR) in the prime field. A two-stage modular reduction (TSMR) algorithm in the SCA-256 prime field is introduced to achieve low latency, which avoids more iterative subtraction operations than traditional algorithms. To cut down the run time, a schedule is put forward when exploiting the parallelism of multiplication and MR inside PM. Synthesized with a 0.13 um CMOS standard cell library, the proposed processor consumes 341.98k gate areas, and each PM takes 0.092 ms.

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High performance hardware support for elliptic curve cryptography over general prime field

Microprocessors and Microsystems ◽

10.1016/j.micpro.2016.12.005 ◽

2017 ◽

Vol 51 ◽

pp. 331-342 ◽

Cited By ~ 16

Author(s):

Khalid Javeed ◽

Xiaojun Wang ◽

Mike Scott

Keyword(s):

Elliptic Curve ◽

Elliptic Curve Cryptography ◽

High Performance ◽

Hardware Support ◽

Prime Field

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A COMPARATIVE ANALYSIS OF ECDSA V/S RSA ALGORITHM

International Journal of Computer Science and Informatics ◽

10.47893/ijcsi.2014.1165 ◽

2014 ◽

pp. 7-9

Author(s):

AMANPREET KAUR ◽

VIKAS GOYAL

Keyword(s):

Elliptic Curve ◽

Finite Fields ◽

Elliptic Curve Cryptography ◽

Public Key Cryptography ◽

Cryptographic Primitives ◽

Prime Field ◽

Rsa Algorithm ◽

Encryption And Decryption ◽

Security Problem ◽

Accurate Observation

Elliptic curve Cryptography with its various protocols implemented in terms of accuracy and fast observation of results for better security solution. ECC applied on two finite fields: prime field and binary field. Because it is public key cryptography so, it also focus on generation of elliptic curve and shows why finite fields are introduced. But for accurate observation we do analysis on category of cryptographic primitives used to solve given security problem. RSA & ECDSA both have basic criteria of production of keys and method of encryption and decryption in basic application as per security and other properties which are authentication, non-repudiation, privacy, integrity.

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Analysis of the Fault Attack ECDLP over Prime Field

Journal of Applied Mathematics ◽

10.1155/2011/580749 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

Mingqiang Wang ◽

Tao Zhan

Keyword(s):

Elliptic Curve ◽

Elliptic Curve Cryptography ◽

Discrete Logarithm ◽

Base Point ◽

Prime Field ◽

Smooth Numbers ◽

Fault Attack ◽

Subexponential Time

In 2000, Biehl et al. proposed a fault-based attack on elliptic curve cryptography. In this paper, we refined the fault attack method. An elliptic curveEis defined over prime field𝔽pwith base pointP∈E(𝔽p). Applying the fault attack on these curves, the discrete logarithm on the curve can be computed in subexponential time ofLp(1/2,1+o(1)). The runtime bound relies on heuristics conjecture about smooth numbers similar to the ones used by Lenstra, 1987.

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HIGH PERFORMANCE MONTGOMERY MODULAR MULTIPLIER WITH A NEW RECODING METHOD

Journal of Circuits System and Computers ◽

10.1142/s0218126611007438 ◽

2011 ◽

Vol 20 (03) ◽

pp. 531-548 ◽

Cited By ~ 1

Author(s):

KOOROUSH MANOCHEHRI ◽

BABAK SADEGHIYAN ◽

SAADAT POURMOZAFARI

Keyword(s):

High Performance ◽

Hardware Implementation ◽

Computer Arithmetic ◽

Public Key Cryptography ◽

Modular Exponentiation ◽

Modular Multiplication ◽

Area Reduction ◽

Montgomery Modular Multiplication ◽

Modular Multiplier ◽

Reduction In Area

Modular calculations are widely used in many applications, especially in public key cryptography. Such operations are very time consuming, due to their long operands. To improve the performance of these calculations, many methods have been introduced. Montgomery modular multiplication is an example of such a solution to enhance the performance of modular multiplication and modular exponentiation. The radix-2 version of this method is simple and fast for hardware implementation, where multi-operand adders are required for its implementation. So far, Carry-Save-Adder (CSA) gives the best performance for multi-addition. In this paper, we propose a new recoding method for the Montgomery modular multiplier to enhance its performance. This is done through replacing CSA blocks with new blocks that have better performances than CSA in multi-addition calculations. With this replacement, we can theoretically have up to 40% reduction in area gates. In our experiments, we obtained 5.8% area reduction and 3% speed improvement in a hardware implementation. The idea behind our proposed method is the use of bitwise subtraction operator, where no carry propagation is needed. This recoding method of operands can also be used in many aspects of computer arithmetic, algorithms and computational hardware, such as multiplication, exponentiation and etc., in order to enhance their performances.

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Twisted Edwards Elliptic Curves for Zero-Knowledge Circuits

Mathematics ◽

10.3390/math9233022 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3022

Author(s):

Marta Bellés-Muñoz ◽

Barry Whitehat ◽

Jordi Baylina ◽

Vanesa Daza ◽

Jose Luis Muñoz-Tapia

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Elliptic Curve Cryptography ◽

Public Key Cryptography ◽

Deterministic Algorithm ◽

Public Key ◽

Security Attacks ◽

Zero Knowledge ◽

Prime Field ◽

Encryption Schemes

Circuit-based zero-knowledge proofs have arose as a solution to the implementation of privacy in blockchain applications, and to current scalability problems that blockchains suffer from. The most efficient circuit-based zero-knowledge proofs use a pairing-friendly elliptic curve to generate and validate proofs. In particular, the circuits are built connecting wires that carry elements from a large prime field, whose order is determined by the number of elements of the pairing-friendly elliptic curve. In this context, it is important to generate an inner curve using this field, because it allows to create circuits that can verify public-key cryptography primitives, such as digital signatures and encryption schemes. To this purpose, in this article, we present a deterministic algorithm for generating twisted Edwards elliptic curves defined over a given prime field. We also provide an algorithm for checking the resilience of this type of curve against most common security attacks. Additionally, we use our algorithms to generate Baby Jubjub, a curve that can be used to implement elliptic-curve cryptography in circuits that can be validated in the Ethereum blockchain.

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