Revisiting scalable modular multiplication over GF(2/sup m/) for elliptic curve cryptography

2006 ◽  
Author(s):  
Yier Jin ◽  
Haibin Shen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Modular Multiplication

scholarly journals Efficient implementation of modular multiplication over 192-bit NIST prime for 8-bit AVR-based sensor node

2020 ◽  
Author(s):  
Dong-won Park ◽  
Seokhie Hong ◽  
Nam Su Chang ◽  
Sung Min Cho
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Sensor Node ◽  
Efficient Implementation ◽  
Memory Access ◽  
Modular Multiplication ◽  
Reduction Operation ◽  
Computational Overhead ◽  
Integer Representation ◽  
Constrained Devices

Abstract Modular multiplication is one of the most time-consuming operations that account for almost 80% of computational overhead in a scalar multiplication in elliptic curve cryptography. In this paper, we present a new speed record for modular multiplication over 192-bit NIST prime P-192 on 8-bit AVR ATmega microcontrollers. We propose a new integer representation named Range Shifted Representation (RSR) which enables an efficient merging of the reduction operation into the subtractive Karatsuba multiplication. This merging results in a dramatic optimization in the intermediate accumulation of modular multiplication by reducing a significant amount of unnecessary memory access as well as the number of addition operations. Our merged modular multiplication on RSR is designed to have two duplicated groups of 96-bit intermediate values during accumulation. Hence, only one accumulation of the group is required and the result can be used twice. Consequently, we significantly reduce the number of load/store instructions which are known to be one of the most time-consuming operations for modular multiplication on constrained devices. Our implementation requires only 2888 cycles for the modular multiplication of 192-bit integers and outperforms the previous best result for modular multiplication over P-192 by a factor of 17%. In addition, our modular multiplication is even faster than the Karatsuba multiplication (without reduction) which achieved a speed record for multiplication on AVR processor.

scholarly journals Area-Time Efficient Hardware Implementation of Modular Multiplication for Elliptic Curve Cryptography

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 73898-73906 ◽  
Author(s):  
MD. Mainul Islam ◽  
MD. Selim Hossain ◽  
MD. Shahjalal ◽  
MOH. Khalid Hasan ◽  
Yeong Min Jang
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Hardware Implementation ◽  
Modular Multiplication

scholarly journals A Hardware-Efficient Elliptic Curve Cryptographic Architecture over GF (p)

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chao Cui ◽  
Yun Zhao ◽  
Yong Xiao ◽  
Weibin Lin ◽  
Di Xu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Scalar Multiplication ◽  
Time Range ◽  
Modular Multiplication ◽  
Hardware Resource ◽  
Data Register

This paper proposes a hardware-efficient elliptic curve cryptography (ECC) architecture over GF(p), which uses adders to achieve scalar multiplication (SM) through hardware-reuse method. In terms of algorithm, the improvement of the interleaved modular multiplication (IMM) algorithm and the binary modular inverse (BMI) algorithm needs two adders. In addition to the adder, the data register is another optimize target. The design compiler is synthesized on 0.13 µm CMOS ASIC platform. The time range of performing scalar multiplication over 160, 192, 224, and 256 field orders under 150 MHz frequency is 1.99–3.17 ms. Moreover, the gate area required for different field orders in this design is in the range of 35.65k–59.14k, with 50%–91% hardware resource less than other processors.

scholarly journals Fast Implementation of NIST P-256 Elliptic Curve Cryptography on 8-Bit AVR Processor

2020 ◽  
Vol 10 (24) ◽  
pp. 8816
Author(s):  
Dong-won Park ◽  
Nam Su Chang ◽  
Sangyub Lee ◽  
Seokhie Hong
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Execution Time ◽  
State Of The Art ◽  
Optimization Technique ◽  
Scalar Multiplication ◽  
Modular Multiplication ◽  
Avr Microcontroller ◽  
High Degree ◽  
The Impact

In this paper, we present a highly optimized implementation of elliptic curve cryptography (ECC) over NIST P-256 curve for an 8-bit AVR microcontroller. For improving the performance of ECC implementation, we focus on optimizing field arithmetics. In particular, we optimize the modular multiplication and squaring method exploiting the state-of-the-art optimization technique, namely range shifted representation (RSR). With optimized field arithmetics, we significantly improve the performance of scalar multiplication and set the speed record for execution time of variable base scalar multiplication over NIST P-256 curve. When compared with previous works, we achieve a performance gain of 17.3% over the best previous result on the same platform. Moreover, the execution time of our result is even faster than that over the NIST P-192 curve of the well-known TinyECC library. Our result shows that RSR can be applied to all field arithmetics and evaluate the impact of the adoption of RSR over the performance of scalar multiplication. Additionally, our implementation provides a high degree of regularity to withstand side-channel attacks.

scholarly journals A Hardware-Accelerated ECDLP with High-Performance Modular Multiplication

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Lyndon Judge ◽  
Suvarna Mane ◽  
Patrick Schaumont
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
High Performance ◽  
Design Space ◽  
Discrete Logarithm ◽  
Public Key Cryptography ◽  
Modular Multiplication ◽  
Polynomial Representation ◽  
Prime Field ◽  
Modular Multiplier

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.

scholarly journals High-Performance Elliptic Curve Cryptography by Using the CIOS Method for Modular Multiplication

2017 ◽  
pp. 185-198 ◽  
Author(s):  
Amine Mrabet ◽  
Nadia El-Mrabet ◽  
Ronan Lashermes ◽  
Jean-Baptiste Rigaud ◽  
Belgacem Bouallegue ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
High Performance ◽  
Modular Multiplication

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer
Sign in / Sign up

Export Citation Format

Share Document