一种Montgomery型椭圆曲线的高效标量乘算法

椭圆曲线标量乘法是椭圆曲线密码系统的基本运算,安全高效的标量乘法将直接提高椭圆曲线密码系统的效率和安全性.本文将Fibonacei数列的概念进行了扩展,提出了Fibonacci型数列的概念,并用Fibonaeei型数列将Montgomery型曲线上点的加法运算公式进行了简化,得到了新的点加公式fibAdd.利用黄金比率...     

抗SPA攻击的椭圆曲线NAF标量乘实现算法

针对椭圆曲线非相邻形式（NAF）标量乘法不能很好地抵抗简单功耗分析攻击（SPA）的问题,对NAF标量乘的实现算法以及对NAF标量乘的SPA攻击原理进行了分析,提出一种新的标量乘实现算法——平衡能量NAF标量乘法。通过对智能卡功耗分析平台的实测波形进行分析验证,平衡能量NAF标量乘法不仅继承了NAF标量乘法运算效率高的优点,而且能够很好地抵抗SPA攻击,提高密码芯片的安全性。     

基于Markov链的椭圆曲线标量乘法算法性能分析

在椭圆曲线密码系统中,采用规范重编码、滑动窗口等优化技术可以有效提高椭圆曲线上点的标量乘法k·P的运算性能,但在实现中,需要对不同优化技术的算法性能进行定量分析,才能确定标量乘法的最优实现.本文运用Markov链对标量k规范重编码表示的滑动窗口划分过程进行了建模,提出了一种对椭圆曲线标量乘法的平均算法性能进行定量分析的方法,并运用该方法分析了不同参数下标量乘法运算的平均性能,计算了滑动窗口的最优窗口大小.最后,通过比较说明,采用规范重编码和滑动窗口技术的椭圆曲线标量乘法的运算开销比用m-ary法少10.32~17.32％,比单纯采用滑动窗口法也要少4.53~8.40％.     

椭圆曲线密码体制中快速标量乘算法实现

椭圆曲线密码(ECC),是一种以椭圆曲线离散对数问题为出发点而制定出的各种公钥密码体制,在1985年由学者Koblitz和Miller两人分别独立提出。ECC的主要特征是采用有限域上的椭圆曲线有限点群而非是传统的基于离散对数问题密码体制中所采用的有限循环群。因为标量乘算法是ECC中最耗时同时也是最为重要的算法,因为其运算效率的高低将直接影响到ECC实现的效率。本篇论文即是研究椭圆曲线密码中的标量乘法,以期能够探寻出一种快速安全的标量乘算法。     

一类安全椭圆曲线的选取及其标量乘法的快速计算

安全椭圆曲线的选取和标量乘法的快速计算是有效实现椭圆曲线密码体制的两个主要问题.本文将二者结合起来考虑给出了一类适合普通PC机实现的安全椭圆曲线,并详细给出了选取这类曲线的具体步骤和基于"大步-小步法"思想构造了一种新的计算这类曲线上标量乘法的快速算法.这类曲线不仅选取容易而且利用本文所提出方法计算其标量乘法时能使所需椭圆曲线运算次数大大减少.此外,选用这类曲线后基域中元素不再需要专门的表示方法,各种运算能非常快地得到实现,从而能极大地提高体制的整体实现速度.     

针对密码芯片的电磁模板分析攻击

给出一种简单的电磁信号的获取办法,说明密码芯片的电磁信号能够用一个手工绕制的金属线圈获取,并且其信号幅度和操作数的汉明重量相关.在描述模板攻击原理和步骤的基础上,介绍了针对密码芯片的电磁模板分析攻击,并且针对一个单片机(AT89C52)上实现的DES密码系统进行了电磁模板分析攻击实验,实验成功恢复了DES第16轮使用的48位子密钥.     

Koblitz曲线上改进的ECDSA算法

标量乘法的效率决定着椭圆曲线密码体制的性能,而Koblitz曲线上的快速标量乘算法,是标量乘法研究的重要课题.Lee et al算法采用Frobenius映射扩展正整数k,并将其扩展后的系数改写成二进制形式,有效地提高标量乘算法效率.文中将JSF应用到扩展后的系数中,以较小存储空间为代价来提高算法效率k并将算法用到改进...     

抗SPA攻击的Twisted Edwards曲线标量乘法实现

椭圆曲线密码（Elliptic Curve Cryptography,ECC）是一种非对称密码，在信息安全领域中扮演着越来越重要的角色。目前对椭圆曲线密码的研究大多针对Weierstrass曲线，对于Twisted Edwards曲线的研究较少。针对Twisted Edwards曲线上标量乘法的效率及安全性，将Twisted Edwards曲线转换为Montgomery曲线，并采用Montgomery标量乘法在每次循环中都固定进行点加和倍点计算，从而能够抵抗简单能量攻击（Simple Power Analysis,SPA）。最后在复旦微电子公司型号为JFM7K325T的现场可编程门阵列（Field Programmable Gate Array,FPGA）中进行了实现和测试。结果表明，该方法能达到较理想的效果。     

二进制域上椭圆曲线加密核的设计

为了提高椭圆曲线加密速度,介绍椭圆曲线密码体制发展优势,确定了椭圆曲线加密系统的体系结构,对二进制域上的加法、平方、乘法和逆运算用最新研究成果做了硬件设计并用Verilog实现,在这些底层运算基础上完成标量乘控制,实现了标量乘运算,最终实现椭圆曲线加密和解密的功能.在验证中编写验证模型和验证平台,对设计进行了功能验证并做了覆盖率统计,功能验证结果正确,覆盖率达到100%.对椭圆曲线加密核进行了逻辑综合和门级仿真,综合结果表明该核的运算频率可达125 MHz,门级仿真结果与功能仿真结果相同.     

基于二进制Edwards曲线的椭圆曲线加密多标量乘结构设计与实现

标量乘及多标量乘算法是影响椭圆曲线密码系统性能的关键.基于二进制Edwards曲线提出并实现了一种新型的椭圆曲线标量乘法器.由于Edwards曲线的完备性,这种乘法器可对曲线上任意一点进行计算,而不用区分倍乘或者负元,实现较简单,有很高的运算速度和很强的抗侧信道攻击的能力.     

基于快速标量乘算法的椭圆曲线数字签名方案

计算标量乘kP是ECC快速实现的关键,也是ECC研究的热点问题。文中介绍了基于Montgomery思想的快速标量乘算法,重点介绍了白国强等人的运算多标量乘kP+lQ的算法,并分析了其局限性,同时对其进行了改进。在此基础上,设计了一种分段快速标量乘算法,将改进的算法与分段标量乘算法运用到ECDSA中。经分析验证,分段快速标量乘算法,提高了效率,对ECDSA的快速实现具有一定意义。     

Accelerated precomputation points–based scalar reduction on elliptic curve cryptography for wireless sensor networks

Sensor devices are limited resource power and energy, thus providing security services for sensor networks is very difficult. Elliptic curve cryptography (ECC) is one of the most famous asymmetric cryptographic schemes, which offers the same level of security with much shorter keys compared to the other widely used asymmetric cryptographic algorithm, RSA (Rivest, Shamir, and Adleman). In ECC, the main and most‐heavily used operation is the scalar multiplication kP , where the scalar value k is a private integer and must be secured. In this work, we present a new approach to accelerate the main scalar multiplication on ECC over prime fields for sensor networks. This approach uses an equivalent representation of points and can act as a support for existing schemes in a selected interval. The simulation results showed that the proposed technique increases the efficiency of the computation time. For example, on this scalar multiplication, we obtain a gain of 4 bits in 161 bits for 6.25% of the scalars. This gain can sometimes reach 100% in some cases. After this significant reduction of the scalar k , we present a fast precomputation algorithm in a distributed scalar multiplication on kP to avoid storage of precomputation points, which requires extra memory.     

高性能可扩展公钥密码协处理器研究与设计

 本文提出了一种高效的点乘调度策略和改进的双域高基Montgomery模乘算法,在此基础上设计了一种新型高性能可扩展公钥密码协处理器体系结构,并采用0.18μm 1P6M标准CMOS工艺实现了该协处理器,以支持RSA和ECC等公钥密码算法的计算加速.该协处理器通过扩展片上高速存储器和使用以基数为处理字长的方法,具有良好的可扩展性和较强的灵活性,支持2048位以内任意大数模幂运算以及576位以内双域任意椭圆曲线标量乘法运算.芯片测试结果表明其具有很好的加速性能,完成一次1024位模幂运算仅需197μs、GF(p)域192位标量乘法运算仅需225μs、GF(2^m)域163位标量乘法运算仅需200.7μs.     

Pseudo 4D projective coordinate-based multi-base scalar multiplication

In order to address the problem of elliptic curve cryptosystem (ECC) for the expensive cost in scalar multiplication and the vulnerability to the power analysis attacks,a pseudo 4D projective coordinate-based multi-base scalar multiplication was proposed to optimize group operation layer and scalar multiplication operation layer,which aimed at increasing the performance of ECC and resisting common power analysis attacks.Experimental results show that compared with the state-of-the-art algorithms,the proposed algorithm decreases 5.71% of point doubling cost,3.17% of point tripling cost,and 8.74% of point quintupling cost under discrete group operations.When the key length is 160 bit,the proposed algorithm decreases 36.32% of point tripling cost,17.42% of point quintupling cost,and 8.70% of the system cost under continuous group operations.The analyzing of power consumption wave shows that the proposed algorithm can resist SPA and DPA attack.     

一种针对特征2域椭圆曲线密码芯片的差分功耗分析

文章对一款基于特征2域实现的椭圆曲线密码ASIC芯片进行了差分功耗分析。其中分析的目标为实现椭圆曲线层多倍点运算的Montgomery Ladder算法。通过详细的差分功耗分析发现，Montgomery Ladder算法并不能抗MESD差分功耗分析，从而从实践的角度证明Montgomery Ladder算法并不安全，椭圆曲线密码芯片的实际应用还需要其它抗功耗分析手段来保证其安全。     

Hardware Elliptic Curve Cryptographic Processor Over$rm GF(p)$

A novel hardware architecture for elliptic curve cryptography (ECC) over$ GF(p)$is introduced. This can perform the main prime field arithmetic functions needed in these cryptosystems including modular inversion and multiplication. This is based on a new unified modular inversion algorithm that offers considerable improvement over previous ECC techniques that use Fermat's Little Theorem for this operation. The processor described uses a full-word multiplier which requires much fewer clock cycles than previous methods, while still maintaining a competitive critical path delay. The benefits of the approach have been demonstrated by utilizing these techniques to create a field-programmable gate array (FPGA) design. This can perform a 256-bit prime field scalar point multiplication in 3.86 ms, the fastest FPGA time reported to date. The ECC architecture described can also perform four different types of modular inversion, making it suitable for use in many different ECC applications.     

Flexible Hardware Processor for Elliptic Curve Cryptography Over NIST Prime Fields

Exchange of private information over a public medium must incorporate a method for data protection against unauthorized access. Elliptic curve cryptography (ECC) has become widely accepted as an efficient mechanism to secure sensitive data. The main ECC computation is a scalar multiplication, translating into an appropriate sequence of point operations, each involving several modular arithmetic operations. We describe a flexible hardware processor for performing computationally expensive modular addition, subtraction, multiplication, and inversion over prime finite fields $GF(p)$ . The proposed processor supports all five primes $p$ recommended by NIST, whose sizes are 192, 224, 256, 384, and 521 bits. It can also be programmed to automatically execute sequences of modular arithmetic operations. Our field-programmable gate-array implementation runs at 60 MHz and takes between 4 and 40 ms (depending on the used prime) to perform a typical scalar multiplication.      

基于Montgomery曲线改进ECDSA算法的研究

提出了一种基于Montgomery曲线改进ECDSA算法,并重点改进异步点乘问题.改进的ECDSA具有更快的计算速度并能有效地抵御时间攻击和能量攻击,将验证签名与产生签名时间之比从2倍降低到约1.2倍,减少约40%,算法对提高椭圆曲线密码的实现效率有一定意义.