Towards Efficient Hardware Implementation of Elliptic and Hyperelliptic Curve Cryptography

Implementation of elliptic and hyperelliptic curve cryptographic algorithms has been the focus of a great deal of recent research directed at increasing efficiency. Elliptic curve cryptography (ECC) was introduced independently by Koblitz and Miller in the 1980s. Hyperelliptic curve cryptography (HECC), a generalization of the elliptic curve case, allows a decreasing field size as the genus increases. The work presented in this thesis examines the problems created by limited area, power, and computation time when elliptic and hyperelliptic curves are integrated into constrained devices such as wireless sensor network (WSN) and smart cards. The lack of a battery in wireless sensor network limits the processing power of these devices, but they still require security. It was widely believed that devices with such constrained resources cannot incorporate a strong HECC processor for performing cryptographic operations such as elliptic curve scalar multiplication (ECSM) or hyperelliptic curve divisor multiplication (HCDM). However, the work presented in this thesis has demonstrated the feasibility of integrating an HECC processor into such devices through the use of the proposed architecture synthesis and optimization techniques for several inversion-free algorithms. The goal of this work is to develop a hardware implementation of binary elliptic and hyperelliptic curves. The focus is on the modeling of three factors: register allocation, operation scheduling, and storage binding. These factors were then integrated into architecture synthesis and optimization techniques in order to determine the best overall implementation suitable for constrained devices. The main purpose of the optimization is to reduce the area and power. Through analysis of the architecture optimization techniques for both datapath and control unit synthesis, the number of registers was reduced by an average of 30%. The use of the proposed efficient explicit formula for the different algorithms also enabled a reduction in the number of read/write operations from/to the register file, which reduces the processing power consumption. As a result, an overall HECC processor requires from 1843 to 3595 slices for a Xilinix XC4VLX200 and the total computation time is limited to between 10.08 ms to 15.82 ms at a maximum frequency of 50 MHz for a varity of inversion-free coordinate systems in hyperelliptic curves. The value of the new model has been demonstrated with respect to its implementation in elliptic and hyperelliptic curve crypogrpahic algorithms, through both synthesis and simulations. In summary, a framework has been provided for consideration of interactions with synthesis and optimization through architecture modeling for constrained enviroments. Insights have also been presented with respect to improving the design process for cryptogrpahic algorithms through datapath and control unit analysis

Recent Application in Biometrics

In the recent years, a number of recognition and authentication systems based on biometric measurements have been proposed. Algorithms and sensors have been developed to acquire and process many different biometric traits. Moreover, the biometric technology is being used in novel ways, with potential commercial and practical implications to our daily activities. The key objective of the book is to provide a collection of comprehensive references on some recent theoretical development as well as novel applications in biometrics. The topics covered in this book reflect well both aspects of development. They include biometric sample quality, privacy preserving and cancellable biometrics, contactless biometrics, novel and unconventional biometrics, and the technical challenges in implementing the technology in portable devices. The book consists of 15 chapters. It is divided into four sections, namely, biometric applications on mobile platforms, cancelable biometrics, biometric encryption, and other applications. The book was reviewed by editors Dr. Jucheng Yang and Dr. Norman Poh. We deeply appreciate the efforts of our guest editors: Dr. Girija Chetty, Dr. Loris Nanni, Dr. Jianjiang Feng, Dr. Dongsun Park and Dr. Sook Yoon, as well as a number of anonymous reviewers

On the Analysis of Public-Key Cryptologic Algorithms

The RSA cryptosystem introduced in 1977 by Ron Rivest, Adi Shamir and Len Adleman is the most commonly deployed public-key cryptosystem. Elliptic curve cryptography (ECC) introduced in the mid 80's by Neal Koblitz and Victor Miller is becoming an increasingly popular alternative to RSA offering competitive performance due the use of smaller key sizes. Most recently hyperelliptic curve cryptography (HECC) has been demonstrated to have comparable and in some cases better performance than ECC. The security of RSA relies on the integer factorization problem whereas the security of (H)ECC is based on the (hyper)elliptic curve discrete logarithm problem ((H)ECDLP). In this thesis the practical performance of the best methods to solve these problems is analyzed and a method to generate secure ephemeral ECC parameters is presented. The best publicly known algorithm to solve the integer factorization problem is the number field sieve (NFS). Its most time consuming step is the relation collection step. We investigate the use of graphics processing units (GPUs) as accelerators for this step. In this context, methods to efficiently implement modular arithmetic and several factoring algorithms on GPUs are presented and their performance is analyzed in practice. In conclusion, it is shown that integrating state-of-the-art NFS software packages with our GPU software can lead to a speed-up of 50%. In the case of elliptic and hyperelliptic curves for cryptographic use, the best published method to solve the (H)ECDLP is the Pollard rho algorithm. This method can be made faster using classes of equivalence induced by curve automorphisms like the negation map. We present a practical analysis of their use to speed up Pollard rho for elliptic curves and genus 2 hyperelliptic curves defined over prime fields. As a case study, 4 curves at the 128-bit theoretical security level are analyzed in our software framework for Pollard rho to estimate their practical security level. In addition, we present a novel many-core architecture to solve the ECDLP using the Pollard rho algorithm with the negation map on FPGAs. This architecture is used to estimate the cost of solving the Certicom ECCp-131 challenge with a cluster of FPGAs. Our design achieves a speed-up factor of about 4 compared to the state-of-the-art. Finally, we present an efficient method to generate unique, secure and unpredictable ephemeral ECC parameters to be shared by a pair of authenticated users for a single communication. It provides an alternative to the customary use of fixed ECC parameters obtained from publicly available standards designed by untrusted third parties. The effectiveness of our method is demonstrated with a portable implementation for regular PCs and Android smartphones. On a Samsung Galaxy S4 smartphone our implementation generates unique 128-bit secure ECC parameters in 50 milliseconds on average

On Error Detection and Recovery in Elliptic Curve Cryptosystems

Fault analysis attacks represent a serious threat to a wide range of cryptosystems including those based on elliptic curves. With the variety and demonstrated practicality of these attacks, it is essential for cryptographic implementations to handle different types of errors properly and securely. In this work, we address some aspects of error detection and recovery in elliptic curve cryptosystems. In particular, we discuss the problem of wasteful computations performed between the occurrence of an error and its detection and propose solutions based on frequent validation to reduce that waste. We begin by presenting ways to select the validation frequency in order to minimize various performance criteria including the average and worst-case costs and the reliability threshold. We also provide solutions to reduce the sensitivity of the validation frequency to variations in the statistical error model and its parameters. Then, we present and discuss adaptive error recovery and illustrate its advantages in terms of low sensitivity to the error model and reduced variance of the resulting overhead especially in the presence of burst errors. Moreover, we use statistical inference to evaluate and fine-tune the selection of the adaptive policy. We also address the issue of validation testing cost and present a collection of coherency-based, cost-effective tests. We evaluate variations of these tests in terms of cost and error detection effectiveness and provide infective and reduced-cost, repeated-validation variants. Moreover, we use coherency-based tests to construct a combined-curve countermeasure that avoids the weaknesses of earlier related proposals and provides a flexible trade-off between cost and effectiveness

An FPGA-based programmable processor for bilinear pairings

Bilinear pairings on elliptic curves are an active research field in cryptography. First cryptographic protocols based on bilinear pairings were proposed by the year 2000 and they are promising solutions to security concerns in different domains, as in Pervasive Computing and Cloud Computing. The computation of bilinear pairings that relies on arithmetic over finite fields is the most time-consuming in Pairing-based cryptosystems. That has motivated the research on efficient hardware architectures that improve the performance of security protocols. In the literature, several works have focused in the design of custom hardware architectures for pairings, however, flexible designs provide advantages due to the fact that there are several types of pairings and algorithms to compute them. This work presents the design and implementation of a novel programmable cryptoprocessor for computing bilinear pairings over binary fields in FPGAs, which is able to support different pairing algorithms and parameters as the elliptic curve, the tower field and the distortion map. The results show that high flexibility is achieved by the proposed cryptoprocessor at a competitive timing and area usage when it is compared to custom designs for pairings defined over singular/supersingular elliptic curves at a 128-bit security level

Curves, codes, and cryptography

This thesis deals with two topics: elliptic-curve cryptography and code-based cryptography. In 2007 elliptic-curve cryptography received a boost from the introduction of a new way of representing elliptic curves. Edwards, generalizing an example from Euler and Gauss, presented an addition law for the curves x2 + y2 = c2(1 + x2y2) over non-binary fields. Edwards showed that every elliptic curve can be expressed in this form as long as the underlying field is algebraically closed. Bernstein and Lange found fast explicit formulas for addition and doubling in coordinates (X : Y : Z) representing (x, y) = (X/Z, Y/Z) on these curves, and showed that these explicit formulas save time in elliptic-curve cryptography. It is easy to see that all of these curves are isomorphic to curves x2 + y2 = 1 + dx2y2 which now are called "Edwards curves" and whose shape covers considerably more elliptic curves over a finite field than x2 + y2 = c2(1 + x2y2). In this thesis the Edwards addition law is generalized to cover all curves ax2 +y2 = 1+dx2y2 which now are called "twisted Edwards curves." The fast explicit formulas for addition and doubling presented here are almost as fast in the general case as they are for the special case a = 1. This generalization brings the speed of the Edwards addition law to every Montgomery curve. Tripling formulas for Edwards curves can be used for double-base scalar multiplication where a multiple of a point is computed using a series of additions, doublings, and triplings. The use of double-base chains for elliptic-curve scalar multiplication for elliptic curves in various shapes is investigated in this thesis. It turns out that not only are Edwards curves among the fastest curve shapes, but also that the speed of doublings on Edwards curves renders double bases obsolete for this curve shape. Elliptic curves in Edwards form and twisted Edwards form can be used to speed up the Elliptic-Curve Method for integer factorization (ECM). We show how to construct elliptic curves in Edwards form and twisted Edwards form with large torsion groups which are used by the EECM-MPFQ implementation of ECM. Code-based cryptography was invented by McEliece in 1978. The McEliece public-key cryptosystem uses as public key a hidden Goppa code over a finite field. Encryption in McEliece’s system is remarkably fast (a matrix-vector multiplication). This system is rarely used in implementations. The main complaint is that the public key is too large. The McEliece cryptosystem recently regained attention with the advent of post-quantum cryptography, a new field in cryptography which deals with public-key systems without (known) vulnerabilities to attacks by quantum computers. The McEliece cryptosystem is one of them. In this thesis we underline the strength of the McEliece cryptosystem by improving attacks against it and by coming up with smaller-key variants. McEliece proposed to use binary Goppa codes. For these codes the most effective attacks rely on information-set decoding. In this thesis we present an attack developed together with Daniel J. Bernstein and Tanja Lange which uses and improves Stern’s idea of collision decoding. This attack is faster by a factor of more than 150 than previous attacks, bringing it within reach of a moderate computer cluster. We were able to extract a plaintext from a ciphertext by decoding 50 errors in a [1024, 524] binary code. The attack should not be interpreted as destroying the McEliece cryptosystem. However, the attack demonstrates that the original parameters were chosen too small. Building on this work the collision-decoding algorithm is generalized in two directions. First, we generalize the improved collision-decoding algorithm for codes over arbitrary fields and give a precise analysis of the running time. We use the analysis to propose parameters for the McEliece cryptosystem with Goppa codes over fields such as F31. Second, collision decoding is generalized to ball-collision decoding in the case of binary linear codes. Ball-collision decoding is asymptotically faster than any previous attack against the McEliece cryptosystem. Another way to strengthen the system is to use codes with a larger error-correction capability. This thesis presents "wild Goppa codes" which contain the classical binary Goppa codes as a special case. We explain how to encrypt and decrypt messages in the McEliece cryptosystem when using wild Goppa codes. The size of the public key can be reduced by using wild Goppa codes over moderate fields which is explained by evaluating the security of the "Wild McEliece" cryptosystem against our generalized collision attack for codes over finite fields. Code-based cryptography not only deals with public-key cryptography: a code-based hash function "FSB"was submitted to NIST’s SHA-3 competition, a competition to establish a new standard for cryptographic hashing. Wagner’s generalized birthday attack is a generic attack which can be used to find collisions in the compression function of FSB. However, applying Wagner’s algorithm is a challenge in storage-restricted environments. The FSBday project showed how to successfully mount the generalized birthday attack on 8 nodes of the Coding and Cryptography Computer Cluster (CCCC) at Technische Universiteit Eindhoven to find collisions in the toy version FSB48 which is contained in the submission to NIST

Proceedings of the 5th International Workshop on Reconfigurable Communication-centric Systems on Chip 2010 - ReCoSoC\u2710 - May 17-19, 2010 Karlsruhe, Germany. (KIT Scientific Reports ; 7551)

ReCoSoC is intended to be a periodic annual meeting to expose and discuss gathered expertise as well as state of the art research around SoC related topics through plenary invited papers and posters. The workshop aims to provide a prospective view of tomorrow\u27s challenges in the multibillion transistor era, taking into account the emerging techniques and architectures exploring the synergy between flexible on-chip communication and system reconfigurability