A Survey Report On Elliptic Curve Cryptography

The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve  and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents  different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve  provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer  Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8

Cofactorization on Graphics Processing Units

We show how the cofactorization step, a compute-intensive part of the relation collection phase of the number field sieve (NFS), can be farmed out to a graphics processing unit. Our implementation on a GTX 580 GPU, which is integrated with a state-of-the-art NFS implementation, can serve as a cryptanalytic co-processor for several Intel i7-3770K quad-core CPUs simultaneously. This allows those processors to focus on the memory-intensive sieving and results in more useful NFS-relations found in less time

Virtualized Reconfigurable Resources and Their Secured Provision in an Untrusted Cloud Environment

The cloud computing business grows year after year. To keep up with increasing demand and to offer more services, data center providers are always searching for novel architectures. One of them are FPGAs, reconfigurable hardware with high compute power and energy efficiency. But some clients cannot make use of the remote processing capabilities. Not every involved party is trustworthy and the complex management software has potential security flaws. Hence, clients’ sensitive data or algorithms cannot be sufficiently protected. In this thesis state-of-the-art hardware, cloud and security concepts are analyzed and com- bined. On one side are reconfigurable virtual FPGAs. They are a flexible resource and fulfill the cloud characteristics at the price of security. But on the other side is a strong requirement for said security. To provide it, an immutable controller is embedded enabling a direct, confidential and secure transfer of clients’ configurations. This establishes a trustworthy compute space inside an untrusted cloud environment. Clients can securely transfer their sensitive data and algorithms without involving vulnerable software or a data center provider. This concept is implemented as a prototype. Based on it, necessary changes to current FPGAs are analyzed. To fully enable reconfigurable yet secure hardware in the cloud, a new hybrid architecture is required.Das Geschäft mit dem Cloud Computing wächst Jahr für Jahr. Um mit der steigenden Nachfrage mitzuhalten und neue Angebote zu bieten, sind Betreiber von Rechenzentren immer auf der Suche nach neuen Architekturen. Eine davon sind FPGAs, rekonfigurierbare Hardware mit hoher Rechenleistung und Energieeffizienz. Aber manche Kunden können die ausgelagerten Rechenkapazitäten nicht nutzen. Nicht alle Beteiligten sind vertrauenswürdig und die komplexe Verwaltungssoftware ist anfällig für Sicherheitslücken. Daher können die sensiblen Daten dieser Kunden nicht ausreichend geschützt werden. In dieser Arbeit werden modernste Hardware, Cloud und Sicherheitskonzept analysiert und kombiniert. Auf der einen Seite sind virtuelle FPGAs. Sie sind eine flexible Ressource und haben Cloud Charakteristiken zum Preis der Sicherheit. Aber auf der anderen Seite steht ein hohes Sicherheitsbedürfnis. Um dieses zu bieten ist ein unveränderlicher Controller eingebettet und ermöglicht eine direkte, vertrauliche und sichere Übertragung der Konfigurationen der Kunden. Das etabliert eine vertrauenswürdige Rechenumgebung in einer nicht vertrauenswürdigen Cloud Umgebung. Kunden können sicher ihre sensiblen Daten und Algorithmen übertragen ohne verwundbare Software zu nutzen oder den Betreiber des Rechenzentrums einzubeziehen. Dieses Konzept ist als Prototyp implementiert. Darauf basierend werden nötige Änderungen von modernen FPGAs analysiert. Um in vollem Umfang eine rekonfigurierbare aber dennoch sichere Hardware in der Cloud zu ermöglichen, wird eine neue hybride Architektur benötigt

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

ECM at Work

The performance of the elliptic curve method (ECM) for integer factorization plays an important role in the security assessment of RSA-based protocols as a cofactorization tool inside the number field sieve. The efficient arithmetic for Edwards curves found an application by speeding up ECM. We propose techniques based on generating and combining addition-subtracting chains to optimize Edwards ECM in terms of both performance and memory requirements. This makes our approach very suitable for memory-constrained devices such as graphics processing units (GPU). For commonly used ECM parameters we are able to lower the required memory up to a factor 55 compared to the state-of-the-art Edwards ECM approach. Our ECM implementation on a GTX 580 GPU sets a new throughput record, outperforming the best GPU, CPU and FPGA results reported in literature

Efficient SIMD arithmetic modulo a Mersenne number

This paper describes carry-less arithmetic operations modulo an integer 2^M − 1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization

A Horizontally Reconfigurable Architecture for Extended Precision Arithmetic (Parallel Computing, Condition Codes Factoring).

A special computer for high-precision arithmetic and parallel processing which features an ALU that is dynamically reconfigurable under program control has been designed and a prototype machine constructed. The 256-bit ALU consists of eight 32-bit slices each of which has its own ALU operation code in each microinstruction. The slices can remain logically separated from each other, or can be dynamically connected to either or both of their neighbors under control of a segment control code that is part of each microinstruction. The result is a unique parallel architecture which provides real parallelism to user programs at the instruction level while globally retaining a sequential control structure. Management of parallelism is achieved through a two level hierarchy of condition codes and extended instruction sets to support conditional instruction execution. New types of parallel micro-programming tools introduce a system for reconfiguration management and parallel programming. An assembler, debug simulator, and interactive operating environment have been implemented. An analysis of the instruction times to execute arithmetic operations on the machine show that it will be exceptionally fast for problems in computational number theory and factoring of integers

Optimization of Supersingular Isogeny Cryptography for Deeply Embedded Systems

Public-key cryptography in use today can be broken by a quantum computer with sufficient resources. Microsoft Research has published an open-source library of quantum-secure supersingular isogeny (SI) algorithms including Diffie-Hellman key agreement and key encapsulation in portable C and optimized x86 and x64 implementations. For our research, we modified this library to target a deeply-embedded processor with instruction set extensions and a finite-field coprocessor originally designed to accelerate traditional elliptic curve cryptography (ECC). We observed a 6.3-7.5x improvement over a portable C implementation using instruction set extensions and a further 6.0-6.1x improvement with the addition of the coprocessor. Modification of the coprocessor to a wider datapath further increased performance 2.6-2.9x. Our results show that current traditional ECC implementations can be easily refactored to use supersingular elliptic curve arithmetic and achieve post-quantum security

On the Cryptanalysis of Public-Key Cryptography

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient "sloppy reduction." The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units