Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

A microcoded elliptic curve cryptographic processor.

Leung Ka Ho.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.Includes bibliographical references (leaves [85]-90).Abstracts in English and Chinese.Abstract --- p.iAcknowledgments --- p.iiiList of Figures --- p.ixList of Tables --- p.xiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Motivation --- p.1Chapter 1.2 --- Aims --- p.3Chapter 1.3 --- Contributions --- p.3Chapter 1.4 --- Thesis Outline --- p.4Chapter 2 --- Cryptography --- p.6Chapter 2.1 --- Introduction --- p.6Chapter 2.2 --- Foundations --- p.6Chapter 2.3 --- Secret Key Cryptosystems --- p.8Chapter 2.4 --- Public Key Cryptosystems --- p.9Chapter 2.4.1 --- One-way Function --- p.10Chapter 2.4.2 --- Certification Authority --- p.10Chapter 2.4.3 --- Discrete Logarithm Problem --- p.11Chapter 2.4.4 --- RSA vs. ECC --- p.12Chapter 2.4.5 --- Key Exchange Protocol --- p.13Chapter 2.4.6 --- Digital Signature --- p.14Chapter 2.5 --- Secret Key vs. Public Key Cryptography --- p.16Chapter 2.6 --- Summary --- p.18Chapter 3 --- Mathematical Background --- p.19Chapter 3.1 --- Introduction --- p.19Chapter 3.2 --- Groups and Fields --- p.19Chapter 3.3 --- Finite Fields --- p.21Chapter 3.4 --- Modular Arithmetic --- p.21Chapter 3.5 --- Polynomial Basis --- p.21Chapter 3.6 --- Optimal Normal Basis --- p.22Chapter 3.6.1 --- Addition --- p.23Chapter 3.6.2 --- Squaring --- p.24Chapter 3.6.3 --- Multiplication --- p.24Chapter 3.6.4 --- Inversion --- p.30Chapter 3.7 --- Summary --- p.33Chapter 4 --- Literature Review --- p.34Chapter 4.1 --- Introduction --- p.34Chapter 4.2 --- Hardware Elliptic Curve Implementation --- p.34Chapter 4.2.1 --- Field Processors --- p.34Chapter 4.2.2 --- Curve Processors --- p.36Chapter 4.3 --- Software Elliptic Curve Implementation --- p.36Chapter 4.4 --- Summary --- p.38Chapter 5 --- Introduction to Elliptic Curves --- p.39Chapter 5.1 --- Introduction --- p.39Chapter 5.2 --- Historical Background --- p.39Chapter 5.3 --- Elliptic Curves over R2 --- p.40Chapter 5.3.1 --- Curve Addition and Doubling --- p.41Chapter 5.4 --- Elliptic Curves over Finite Fields --- p.44Chapter 5.4.1 --- Elliptic Curves over Fp with p>〉3 --- p.44Chapter 5.4.2 --- Elliptic Curves over F2n --- p.45Chapter 5.4.3 --- Operations of Elliptic Curves over F2n --- p.46Chapter 5.4.4 --- Curve Multiplication --- p.49Chapter 5.5 --- Elliptic Curve Discrete Logarithm Problem --- p.51Chapter 5.6 --- Public Key Cryptography --- p.52Chapter 5.7 --- Elliptic Curve Diffie-Hellman Key Exchange --- p.54Chapter 5.8 --- Summary --- p.55Chapter 6 --- Design Methodology --- p.56Chapter 6.1 --- Introduction --- p.56Chapter 6.2 --- CAD Tools --- p.56Chapter 6.3 --- Hardware Platform --- p.59Chapter 6.3.1 --- FPGA --- p.59Chapter 6.3.2 --- Reconfigurable Hardware Computing --- p.62Chapter 6.4 --- Elliptic Curve Processor Architecture --- p.63Chapter 6.4.1 --- Arithmetic Logic Unit (ALU) --- p.64Chapter 6.4.2 --- Register File --- p.68Chapter 6.4.3 --- Microcode --- p.69Chapter 6.5 --- Parameterized Module Generator --- p.72Chapter 6.6 --- Microcode Toolkit --- p.73Chapter 6.7 --- Initialization by Bitstream Reconfiguration --- p.74Chapter 6.8 --- Summary --- p.75Chapter 7 --- Results --- p.76Chapter 7.1 --- Introduction --- p.76Chapter 7.2 --- Elliptic Curve Processor with Serial Multiplier (p = 1) --- p.76Chapter 7.3 --- Projective verses Affine Coordinates --- p.78Chapter 7.4 --- Elliptic Curve Processor with Parallel Multiplier (p > 1) --- p.79Chapter 7.5 --- Summary --- p.80Chapter 8 --- Conclusion --- p.82Chapter 8.1 --- Recommendations for Future Research --- p.83Bibliography --- p.85Chapter A --- Elliptic Curves in Characteristics 2 and3 --- p.91Chapter A.1 --- Introduction --- p.91Chapter A.2 --- Derivations --- p.91Chapter A.3 --- "Elliptic Curves over Finite Fields of Characteristic ≠ 2,3" --- p.92Chapter A.4 --- Elliptic Curves over Finite Fields of Characteristic = 2 --- p.94Chapter B --- Examples of Curve Multiplication --- p.95Chapter B.1 --- Introduction --- p.95Chapter B.2 --- Numerical Results --- p.9

ElGamal-type signature schemes in modular arithmetic and Galois fields

A digital signature is like a handwritten signature for a file, such that it ensures the identity of the person responsible for the file and prevents any unauthorized changes to the original file. Digital signatures use the same technology as most public key cryptosystems in which there is a public and private key. Most mathematical operations are done over a field Zp where p is some large prime. It is possible to do the same operations over other finite fields. My project explains and studies the different finite fields that can be used as well as ways to implement and experiment with them. It turns out that operations over Zp run the fastest, but with polynomial basis in a close second. Normal basis did not prove to be efficient at all. These results turned out to be against most claims of others, especially in hardware implementations. Large integer libraries are so efficient and fast that is was hard to beat the times with custom bit manipulation structures. Various secure signature schemes have proven to be practical and it is likely that they will be used much more in the near future in many applications

Identity based cryptography from bilinear pairings

This report contains an overview of two related areas of research in cryptography which have been prolific in significant advances in recent years. The first of these areas is pairing based cryptography. Bilinear pairings over elliptic curves were initially used as formal mathematical tools and later as cryptanalysis tools that rendered supersingular curves insecure. In recent years, bilinear pairings have been used to construct many cryptographic schemes. The second area covered by this report is identity based cryptography. Digital certificates are a fundamental part of public key cryptography, as one needs a secure way of associating an agent’s identity with a random (meaningless) public key. In identity based cryptography, public keys can be arbitrary bit strings, including readable representations of one’s identity.Fundação para a Ci~Encia e Tecnologia - SFRH/BPD/20528/2004

Computing cardinalities of Q-curve reductions over finite fields

We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of Drew Sutherlan

Efficient Implementation of Elliptic Curve Cryptography on FPGAs

This work presents the design strategies of an FPGA-based elliptic curve co-processor. Elliptic curve cryptography is an important topic in cryptography due to its relatively short key length and higher efficiency as compared to other well-known public key crypto-systems like RSA. The most important contributions of this work are: - Analyzing how different representations of finite fields and points on elliptic curves effect the performance of an elliptic curve co-processor and implementing a high performance co-processor. - Proposing a novel dynamic programming approach to find the optimum combination of different recursive polynomial multiplication methods. Here optimum means the method which has the smallest number of bit operations. - Designing a new normal-basis multiplier which is based on polynomial multipliers. The most important part of this multiplier is a circuit of size $O(n \log n)$ for changing the representation between polynomial and normal basis

A usability study of elliptic curves

In the recent years, the need of information security has rapidly increased due to an enormous growth of data transmission. In this thesis, we study the uses of elliptic curves in the cryptography. We discuss the elliptic curves over finite fields, attempts to attack; discrete logarithm, Pollard’s rho algorithm, baby-step giant-step algorithm, Pohlig-Hellman algorithm, function field sieve, and number field sieve. The main cryptographic reason to use elliptic curves over finite fields is to provide arbitrarily large finite cyclic groups having a computationally difficult discrete logarithm problem

Efficient Indifferentiable Hashing into Ordinary Elliptic Curves

We provide the first construction of a hash function into ordinary elliptic curves that is indifferentiable from a random oracle, based on Icart\u27s deterministic encoding from Crypto 2009. While almost as efficient as Icart\u27s encoding, this hash function can be plugged into any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle model. We also describe a more general (but less efficient) construction that works for a large class of encodings into elliptic curves, for example the Shallue-Woestijne-Ulas (SWU) algorithm. Finally we describe the first deterministic encoding algorithm into elliptic curves in characteristic 3

BINARY EDWARDS CURVES IN ELLIPTIC CURVE CRYPTOGRAPHY

Edwards curves are a new normal form for elliptic curves that exhibit some cryp- tographically desirable properties and advantages over the typical Weierstrass form. Because the group law on an Edwards curve (normal, twisted, or binary) is complete and unified, implementations can be safer from side channel or exceptional procedure attacks. The different types of Edwards provide a better platform for cryptographic primitives, since they have more security built into them from the mathematic foun- dation up. Of the three types of Edwards curves—original, twisted, and binary—there hasn’t been as much work done on binary curves. We provide the necessary motivation and background, and then delve into the theory of binary Edwards curves. Next, we examine practical considerations that separate binary Edwards curves from other recently proposed normal forms. After that, we provide some of the theory for bi- nary curves that has been worked on for other types already: pairing computations. We next explore some applications of elliptic curve and pairing-based cryptography wherein the added security of binary Edwards curves may come in handy. Finally, we finish with a discussion of e2c2, a modern C++11 library we’ve developed for Edwards Elliptic Curve Cryptography

Compiler assisted elliptic curve cryptography

Although cryptographic software implementation is often performed by expert programmers, the range of performance and secu- rity driven options, as well as more mundane software engineering issues, still make it a challenge. The use of domain specific language and com- piler techniques to assist in description and optimisation of cryptographic software is an interesting research challenge. Our results, which focus on Elliptic Curve Cryptography (ECC), show that a suitable language allows description of ECC based software in a manner close to the original mathe- matics; the corresponding compiler allows automatic production of an ex- ecutable whose performance is competitive with that of a hand-optimised implementation. Our work are set within the context of CACE, an ongo- ing EU funded pro ject on this general topic