Definition 1. Let k be a field. An algebraic variety over k is a k-scheme X such that there exists a covering by a finite number of affine open subschemes Xi which are affine varieties over k, i.e. each Xi is the affine scheme associated to a finitely generated algebra over k. A projective variety over k is a projective scheme over k, i.e. a k-scheme isomorphic to Proj k[T0, . . . , Tn]/I for a...

Although cryptographic implementation tasks are often undertaken by expert programmers, a plethora of performance and security driven options, as well as more mundane software engineering issues, still make this a challenge. In an attempt to transfer expert knowledge into automated tools, we investigate the use of domain specific language and compilation techniques for cryptographic software, f...

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This M.S. thesis introduces new modulus scaling techniques for transforming a class of primes into special forms which enable efficient arithmetic. The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of a scaled modulus. Our inversion algorithm exhibits superior performance to the Euc...

This paper presents an extensive and careful study of the software implementation on workstations of the NIST-recommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation.

This paper introduces “hyper-and-elliptic-curve cryptography”, in which a single high-security group supports fast genus-2-hyperelliptic-curve formulas for variable-base-point single-scalar multiplication (e.g., Diffie–Hellman shared-secret computation) and at the same time supports fast elliptic-curve formulas for fixed-base-point scalar multiplication (e.g., key generation) and multi-scalar m...

The increased vulnerability of network attached computers has elevated the need for strong security methods. At the same time, advancements in the the area of cryptography have risen to meet this challenge. This report concerns one particular form of cryptography which uses the mathematics of elliptic curves. In particular, the suitability of enhancing current methods with specialized hardware ...

In this paper we perform a review of elliptic curve cryptography (ECC) as it is used in practice today in order to reveal unique mistakes and vulnerabilities that arise in implementations of ECC. We study four popular protocols that make use of this type of public-key cryptography: Bitcoin, secure shell (SSH), transport layer security (TLS), and the Austrian e-ID card. We are pleased to observe...

Pairing-based cryptosystems rely on bilinear non-degenerate maps called pairings, such as the Tate and Weil pairings defined over certain elliptic curve groups. In this paper we show how to compress pairing values, how to couple this technique with that of point compression, and how to benefit from the compressed representation to speed up exponentiations involving pairing values, as required i...