45 research outputs found
Elliptic Curve Cryptography on Modern Processor Architectures
Abstract
Elliptic Curve Cryptography (ECC) has been adopted by the US National Security Agency (NSA) in Suite "B" as part of its "Cryptographic Modernisation Program ". Additionally,
it has been favoured by an entire host of mobile devices due to its superior performance characteristics. ECC is also the building block on which the exciting field of pairing/identity based cryptography is based. This widespread use means that there is potentially a lot to be gained by researching efficient implementations on modern processors such as IBM's Cell Broadband Engine and Philip's next generation smart card cores. ECC operations can be thought of as a pyramid of building blocks, from instructions on a core, modular operations on a finite field, point addition & doubling, elliptic curve scalar
multiplication to application level protocols. In this thesis we examine an implementation of these components for ECC focusing on a range of optimising techniques for the Cell's SPU and the MIPS smart card. We show significant performance improvements that can be achieved through of adoption of EC
Cryptographic Pairings: Efficiency and DLP security
This thesis studies two important aspects of the use of pairings in cryptography, efficient
algorithms and security.
Pairings are very useful tools in cryptography, originally used for the cryptanalysis of
elliptic curve cryptography, they are now used in key exchange protocols, signature schemes
and Identity-based cryptography.
This thesis comprises of two parts: Security and Efficient Algorithms.
In Part I: Security, the security of pairing-based protocols is considered, with a thorough
examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the
relationship between the two instances of the DLP will be presented along with a discussion
about the appropriate selection of parameters to ensure particular security level.
In Part II: Efficient Algorithms, some of the computational issues which arise when using
pairings in cryptography are addressed. Pairings can be computationally expensive, so
the Pairing-Based Cryptography (PBC) research community is constantly striving to find
computational improvements for all aspects of protocols using pairings. The improvements
given in this section contribute towards more efficient methods for the computation of pairings,
and increase the efficiency of operations necessary in some pairing-based protocol
Developing an Automatic Generation Tool for Cryptographic Pairing Functions
Pairing-Based Cryptography is receiving steadily more attention from industry, mainly
because of the increasing interest in Identity-Based protocols. Although there are plenty of
applications, efficiently implementing the pairing functions is often difficult as it requires
more knowledge than previous cryptographic primitives. The author presents a tool for
automatically generating optimized code for the pairing functions which can be used in the
construction of such cryptographic protocols.
In the following pages I present my work done on the construction of pairing function
code, its optimizations and how their construction can be automated to ease the work of the
protocol implementer.
Based on the user requirements and the security level, the created cryptographic compiler
chooses and constructs the appropriate elliptic curve. It identifies the supported pairing
function: the Tate, ate, R-ate or pairing lattice/optimal pairing, and its optimized parameters.
Using artificial intelligence algorithms, it generates optimized code for the final exponentiation
and for hashing a point to the required group using the parametrisation of the
chosen family of curves.
Support for several multi-precision libraries has been incorporated: Magma, MIRACL
and RELIC are already included, but more are possible
Computing Bilinear Pairings on Elliptic Curves with Automorphisms
In this paper, we present a novel method for constructing a
super-optimal pairing with great efficiency, which we call the omega
pairing. The computation of the omega pairing requires the simple
final exponentiation and short loop length in Miller\u27s algorithm
which leads to a significant improvement over the previously known
techniques on certain pairing-friendly curves. Experimental results
show that the omega pairing is about 22% faster and 19% faster
than the super-optimal pairing proposed by Scott at security level
of AES 80 bits on certain pairing-friendly curves in affine
coordinate systems and projective coordinate systems, respectively
Pairings in Cryptology: efficiency, security and applications
Abstract
The study of pairings can be considered in so many di�erent ways that it
may not be useless to state in a few words the plan which has been adopted,
and the chief objects at which it has aimed. This is not an attempt to write
the whole history of the pairings in cryptology, or to detail every discovery,
but rather a general presentation motivated by the two main requirements
in cryptology; e�ciency and security.
Starting from the basic underlying mathematics, pairing maps are con-
structed and a major security issue related to the question of the minimal
embedding �eld [12]1 is resolved. This is followed by an exposition on how
to compute e�ciently the �nal exponentiation occurring in the calculation
of a pairing [124]2 and a thorough survey on the security of the discrete log-
arithm problem from both theoretical and implementational perspectives.
These two crucial cryptologic requirements being ful�lled an identity based
encryption scheme taking advantage of pairings [24]3 is introduced. Then,
perceiving the need to hash identities to points on a pairing-friendly elliptic
curve in the more general context of identity based cryptography, a new
technique to efficiently solve this practical issue is exhibited.
Unveiling pairings in cryptology involves a good understanding of both
mathematical and cryptologic principles. Therefore, although �rst pre-
sented from an abstract mathematical viewpoint, pairings are then studied
from a more practical perspective, slowly drifting away toward cryptologic
applications