1,658 research outputs found
The ElGamal cryptosystem over circulant matrices
In this paper we study extensively the discrete logarithm problem in the
group of non-singular circulant matrices. The emphasis of this study was to
find the exact parameters for the group of circulant matrices for a secure
implementation. We tabulate these parameters. We also compare the discrete
logarithm problem in the group of circulant matrices with the discrete
logarithm problem in finite fields and with the discrete logarithm problem in
the group of rational points of an elliptic curve
Analysis of Parallel Montgomery Multiplication in CUDA
For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
Cloud-based Quadratic Optimization with Partially Homomorphic Encryption
The development of large-scale distributed control systems has led to the
outsourcing of costly computations to cloud-computing platforms, as well as to
concerns about privacy of the collected sensitive data. This paper develops a
cloud-based protocol for a quadratic optimization problem involving multiple
parties, each holding information it seeks to maintain private. The protocol is
based on the projected gradient ascent on the Lagrange dual problem and
exploits partially homomorphic encryption and secure multi-party computation
techniques. Using formal cryptographic definitions of indistinguishability, the
protocol is shown to achieve computational privacy, i.e., there is no
computationally efficient algorithm that any involved party can employ to
obtain private information beyond what can be inferred from the party's inputs
and outputs only. In order to reduce the communication complexity of the
proposed protocol, we introduced a variant that achieves this objective at the
expense of weaker privacy guarantees. We discuss in detail the computational
and communication complexity properties of both algorithms theoretically and
also through implementations. We conclude the paper with a discussion on
computational privacy and other notions of privacy such as the non-unique
retrieval of the private information from the protocol outputs
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