We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.

Perfect, Efficent, State Transfer and its Application as a Constructive Tool

State transfer on graphs

We review the subject of perfect state transfer; how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.

A Review of Perfect State Transfer and its Application as a Constructive Tool

Let A be a thin set in Z n , and A N the intersection of A with the ball of diameter N centred at the origin. When n = 1 we have seen in §9.7 (as a consequence of Siegel’s theorem) that |A N | = O(N 1/2) when N → ∞. To prove a similar result when n ≥ 2 one needs a different method, based on the large sieve inequality (cf. [Co]).

The Large Sieve

In this paper we describe an efficient implementation of the Tate and Ate pairings using Barreto-Naehrig pairing-friendly curves, on both a standard PC and on a 32-bit smartcard. First we introduce a subfamily of such curves with a particularly simple representation. Next we consider the issues that arise in the efficient implemention of field arithmetic in F p12, which is crucial to good performance. Various optimisations are suggested, including a novel approach to the 'final exponentiation', which is faster and requires less memory than the methods previously recommended.

/pdf/implementing-cryptographic-pairings-over-barreto-naehrig-jlpcaccrkv.pdf

Implementing cryptographic pairings over barreto-naehrig curves

http://www.tau.ac.il/~nogaa/PDFS/integergraphs1.pdf

Graphs with integral spectrum

Fast arithmetic for characteristic three finite fields is desirable in pairing-based cryptography because there is a suitable family of elliptic curves over having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of , and use the results to devise faster multiplication algorithms. We carefully compare multiplication in using polynomial bases and Gaussian normal bases. Finally, we compare the speed of encryption and decryption for the Boneh-Franklin and Sakai-Kasahara identity-based encryption schemes at the 128-bit security level, in the case where supersingular elliptic curves with embedding degrees 2, 4 and 6 are employed.

Software Implementation of Arithmetic in

We present an algorithm that by using the and 1 Frobenius operators concurrently allows us to obtain a parallelized version of the classical -and-add scalar multiplication algorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to efficient implementations of both and 1 , thus showing that our algorithm can be effectively applied on all the NIST-recommended curves. We also present design details of software and hardware implementations of our procedure. In a two-processor workstation software implementation, we report experimental data showing that our parallel algorithm is able to achieve a speedup factor of almost 2 when compared with the standard sequential point multiplication. In our hardware implementation, the parallel version yields a more modest acceleration of 17% when compared with the traditional point multiplication algorithm. Although the focus is on Koblitz curves, analogous strategies are discussed for other curves, in particular for random curves over binary fields.

/pdf/parallel-formulations-of-scalar-multiplication-on-koblitz-11svw1yz7k.pdf

Parallel Formulations of Scalar Multiplication on Koblitz Curves

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.

Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.

http://mathsci.ucd.ie/~gmg/Fq9Talks/Ahmadi.pdf

Omran Ahmadi

Papers

Graphs with integral spectrum

Software Implementation of Arithmetic in

Parallel Formulations of Scalar Multiplication on Koblitz Curves

Multiplicative Order of Gauss Periods

Multiplicative order of Gauss periods