403 research outputs found
Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets
We generalize an efficient exact synthesis algorithm for single-qubit
unitaries over the Clifford+T gate set which was presented by Kliuchnikov,
Maslov and Mosca. Their algorithm takes as input an exactly synthesizable
single-qubit unitary--one which can be expressed without error as a product of
Clifford and T gates--and outputs a sequence of gates which implements it. The
algorithm is optimal in the sense that the length of the sequence, measured by
the number of T gates, is smallest possible. In this paper, for each positive
even integer we consider the "Clifford-cyclotomic" gate set consisting of
the Clifford group plus a z-rotation by . We present an
efficient exact synthesis algorithm which outputs a decomposition using the
minimum number of z-rotations. For the Clifford+T case
the group of exactly synthesizable unitaries was shown to be equal to the group
of unitaries with entries over the ring .
We prove that this characterization holds for a handful of other small values
of but the fraction of positive even integers for which it fails to hold is
100%.Comment: v2: published versio
Fast polynomial arithmetic in homomorphic encryption with cyclo-multiquadratic fields
This work provides refined polynomial upper bounds for the condition number
of the transformation between RLWE/PLWE for cyclotomic number fields with up to
6 primes dividing the conductor. We also provide exact expressions of the
condition number for any cyclotomic field, but under what we call the twisted
power basis. Finally, from a more practical perspective, we discuss the
advantages and limitations of cyclotomic fields to have fast polynomial
arithmetic within homomorphic encryption, for which we also study the RLWE/PLWE
equivalence of a concrete non-cyclotomic family of number fields. We think this
family could be of particular interest due to its arithmetic efficiency
properties
The Langlands Program and String Modular K3 Surfaces
A number theoretic approach to string compactification is developed for
Calabi-Yau hypersurfaces in arbitrary dimensions. The motivic strategy involved
is illustrated by showing that the Hecke eigenforms derived from Galois group
orbits of the holomorphic two-form of a particular type of K3 surfaces can be
expressed in terms of modular forms constructed from the worldsheet theory. The
process of deriving string physics from spacetime geometry can be reversed,
allowing the construction of K3 surface geometry from the string characters of
the partition function. A general argument for K3 modularity follows from
mirror symmetry, in combination with the proof of the Shimura-Taniyama
conjecture.Comment: 33 page
Number Theoretic Transform and Its Applications in Lattice-based Cryptosystems: A Survey
Number theoretic transform (NTT) is the most efficient method for multiplying
two polynomials of high degree with integer coefficients, due to its series of
advantages in terms of algorithm and implementation, and is consequently
widely-used and particularly fundamental in the practical implementations of
lattice-based cryptographic schemes. Especially, recent works have shown that
NTT can be utilized in those schemes without NTT-friendly rings, and can
outperform other multiplication algorithms. In this paper, we first review the
basic concepts of polynomial multiplication, convolution and NTT. Subsequently,
we systematically introduce basic radix-2 fast NTT algorithms in an algebraic
way via Chinese Remainder Theorem. And then, we elaborate recent advances about
the methods to weaken restrictions on parameter conditions of NTT. Furthermore,
we systematically introduce how to choose appropriate strategy of NTT
algorithms for the various given rings. Later, we introduce the applications of
NTT in the lattice-based cryptographic schemes of NIST post-quantum
cryptography standardization competition. Finally, we try to present some
possible future research directions
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