Fast Fourier transform via automorphism groups of rational function fields

The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a smooth number, then the divide-and-conquer approach leads to the fastest known FFT algorithms. Depending on the type of group that the set of evaluation points forms, these algorithms are classified as multiplicative (Math of Comp. 1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a unified framework for FFT algorithms that include both multiplicative and additive FFT algorithms as special cases, and beyond: our framework also works when $q+1$ is smooth, while all known results require $q$ or $q-1$ to be smooth. For the new case where $q+1$ is smooth (this new case was not considered before in literature as far as we know), we show that if $n$ is a divisor of $q+1$ that is $B$-smooth for a real $B>0$, then our FFT needs $O(Bn\log n)$ arithmetic operations in $\mathbb{F}_q$. Our unified framework is a natural consequence of introducing the algebraic function fields into the study of FFT

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2

On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras sl (2, ℝ) or su(2) according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.Instituto de Física La Plat

On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras sl (2, ℝ) or su(2) according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.Instituto de Física La Plat

General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron (LSH) degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The LSH formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantum simulating other complex theories of relevance to nature.Comment: 59+17+7 pages, 16 figure

Design of microprocessor-based hardware for number theoretic transform implementation

Number Theoretic Transforms (NTTs) are defined in a finite ring of integers Z (_M), where M is the modulus. All the arithmetic operations are carried out modulo M. NTTs are similar in structure to DFTs, hence fast FFT type algorithms may be used to compute NTTs efficiently. A major advantage of the NTT is that it can be used to compute error free convolutions, unlike the FFT it is not subject to round off and truncation errors. In 1976 Winograd proposed a set of short length DFT algorithms using a fewer number of multiplications and approximately the same number of additions as the Cooley-Tukey FFT algorithm. This saving is accomplished at the expense of increased algorithm complexity. These short length DFT algorithms may be combined to perform longer transforms. The Winograd Fourier Transform Algorithm (WFTA) was implemented on a TMS9900 microprocessor to compute NTTs. Since multiplication conducted modulo M is very time consuming a special purpose external hardware modular multiplier was designed, constructed and interfaced with the TMS9900 microprocessor. This external hardware modular multiplier allowed an improvement in the transform execution time. Computation time may further be reduced by employing several microprocessors. Taking advantage of the inherent parallelism of the WFTA, a dedicated parallel microprocessor system was designed and constructed to implement a 15-point WFTA in parallel. Benchmark programs were written to choose a suitable microprocessor for the parallel microprocessor system. A master or a host microprocessor is used to control the parallel microprocessor system and provides an interface to the outside world. An analogue to digital (A/D) and a digital to analogue (D/A) converter allows real time digital signal processing