Optimization of new Chinese Remainder theorems using special moduli sets

The residue number system (RNS) is an integer number representation system, which is capable of supporting parallel, high-speed arithmetic. This system also offers some useful properties for error detection, error correction and fault tolerance. It has numerous applications in computation-intensive digital signal processing (DSP) operations, like digital filtering, convolution, correlation, Discrete Fourier Transform, Fast Fourier Transform, direct digital frequency synthesis, etc. The residue to binary conversion is based on Chinese Remainder Theorem (CRT) and Mixed Radix Conversion (MRC). However, the CRT requires a slow large modulo operation while the MRC requires finding the mixed radix digits which is a slow process. The new Chinese Remainder Theorems (CRT I, CRT II and CRT III) make the computations faster and efficient without any extra overheads. But, New CRTs are hardware intensive as they require many inverse modulus operators, modulus operators, multipliers and dividers. Dividers and inverse modulus operators in turn needs many half and full adders and subtractors. So, some kind of optimization is necessary to implement these theorems practically. In this research, for the optimization, new both co-prime and non co-prime multi modulus sets are proposed that simplify the new Chinese Remainder theorems by eliminating the huge summations, inverse modulo operators, and dividers. Furthermore, the proposed hardware optimization removes the multiplication terms in the theorems, which further simplifies the implementation

Efficient convolvers using the Polynomial Residue Number System technique

The problem of computing linear convolution is a very important one because with linear convolution we can mechanize digital filtering. The linear convolution of two N-point sequences can be computed by the cyclic convolution of the following 2N-point sequences. The original sequence padded with N zero’s each. The cyclic convolution of two N-point sequences requires multiplications and additions for its computation. A very efficient way of computing cyclic convolution of two sequences is by using the Polynomial Residue Number System (PRNS) technique. Using this technique the cyclic convolution of two N-point sequences can be computed using only N multiplications instead of N2 multiplications. This can be achieved based on some forward and inverse PRNS transformation mappings. These mappings rely on additions, subtractions and many scaling operations (multiplications by constants). The PRNS technique would lose a lot in value if these many scaling operations were difficultly implemented. In this thesis we will show how to calculate cyclic convolution of two sequences using the PRNS technique based on forward and inverse transformation mapping which rely on complement operations (negations), additions and rotation operations. These rotation operations do not require any computational hardware. Therefore the complicated hardware required for the scaling operations has now been substituted by rotators, which do not require any computational hardware

Exact resolution method for general 1D polynomial Schr\"odinger equation

The stationary 1D Schr\"odinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing spectral determinants of (N+2) generically distinct operators, all the transforms of one quantum Hamiltonian under a cyclic group of complex scalings. The determinants' zeros define (N+2) semi-infinite chains of points in the complex spectral plane, and they encode the original quantum problem. Each chain can now be described by an exact quantization condition which constrains it in terms of its neighbors, resulting in closed equilibrium conditions for the global chain system; these are supplemented by the standard (Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each chain asymptotically. This reduced problem is then probed numerically for effective solvability upon test cases (mostly, symmetric quartic oscillators): we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues/eigenfunctions. We conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable (contractive) and thus determine the exact quantum data as their fixed point. (To date, this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials $V(q)$.)Comment: flatex text.tex, 4 files Submitted to: J. Phys. A: Math. Ge

Efficient implementation of the Hardy-Ramanujan-Rademacher formula

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.Comment: updated version containing an unconditional complexity proof; accepted for publication in LMS Journal of Computation and Mathematic

Attacks on the Search-RLWE problem with small errors

The Ring Learning-With-Errors (RLWE) problem shows great promise for post-quantum cryptography and homomorphic encryption. We describe a new attack on the non-dual search RLWE problem with small error widths, using ring homomorphisms to finite fields and the chi-squared statistical test. In particular, we identify a "subfield vulnerability" (Section 5.2) and give a new attack which finds this vulnerability by mapping to a finite field extension and detecting non-uniformity with respect to the number of elements in the subfield. We use this attack to give examples of vulnerable RLWE instances in Galois number fields. We also extend the well-known search-to-decision reduction result to Galois fields with any unramified prime modulus q, regardless of the residue degree f of q, and we use this in our attacks. The time complexity of our attack is O(nq2f), where n is the degree of K and f is the residue degree of q in K. We also show an attack on the non-dual (resp. dual) RLWE problem with narrow error distributions in prime cyclotomic rings when the modulus is a ramified prime (resp. any integer). We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks

Abelian Surfaces over totally real fields are Potentially Modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.Comment: 285 page