The Study of Monotonic Core Functions and Their Use to Build RNS Number Comparators

A non-positional residue number system (RNS) enjoys particularly efficient implementation of addition and multiplication, but non-modular arithmetic operations in RNS-like number comparison are known to be difficult. In this paper, a new technique for designing comparators of RNS numbers represented in an arbitrary moduli set is presented. It is based on using the core function for which it was shown that it must be monotonic to allow for RNS number comparison. The conditions of the monotonicity of the core function were formulated, which also ensured the minimal range of the core function (essential to obtain the best characteristics of the comparator). The best choice is a core function in which only one coefficient corresponding to the largest modulus is set to 1 whereas all other coefficients are set to 0. It is also shown that the already known diagonal function is nothing else but the special case of the core function with all coefficients set to 1. Performance evaluation suggests that the new comparator uses less hardware and in some cases also introduces smaller delay than its counterparts based on diagonal function. The potential applications of the new comparator include some recently developed homomorphic encryption algorithms implemented using RNS.

RNS Number Comparator Based on a Modified Diagonal Function

Number comparison has long been recognized as one of the most fundamental non-modular arithmetic operations to be executed in a non-positional Residue Number System (RNS). In this paper, a new technique for designing comparators of RNS numbers represented in an arbitrary moduli set is presented. It is based on a newly introduced modified diagonal function, whose strictly monotonic properties make it possible to replace the cumbersome operations of finding the remainder of the division by a large and awkward number with significantly simpler computations involving only a power of 2 modulus. Comparators of numbers represented in sample RNSs composed of varying numbers of moduli and offering different dynamic ranges, designed using various methods, were synthesized for the 65 nm technology. The experimental results suggest that the new circuits enjoy a delay reduction ranging from over 11% to over 75% compared to the fastest circuits designed using existing methods. Moreover, it is achieved using less hardware, the reduction of which reaches over 41%, and is accompanied by significantly reduced power-consumption, which in several cases exceeds 100%. Therefore, it seems that the presented method leads to the design of the most efficient current hardware comparators of numbers represented using a general RNS moduli set.

Construction of Residue Number System Using Hardware Efficient Diagonal Function

The residue number system (RNS) is a non-positional number system that allows one to perform addition and multiplication operations fast and in parallel. However, because the RNS is a non-positional number system, magnitude comparison of numbers in RNS form is impossible, so a division operation and an operation of reverse conversion into a positional form containing magnitude comparison operations are impossible too. Therefore, RNS has disadvantages in that some operations in RNS, such as reverse conversion into positional form, magnitude comparison, and division of numbers are problematic. One of the approaches to solve this problem is using the diagonal function (DF). In this paper, we propose a method of RNS construction with a convenient form of DF, which leads to the calculations modulo 2 n , 2 n − 1 or 2 n + 1 and allows us to design efficient hardware implementations. We constructed a hardware simulation of magnitude comparison and reverse conversion into a positional form using RNS with different moduli sets constructed by our proposed method, and used different approaches to perform magnitude comparison and reverse conversion: DF, Chinese remainder theorem (CRT) and CRT with fractional values (CRTf). Hardware modeling was performed on Xilinx Artix 7 xc7a200tfbg484-2 in Vivado 2016.3 and the strategy of synthesis was highly area optimized. The hardware simulation of magnitude comparison shows that, for three moduli, the proposed method allows us to reduce hardware resources by 5.98–49.72% in comparison with known methods. For the four moduli, the proposed method reduces delay by 4.92–21.95% and hardware costs by twice as much by comparison to known methods. A comparison of simulation results from the proposed moduli sets and balanced moduli sets shows that the use of these proposed moduli sets allows up to twice the reduction in circuit delay, although, in several cases, it requires more hardware resources than balanced moduli sets.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.