M.S. Thesis Defense
Computer Science and Electrical Engineering
University of Maryland, Baltimore County

Fast Modular Exponentiation Using Residue Domain Representation:
A Hardware Reference Implementation and Analysis

Christopher D. Nguyen

12:00–2:00pm, Tuesday, 5 November 2013, ITE 228, UMBC

Using field-programmable gate arrays (FPGAs) we engineered and analyzed the first hardware implementation of Phatak’s reduced-precision residue number system (RP-RNS) to perform modular exponentiation.

Residue number systems (RNSs) provide an alternative representation to the binary system for performing integer arithmetic used in applications such as public-key cryptography and digital signal processing. They offer full parallel computation for addition, subtraction, and multiplication increasing performance from O(K) to O(lg K) for a K-bit number. However, base extension, division, and sign detection become harder operations.

RP-RNS is a new set of algorithms that uses approximation and a time-memory trade-off to address the hard operations. The partial reconstruction (PR) algorithm addresses base extension and the quotient-first scaling (QFS) algorithm addresses scaling. RP-RNS modular exponentiation uses the PR and QFS algorithms. RP-RNS improves performance of modular multiplication in an RNS with range [0, M-1] from the O((lg n)^2) delay of current systems (e.g. Cox-Rower) to a theoretical O(lg n) delay where n is the word-length of M.

Our implementation is based on Phatak’s description and recommended architecture diagrams. We found even low-end FPGAs can store over 30 channels of logic. Following the recommendation of parallel look-up table (LUT) access, we distributed the LUTs to be local to each channel. We found this recommendation applied to QFS exceeds the capacity of today’s high-capacity FPGAs (e.g. Xilinx Virtex-7) for modest 2,000-bit divisors. We propose several improvements to increase feasibility; one is to store the LUTs external to the FPGA, which would introduce a performance penalty per look-up.

Committee: Alan Sherman (chair), Dhananjay Phatak, Chintan Patel and
Ryan Robucci

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The Council of Computing Majors (CCM) will meet from Noon to 12:45pm on Monday, November 4 in MP 103. The CCM is a student organization representing undergraduate computer science and computer engineering majors and anyone else with an interest in computing. Everyone is welcome.

At next week’s meeting, three computer science graduate students, Prajit Das, Primal Pappachan and Roberto Yus, will demonstrate Google Glass and talk about how they are using it in their research.

Google glass is the latest cool gadget in town. Developed by Google, it is is a wearable computer with a head mounted display. We will talk about the technical capabilities of Google glass as well as about developing apps for it using the Mirror API and the Android Software Development Kit. We will also give a live demo with the device where we will show some apps we developed.