Algorithms & Architectures for Cryptography

Motivation and Background

Arithmetic operations play a critical role in making cryptographic schemes viable (or not). Especially public-key cryptographic operations require intensive computations which may become too costly for certain applications. Efficient arithmetic operations are especially critical for interactive applications. The proliferation of embedded and ubiqitous computing has driven cryptographer to develop cryptographic schemes using non-integer arithmetic (e.g. algebra over finite field) over more complex mathematical structures (e.g. elliptic curves, lattices etc.). Some of the more popular public-key algorithms and their requirements are summarized in the table below.

Cryptographic Scheme	Type of Arithmetic	Bit-length
RSA & DH	Integer Ring / Field (Z_n & GF(p))	1024	-	2048 Bits
Elliptic Curve Schemes	Finite Field (GF(p) & GF(2^k))	160	-	512 Bits
Hyperelliptic Curve Schemes	Short polynomial over GF(p)	170	-	256 Bits
NTRU	Cyclic ring over Z_q (Z_q[x]/(x^N-1))	1169	-	4024 Bits
XTR	Traces in GF(p⁶) (through GF(p) arithmetic)	1020	-	.. Bits

Our Research

CRIS members have developed and implemented a large number of algorithms and hardware architectures to support a variety of cryptographic schemes in recent years. The research work ranges from purely algorithmic contributionas to full custom ASIC & FPGA implementations. Some of the publications are given below. For a full list please browse through publications on the left-menu.

Selected Publications

Selcuk Baktir and Berk Sunar, Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography. Preprint., Pre-print: (PDF)
Selcuk Baktir, Sandeep Kumar, Christof Paar, Berk Sunar, A State-of-the-art Elliptic Curve Cryptographic Processor Operating in the Frequency Domain, Pre-print: (PDF)
Erdinc Ozturk, Erkay Savas, Berk Sunar, A Versatile Montgomery Multiplier Architecture with Characteristic Three Support, Under Review, Pre-Print (PDF)
S. Baktir and B. Sunar. Finite Field Polynomial Multiplication in the Frequency Domain with Application to Elliptic Curve Cryptography, Proceedings of the 21st International Symposium on Computer and Information Sciences (ISCIS 2006), Lecture Notes in Computer Science, volume 4263, Springer, Heidelberg, pages 991-1001, October, 2006. (PDF).
S. Baktir and B. Sunar. Achieving Efficient Polynomial Multiplication in Fermat Fields Using the Fast Fourier Transform, Proceedings of the 44th ACM Southeast Conference (ACMSE 2006) , ACM Press, pages 549-554, March, 2006. (PDF).
S. Baktir, J. Pelzl, T. Wollinger, B. Sunar and C. Paar. Optimal Tower Fields for Hyperelliptic Curve Cryptosystems, IEEE Proceedings of the 38th Annual Asilomar Conference on Signals, Systems and Computers, November 2004. (PDF)
B. Sunar. A Generalized Method for Constructing Sub-quadratic Complexity Bit-Parallel Multipliers, IEEE Transactions on Computers, 53(9):1097-1105, September 2004.
S. Baktir and B. Sunar. Optimal Tower Fields, IEEE Transactions on Computers, 53(10):1231-1243, October 2004. (PDF)
Selcuk Baktir. Efficient Algorithms for Finite Fields, with Applications in Elliptic Curve Cryptography, Master's Thesis, Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA, April 2003. (PDF)
B. Sunar and C. M. O'Rourke. Achieving NTRU with Montgomery Multiplication, IEEE Transactions on Computers, Special Issue on Cryptographic Hardware and Embedded Systems, 52(4)440-448, April, 2003.
E. Öztürk, B. Sunar, and E. Savaç. Low-power elliptic curve cryptography using scaled modular arithmetic, Cryptographic Hardware and Embedded Systems, CHES 2004, August 2004. (PDF)
B. Sunar and C. K. Koc. An efficient optimal normal basis type II multiplier, IEEE Transactions on Computers, 50(1):83-87, January 2001. (PDF)

Links to Other Research Groups and Individuals

Communication Security Group, Ruhr-Universität Bochum, Germany.
Information Security Laboratory, Oregon State University.
Center for Applied Cryptographic Research, University of Waterloo.
Cryptography and Network Security Implementations Lab, George Mason University

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Last modified: Friday, 02-Feb-2007 18:54:34 EST