### Elliptic Curve Cryptography and Finite Fields

**Funding:**Technicolor

**Research Team:**

Çetin Kaya Koç

Ömer Eğecioğlu

Wangchen Dai

Pedro Sosa

Josephine Vo

Shane Masuda

##### Elliptic Curve Cryptography

Elliptic curve cryptography provides a general methodology for obtaining high-speed, efficient, and scalable implementations of cryptographically strong random number generators, public-key cryptographic and network security protocols. The security of these schemes depends on the difficulty of computing elliptic curve discrete logarithm in the elliptic curve group. The arithmetic of elliptic curve operations depend on the arithmetic on the underlying finite field. The current standards suggest the use of Galois fields of p or 2^k elements, meanwhile researchers are also interested in exploring GF(p^k). In this research project we are developing algorithms, hardware and software realizations of these finite fields, whose elements are represented using the polynomial, normal, optimal normal and Gaussian bases.

**Our recent publications**

“Design of elliptic curve cryptoprocessors over GF(2163) on Koblitz curves”, Realpe-Munoz, Paulo ; Trujillo-Olaya, Vladimir ; Velasco-Medina, Jaime. Circuits and Systems (LASCAS), 2014 IEEE 5th Latin American Symposium on; 25- 28 Feb. 2014, Santiago de Chile.

“Design of elliptic curve cryptoprocessors over GF(2^163) using the Gaussian normal basis”, Paulo Cesar Realpe Muñoz, Vladimir Trujillo-Olaya, Jaime Velasco-Medina. Revista Ingeniería E Investigación vol. 34, Num.2. ISSN: 0120- 5609, 2014.

“Design of an Elliptic Curve Cryptoprocessor using Optimal Normal Basis over GF(2^233)”, Urbano-Molano, F.A. Trujillo-Olaya, V. ; Velasco-Medina, J., 2013 IEEE Fourth Latin American Symposium on Circuits and Systems (LASCAS), Cusco-Peru, Feb. 27 2013-March 2013.

V. Trujillo-Olaya, T. Sherwood, and Ç. K. Koç. Analysis of performance versus security in hardware realizations of small elliptic curves for lightweight applications. *Journal of Cryptographic Engineering*, 2(3):179-188, 2012. pdf

L. A. Tawalbeh, A. F. Tenca, S. Park, and Ç. K. Koç. An efficient hardware architecture of a scalable elliptic curve crypto-processor over GF(2^m). *Advanced Signal Processing Algorithms, Architectures, and Implementations XV, Proceedings of SPIE Conference,*, F. T. Luk, editor, pages 216-226, Vol. 5910, San Diego, California, August 2-4, 2005. pdf

T. Wollinger, J. Pelzl, V. Wittelsberger, C. Paar, G. Saldamlı, and Ç. K. Koç. Elliptic and hyperelliptic curves on embedded µP. *ACM Transactions on Embedded Computing Systems,* 3(3):509-533, August 2004. pdf

E. Savaş and Ç. K. Koç. Architectures for unified field inversion with applications in elliptic curve cryptography. *The 9th IEEE International Conference on Electronics, Circuits and Systems - ICECS 2002,* Vol. 3, pages 1155-1158, Dubrovnik, Croatia, September 15-18, 2002. pdf

E. Savaş, T. A. Schmidt, and Ç. K. Koç. Generating elliptic curves of known order. *Cryptographic Hardware and Embedded Systems - CHES 2001,* Ç. K. Koç, D. Naccache, and C. Paar, editors, Third International Workshop, Paris, France, pages 142-158, Springer, LNCS Nr. 2162, May 14-16, 2001. pdf

M. Aydos, E. Savaş, and Ç. K. Koç. Implementing network security protocols based on elliptic curve cryptography. *Proceedings of the Fourth Symposium on Computer Networks,* S. Oktug, B. Orencik, and E. Harmanci, editors, pages 130-139, Istanbul, Turkey, May 20-21, 1999. pdf

##### Finite Fields

Finite fields underpin the mathematical strength of many symmetric and asymmetric cryptographic operations. Additionally finite fields are important for error detection and correction for communications. Unfortunately, finite field arithmetic can be computationally expensive. We make fundamental contributions to this domain, in both mathematics and efficient implementation.

**Our related workshop**

International Workshop on the Arithmetic of Finite Fields http://waifi.org/ see also:

Ç. K. Koç, S. Mesnager and E. Savaş, editors. *Arithmetic of Finite Fields.* 5th International Workshop, WAIFI 2014. Gebze, Turkey. Springer, LNCS Nr. 9061, September 27-28, 2014. link

**Our recent publications**

S. Contini, Ç. K. Koç, and C. D. Walter. Modular arithmetic. *Encyclopedia of Cryptography and Security,* 2nd Edition, H. C. A. van Tilborg and S. Jajodia, editors, Springer, September 2011.

Ç. K. Koç and C. D. Walter. Montgomery arithmetic. *Encyclopedia of Cryptography and Security,* 2nd Edition, H. C. A. van Tilborg and S. Jajodia, editors, Springer, September 2011.

İ. San and N. At.
Improving the computational efficiency of modular operations
for embedded systems.
*Journal of Systems Architecture*,
Vol. 60, issue.5, pages 440-451, May 2014. link

D. D. Chen, G. X. Yao, R. C. C. Cheung, D. Pao, and Ç. K. Koç. Parameter space for the architecture of FFT-based Montgomery modular multiplication. *IEEE Transactions on Computers*, to appear, 2016. pdf

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