Patents

1. Ç. K. Koç and B. Sunar. Methods and apparatus for multiplication in a Galois field GF(2^m), and encoders and decoders using same. US Patent Nr. 6,343,305. January 29, 2002.   pdf
2. Ç. K. Koç and A. F. Tenca. A scalable architecture for Montgomery multiplication. US Patent Nr. 7,046,800. May 16, 2006.   pdf
3. Ç. K. Koç and E. Savaş. Cryptographic methods and apparatus using word-wise Montgomery multiplication. US Patent Nr. 7,050,579. May 23, 2006.   pdf
4. Ç. K. Koç, T. Yanık, and E. Savaş. Methods and apparatus for incomplete modular arithmetic. US Patent Nr. 7,080,109. July 18, 2006.   pdf
5. Ç. K. Koç, A. F. Tenca, and G. Todorov. Methods and apparatus for variable radix scalable modular multiplication. US Patent Nr. 7,174,015. February 6, 2007.   pdf
6. Ç. K. Koç, E. Savaş, and A. F. Tenca. Scalable and unified multiplication methods and apparatus. US Patent Nr. 7,240,204. July 3, 2007.   pdf
7. Ç. K. Koç and S. S. Erdem. Multiplication of multi-precision numbers having a size of a power of two. US Patent Nr. 7,401,109. July 15, 2008.   pdf
8. Ç. K. Koç and S. S. Erdem. Lean multiplication of multi-precision numbers over GF(2^m). US Patent Nr. 7,447,310. November 4, 2008.   pdf
9. Ç. K. Koç. Systems and methods for providing security for computer systems. US Patent Nr. 8,090,934. January 3, 2012.   pdf
10. G. M. Hammouri, B. Sunar, Ç. K. Koç, and K. Akdemir. Computing-system identifier using software extraction of manufacturing variability. US Patent Nr. 8,694,687. April 8, 2014.   pdf
11. Ç. K. Koç and G. Saldamlı. Spectral modular arithmetic method and apparatus. US Patent Nr. 8,719,324. May 6, 2014.   pdf
12. G. Hammouri, B. Sunar, and Ç. K. Koç. Mobile phone aided operations system and method. US Patent Nr. 8,842,827. September 23, 2014.   pdf
13. M. Hubert, C. Walker, C. Minden, G. Hammouri, and Ç. K. Koç. Systems and methods for authorizing transactions via a digital device. US Patent Nr. 10,013,692. July 3, 2018.   pdf