# A Reduction Attack on Algebraic Surface Public-Key Cryptosystems

## Abstract

An algebraic surface public-key cryptosystem was developed by Akiyama and Goto. Its security is based on a decision randomizing polynomial problem which is related to a problem of finding sections on fibered algebraic surfaces which can be reduced to solving a multivariate equation system known to be NP-complete. In the case that the defining equation of a surface used for public-key is in a certain form, Uchiyama and Tokunaga succeeded in attacking in the sense of getting plain texts from corresponding ciphertexts using reductions efficiently without solving section finding problem. In this paper, two algorithms applicable to all cases are suggested. One is the generalization of Uchiyama-Tokunaga’s attack from polynomial ring over Open image in new window to polynomial ring over rational function field, and the other takes advantages of Gröbner base techniques so as to deal with in the polynomial ring over Open image in new window .

## Keywords

Normal Form Polynomial Ring Base Technique Algebraic Surface Plain Text## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Akiyama, A., Goto, Y.: A Construction of an Algebraic Surface Public-key Cryptosystem. In: CD-ROM 2E4-3, Symposium on Cryptography and Information Security (SCIS 2005), January 2005, pp. 925–930 (2005)Google Scholar
- 2.Algebraic surface public key cryptosystem. opened to the general public at website (February 2005), http://www.toshiba.co.jp/rdc/rd/topics_e_05.htm#050206, http://www.toshiba.co.jp/rdc/rd/detail_j/0502_06.htm
- 3.Akiyama, A., Goto, Y.: A Security Analysis for a Public-key Cryptosystem using Algebraic Surfaces. In: CD-ROM 2A3-1, SCIS 2006 (January 2006)Google Scholar
- 4.Akiyama, K., Goto, Y.: A Public-key Cryptosystem using Algebraic Surfaces. In: Workshop Record of the International Workshop on Post-Quantum Cryptography (PQCrypto 2006), May 2006, pp. 119–138 (2006)Google Scholar
- 5.Uchiyama, S., Tokunaga, H.: On the Security of the Algebraic Surface Public-Key Cryptosystems. In: CD-ROM 2C1-2, SCIS 2007 (January 2007) (written in Japanese)Google Scholar
- 6.Cryptography Research and Evaluation Committees : CRYPTREC Report 2006, Report of the Cryptographic Technique Monitoring Subcommittee (March 2007), http://www2.nict.go.jp/y/y213/cryptrec_publicity/c06_wat_final.pdf
- 7.Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algoriths: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edn. Springer, HeidelbergGoogle Scholar
- 8.Iwami, M.: A Reduction Attack on Algebraic Surface Public-Key Cryptosystems. Workshop of Research Institute for Mathematical Sciences (RIMS) Kyoto University, New development of research on Computer Algebra (held on July 4-6 2007) RIMS Kokyuroku 1572, pp.114–123 (November 2007) (written in Japanese)Google Scholar