Abstract
In this paper, we study some properties of semigroups with presentation \(\left\langle a,b, c\ ;\ a^p=b^s, b^r=c^v, c^u=a^q\right\rangle\). We will determine a complete rewriting system for the semigroup which could be useful for cryptosystems.
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References
Anshel, I., Anshel, M., Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6(3–4), 287–291 (1999)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1999)
Baumslag, G., Fine, B., Xu, X.: Cryptosystems using linear groups. Appl. Algebra Engrg. Commun. Comput. 17, 205–217 (2006)
Baumslag, G., Fine, B., Xu, X.: A proposed public key cryptosystem using the modular groups. Combinatorial group theory, discrete groups, and number theory. Cont. Math. 421, 35–43 (2006)
Cha, J.C., Ko, K.H., Lee, S.J., Cheon, J.H., Han J.W., Cheon, J.H.: An efficient implimentation of braid groups. In: ASIACRYPT 2001, volume 2248 of Lecture Notes in Computer Science, Springer, Berlin, 144–156 (2001)
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inform. Theory IT 22(6), 644–654 (1976)
ElGamal, T.: A public key crytosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31(4), 469–472 (1985)
Ko, K.H., Lee, S.J., Cheon, J.H., Han, J.W., Kang, J.S., Park, C.: New public-key cryptosystem using braid groups. Advances in Cryptology—CRYPTO 2000 (Santa Barbara, CA), volume 1880 of Lecture Notes in Computer Science. Springer, Berlin, 166–183 (2000)
Kropholler, P.H., Pride, S.J., Othman, W.A.M., Wong, K.B., Wong, P.C.: Properties of certain semigroups and their potential as platforms for cryptosystems. Semigroup Forum 81, 172–186 (2010)
Maze, G., Monico, C., Rosenthal, J.: Public key cryptography based on semigroup actions. Adv. Math. Commun. 1(4), 489–507 (2007)
Shpilrain, V., Zapata, G.: Combinatorial group theory and and public key cryptography. Appl. Algebra Engrg. Commun. Comput. 17, 291–302 (2006)
Shpilrain, V., Ushakov, A.: The conjugacy search problem in public key cryptography: unnecessary and insufficient. Appl. Algebra Engrg. Commun. Comput. 17, 285–289 (2006)
Shpilrain, V., Ushakov, A.: Thompson’s group and public key cryptography. volume 3531 of Lecture Notes in Computer Science. Springer Verlag, Berlin, 151–164 (2005)
Yamamura, A.: Public-key cryptosystems using the modular group. Public Key Cryptography, volume 1431 of Lecture Notes in Computer Science. Springer, Berlin, 203–216 (1998)
Acknowledgements
We would like to thank the referees for their comments. This project is supported by University of Malaya IIRG0019C.
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Asri, M.S.M., Othman, W.A.M. & Wong, K.B. Rewriting system of certain semigroups with three generators. AAECC 34, 469–487 (2023). https://doi.org/10.1007/s00200-021-00506-7
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DOI: https://doi.org/10.1007/s00200-021-00506-7