November  2016, 10(4): 861-870. doi: 10.3934/amc.2016046

Public key protocols over the ring $E_{p}^{(m)}$

1. 

Departament de Matemàtiques, Universitat d'Alacant, Carretera de Sant Vicent del Raspeig, s/n, E-03690 Sant Vicent del Raspeig, Alacant, Spain

2. 

Departamento de Matemáticas, Universidad de Almería, Carretera de Sacramento, s/n, Almería, 04120, Spain

Received  March 2015 Revised  June 2016 Published  November 2016

In this paper we use the nonrepresentable ring $E_{p}^{(m)}$ to introduce public key cryptosystems in noncommutative settings and based on the Semigroup Action Problem and the Decomposition Problem respectively.
Citation: Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046
References:
[1]

I. Anshel, M. Anshel, B. Fisher and D. Goldfeld, New key agreement protocols in braid group cryptography, in Topics Crypt. - CT-RSA 2001 (ed. D. Naccache), Springer, 2001, 13-27. doi: 10.1007/3-540-45353-9_2.  Google Scholar

[2]

I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett., 6 (1999), 1-5. doi: 10.4310/MRL.1999.v6.n3.a3.  Google Scholar

[3]

G. M. Bergman, Some examples in PI ring theory, Israel J. Math., 18 (1974), 257-277.  Google Scholar

[4]

J.-J. Climent, F. Ferrández, J.-F. Vicent and A. Zamora, A nonlinear elliptic curve cryptosystem based on matrices, Appl. Math. Comput., 174 (2006), 150-164. doi: 10.1016/j.amc.2005.03.032.  Google Scholar

[5]

J.-J. Climent, P. R. Navarro and L. Tortosa, Key exchange protocols over noncommutative rings. The case of End$(\mathbb Z_p\times\mathbb Z_{p^2})$, Proc. 11th Int. Conf. Comput. Math. Methods Sci. Engin. (CMMSE 2011), 2011, 357-364. doi: 10.1080/00207160.2012.696105.  Google Scholar

[6]

J.-J. Climent, P. R. Navarro and L. Tortosa, On the arithmetic of the endomorphisms ring End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Appl. Algebr. Eng, Comm., 22 (2011), 91-108. doi: 10.1007/s00200-011-0138-4.  Google Scholar

[7]

J.-J. Climent, P. R. Navarro and L. Tortosa, Key exchange protocols over noncommutative rings. The case of End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Int. J. Comput. Math., 89(13-14) (2012), 1753-1763. doi: 10.1080/00207160.2012.696105.  Google Scholar

[8]

J.-J. Climent, P. R. Navarro and L. Tortosa, An extension of the noncommutative Bergman's ring with a large number of noninvertible elements, Appl. Algebr. Eng. Comm., 25 (2014), 347-361. doi: 10.1007/s00200-014-0231-6.  Google Scholar

[9]

W. D. Diffie and M. E. Hellman, New Directions in Cryptography, IEEE Trans. Inform. Theory, 22 (1976), 644-654.  Google Scholar

[10]

T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory, 31 (1985), 469-472. doi: 10.1109/TIT.1985.1057074.  Google Scholar

[11]

A. A. Kamal and A. M. Youssef, Cryptanalysis of a key exchange protocol based on the endomorphisms ring End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Appl. Algebr. Eng. Comm., 23) (2012), 143-149. doi: 10.1007/s00200-012-0170-z.  Google Scholar

[12]

K. H. Ko, J. W. Lee and T. Thomas, Towards generating secure keys for braid cryptography, Design. Code. Cryptogr., 45 (2007), 317-333. doi: 10.1007/s10623-007-9123-0.  Google Scholar

[13]

K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J.-S. Kang and C. Park, New public-key cryptosystem using braid groups, Adv. Crypt. - CRYPTO 2000 (ed. M. Bellare), Springer, 2000, 166-183. doi: 10.1007/3-540-44598-6_10.  Google Scholar

[14]

A. Mahalanobis, A simple generalization of the ElGamal cryptosystem to non-abelian groups, Commun. Algebra, 36 (2008), 3878-3889. doi: 10.1080/00927870802160883.  Google Scholar

[15]

A. Mahalanobis, The Diffie-Hellman key exchange and non-abelian nilpotent groups, Israel J. Math., 165 (2008), 161-187. doi: 10.1007/s11856-008-1008-z.  Google Scholar

[16]

A. Mahalanobis, Are matrices useful in public-key cryptography?, Int. Math. Forum, 8 (2013), 1939-1953. doi: 10.12988/imf.2013.310187.  Google Scholar

[17]

A. Mahalanobis, The MOR cryptosystem and finite $p$-groups, in Algorithmic Problems of Group Theory, Their Complexity, and Applications to Cryptography, Amer. Math. Soc., 2015, 81-95. doi: 10.1090/conm/633/12653.  Google Scholar

[18]

G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun., 1 (2007), 489-507. doi: 10.3934/amc.2007.1.489.  Google Scholar

[19]

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, 1996. Google Scholar

[20]

A. G. Myasnikov, V. Shpilrain and A. Ushakov, Group-Based Cryptography, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[21]

S.-H. Paeng, K.-C. Ha, J. H. Kim, S. Chee and C. Park, New public key cryptosystem using finite non abelian groups, in Adv. Crypt. - CRYPTO 2001 (ed. J. Kilian), Springer, 2001, 470-485. doi: 10.1007/3-540-44647-8_28.  Google Scholar

[22]

R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (1978), 120-126. doi: 10.1145/359340.359342.  Google Scholar

[23]

E. Sakalauskas and T. Burba, Basic semigroup primitive for cryptographic session key exchange protocol (SKEP), Inf. Technol. Control, 28 (2003), 76-80. Google Scholar

[24]

V. Shpilrain and A. Ushakov, A new key exchange protocol based on the decomposition problem, Contemp. Math., Amer. Math. Soc., 418 (2006), 161-167. doi: 10.1090/conm/418/07954.  Google Scholar

[25]

V. Shpilrain and G. Zapata, Combinatorial group theory and public key cryptography, Appl. Algebr. Eng. Comm., 17 (2006), 291-302. doi: 10.1007/s00200-006-0006-9.  Google Scholar

[26]

V. M. Sidelnikov, M. A. Cherepnev and V. V. Yashchenko, Systems of open distribution of keys on the basis of noncommutative semigroups, Russ. Ac. Sc. Doklady Math., 48 (1994), 384-386.  Google Scholar

[27]

E. Stickel, A new method for exchanging secret keys, in Proc. 3rd Int. Conf. Inform. Techn. Appl. (ICITA'05), Sidney, 2005, 426-430. Google Scholar

[28]

D. R. Stinson, Cryptography. Theory and Practice, CRC Press, Boca Raton, 1995.  Google Scholar

[29]

T. Thomas and A. K. Lal, A zero-knowledge undeniable signature scheme in non-abelian group setting, Int. J. Netw. Secur., 6 (2008), 265-269. Google Scholar

[30]

H. Yoo, S. Hong, S. Lee, J. Lim, O. Yi and M. Sung, A proposal of a new public key cryptosystem using matrices over a ring, in Information Security and Privacy, Springer, 2000, 41-48. Google Scholar

show all references

References:
[1]

I. Anshel, M. Anshel, B. Fisher and D. Goldfeld, New key agreement protocols in braid group cryptography, in Topics Crypt. - CT-RSA 2001 (ed. D. Naccache), Springer, 2001, 13-27. doi: 10.1007/3-540-45353-9_2.  Google Scholar

[2]

I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett., 6 (1999), 1-5. doi: 10.4310/MRL.1999.v6.n3.a3.  Google Scholar

[3]

G. M. Bergman, Some examples in PI ring theory, Israel J. Math., 18 (1974), 257-277.  Google Scholar

[4]

J.-J. Climent, F. Ferrández, J.-F. Vicent and A. Zamora, A nonlinear elliptic curve cryptosystem based on matrices, Appl. Math. Comput., 174 (2006), 150-164. doi: 10.1016/j.amc.2005.03.032.  Google Scholar

[5]

J.-J. Climent, P. R. Navarro and L. Tortosa, Key exchange protocols over noncommutative rings. The case of End$(\mathbb Z_p\times\mathbb Z_{p^2})$, Proc. 11th Int. Conf. Comput. Math. Methods Sci. Engin. (CMMSE 2011), 2011, 357-364. doi: 10.1080/00207160.2012.696105.  Google Scholar

[6]

J.-J. Climent, P. R. Navarro and L. Tortosa, On the arithmetic of the endomorphisms ring End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Appl. Algebr. Eng, Comm., 22 (2011), 91-108. doi: 10.1007/s00200-011-0138-4.  Google Scholar

[7]

J.-J. Climent, P. R. Navarro and L. Tortosa, Key exchange protocols over noncommutative rings. The case of End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Int. J. Comput. Math., 89(13-14) (2012), 1753-1763. doi: 10.1080/00207160.2012.696105.  Google Scholar

[8]

J.-J. Climent, P. R. Navarro and L. Tortosa, An extension of the noncommutative Bergman's ring with a large number of noninvertible elements, Appl. Algebr. Eng. Comm., 25 (2014), 347-361. doi: 10.1007/s00200-014-0231-6.  Google Scholar

[9]

W. D. Diffie and M. E. Hellman, New Directions in Cryptography, IEEE Trans. Inform. Theory, 22 (1976), 644-654.  Google Scholar

[10]

T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory, 31 (1985), 469-472. doi: 10.1109/TIT.1985.1057074.  Google Scholar

[11]

A. A. Kamal and A. M. Youssef, Cryptanalysis of a key exchange protocol based on the endomorphisms ring End $(\mathbb Z_p\times\mathbb Z_{p^2})$, Appl. Algebr. Eng. Comm., 23) (2012), 143-149. doi: 10.1007/s00200-012-0170-z.  Google Scholar

[12]

K. H. Ko, J. W. Lee and T. Thomas, Towards generating secure keys for braid cryptography, Design. Code. Cryptogr., 45 (2007), 317-333. doi: 10.1007/s10623-007-9123-0.  Google Scholar

[13]

K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J.-S. Kang and C. Park, New public-key cryptosystem using braid groups, Adv. Crypt. - CRYPTO 2000 (ed. M. Bellare), Springer, 2000, 166-183. doi: 10.1007/3-540-44598-6_10.  Google Scholar

[14]

A. Mahalanobis, A simple generalization of the ElGamal cryptosystem to non-abelian groups, Commun. Algebra, 36 (2008), 3878-3889. doi: 10.1080/00927870802160883.  Google Scholar

[15]

A. Mahalanobis, The Diffie-Hellman key exchange and non-abelian nilpotent groups, Israel J. Math., 165 (2008), 161-187. doi: 10.1007/s11856-008-1008-z.  Google Scholar

[16]

A. Mahalanobis, Are matrices useful in public-key cryptography?, Int. Math. Forum, 8 (2013), 1939-1953. doi: 10.12988/imf.2013.310187.  Google Scholar

[17]

A. Mahalanobis, The MOR cryptosystem and finite $p$-groups, in Algorithmic Problems of Group Theory, Their Complexity, and Applications to Cryptography, Amer. Math. Soc., 2015, 81-95. doi: 10.1090/conm/633/12653.  Google Scholar

[18]

G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun., 1 (2007), 489-507. doi: 10.3934/amc.2007.1.489.  Google Scholar

[19]

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, 1996. Google Scholar

[20]

A. G. Myasnikov, V. Shpilrain and A. Ushakov, Group-Based Cryptography, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[21]

S.-H. Paeng, K.-C. Ha, J. H. Kim, S. Chee and C. Park, New public key cryptosystem using finite non abelian groups, in Adv. Crypt. - CRYPTO 2001 (ed. J. Kilian), Springer, 2001, 470-485. doi: 10.1007/3-540-44647-8_28.  Google Scholar

[22]

R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (1978), 120-126. doi: 10.1145/359340.359342.  Google Scholar

[23]

E. Sakalauskas and T. Burba, Basic semigroup primitive for cryptographic session key exchange protocol (SKEP), Inf. Technol. Control, 28 (2003), 76-80. Google Scholar

[24]

V. Shpilrain and A. Ushakov, A new key exchange protocol based on the decomposition problem, Contemp. Math., Amer. Math. Soc., 418 (2006), 161-167. doi: 10.1090/conm/418/07954.  Google Scholar

[25]

V. Shpilrain and G. Zapata, Combinatorial group theory and public key cryptography, Appl. Algebr. Eng. Comm., 17 (2006), 291-302. doi: 10.1007/s00200-006-0006-9.  Google Scholar

[26]

V. M. Sidelnikov, M. A. Cherepnev and V. V. Yashchenko, Systems of open distribution of keys on the basis of noncommutative semigroups, Russ. Ac. Sc. Doklady Math., 48 (1994), 384-386.  Google Scholar

[27]

E. Stickel, A new method for exchanging secret keys, in Proc. 3rd Int. Conf. Inform. Techn. Appl. (ICITA'05), Sidney, 2005, 426-430. Google Scholar

[28]

D. R. Stinson, Cryptography. Theory and Practice, CRC Press, Boca Raton, 1995.  Google Scholar

[29]

T. Thomas and A. K. Lal, A zero-knowledge undeniable signature scheme in non-abelian group setting, Int. J. Netw. Secur., 6 (2008), 265-269. Google Scholar

[30]

H. Yoo, S. Hong, S. Lee, J. Lim, O. Yi and M. Sung, A proposal of a new public key cryptosystem using matrices over a ring, in Information Security and Privacy, Springer, 2000, 41-48. Google Scholar

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