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2016 | OriginalPaper | Chapter

A Multivariate Encryption Scheme with Rainbow

Authors : Takanori Yasuda, Kouichi Sakurai

Published in: Information and Communications Security

Publisher: Springer International Publishing

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Abstract

Multivariate Public Key Cryptosystems (MPKC) are a candidate of post-quantum cryptography. The MPKC signature scheme Rainbow is endowed of efficient signature generation and verification, while no major attack has been reported so far. In this paper, we propose a MPKC encryption scheme based on Rainbow. The public key of Rainbow is a surjective polynomial map, whereas the encryption scheme requires an injective polynomial map. We explain how to change the public key of Rainbow to an injective map.

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Appendix
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Literature
1.
go back to reference Bardet, M., Faugére, J.-C., Salvy, B.: On the complexity of gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of International Conference on Polynomial System Solving (ICPSS), pp. 71–75 (2004) Bardet, M., Faugére, J.-C., Salvy, B.: On the complexity of gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of International Conference on Polynomial System Solving (ICPSS), pp. 71–75 (2004)
2.
go back to reference Bernstein, D.J., Buchmann, J., Dahmen, E.: Post Quantum Cryptography. Springer, Heidelberg (2009)MATH Bernstein, D.J., Buchmann, J., Dahmen, E.: Post Quantum Cryptography. Springer, Heidelberg (2009)MATH
3.
go back to reference Berger, T.P., Cayrel, P.-L., Gaborit, P., Otmani, A.: Reducing key length of the McEliece cryptosystem. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 77–97. Springer, Heidelberg (2009) Berger, T.P., Cayrel, P.-L., Gaborit, P., Otmani, A.: Reducing key length of the McEliece cryptosystem. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 77–97. Springer, Heidelberg (2009)
4.
go back to reference Bettale, L., Faugére, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Crypt. 3(3), 177–197 (2010)MathSciNetMATH Bettale, L., Faugére, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Crypt. 3(3), 177–197 (2010)MathSciNetMATH
5.
go back to reference Billet, O., Gilbert, H.: Cryptanalysis of rainbow. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 336–347. Springer, Heidelberg (2006) Billet, O., Gilbert, H.: Cryptanalysis of rainbow. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 336–347. Springer, Heidelberg (2006)
6.
go back to reference Billet, O., Macario-Rat, G.: Cryptanalysis of the square cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 451–468. Springer, Heidelberg (2009) Billet, O., Macario-Rat, G.: Cryptanalysis of the square cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 451–468. Springer, Heidelberg (2009)
7.
go back to reference Clough, C., Baena, J., Ding, J., Yang, B.-Y., Chen, M.: Square, a new multivariate encryption scheme. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 252–264. Springer, Heidelberg (2009) Clough, C., Baena, J., Ding, J., Yang, B.-Y., Chen, M.: Square, a new multivariate encryption scheme. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 252–264. Springer, Heidelberg (2009)
8.
go back to reference Clough, C.L., Ding, J.: Secure variants of the square encryption scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 153–164. Springer, Heidelberg (2010) Clough, C.L., Ding, J.: Secure variants of the square encryption scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 153–164. Springer, Heidelberg (2010)
9.
go back to reference Courtois, N.T., Daum, M., Felke, P.: On the security of HFE, HFEv- and Quartz. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 337–350. Springer, Heidelberg (2002) Courtois, N.T., Daum, M., Felke, P.: On the security of HFE, HFEv- and Quartz. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 337–350. Springer, Heidelberg (2002)
10.
go back to reference Chen, J.M., Yang, B.-Y.: A more secure and efficacious TTS signature scheme. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 320–338. Springer, Heidelberg (2004) Chen, J.M., Yang, B.-Y.: A more secure and efficacious TTS signature scheme. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 320–338. Springer, Heidelberg (2004)
11.
go back to reference Ding, J., Clough, C., Araujo, R.: Inverting square systems algebraically is exponential. Finite Fields Appl. 26, 32–48 (2014)MathSciNetMATH Ding, J., Clough, C., Araujo, R.: Inverting square systems algebraically is exponential. Finite Fields Appl. 26, 32–48 (2014)MathSciNetMATH
12.
go back to reference Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate Public Key Cryptosystems. Advances in Information Security, vol. 25. Springer, New york (2006)MATH Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate Public Key Cryptosystems. Advances in Information Security, vol. 25. Springer, New york (2006)MATH
13.
go back to reference Dubois, V., Fouque, P.-A., Shamir, A., Stern, J.: Practical cryptanalysis of SFLASH. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 1–12. Springer, Heidelberg (2007) Dubois, V., Fouque, P.-A., Shamir, A., Stern, J.: Practical cryptanalysis of SFLASH. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 1–12. Springer, Heidelberg (2007)
14.
go back to reference Ding, J., Petzoldt, A., Wang, L.: The cubic simple matrix encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 76–87. Springer, Heidelberg (2014) Ding, J., Petzoldt, A., Wang, L.: The cubic simple matrix encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 76–87. Springer, Heidelberg (2014)
15.
go back to reference Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005) Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)
16.
go back to reference Ding, J., Schmidt, D., Yin, Z.: Cryptanalysis of the new TTS scheme in CHES 2004. Int. J. Inf. Secur. 5(4), 231–240 (2006)MATH Ding, J., Schmidt, D., Yin, Z.: Cryptanalysis of the new TTS scheme in CHES 2004. Int. J. Inf. Secur. 5(4), 231–240 (2006)MATH
17.
go back to reference Ding, J., Wolf, C., Yang, B.-Y.: \(\ell \)-invertible cycles for multivariate quadratic (MQ) public key cryptography. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 266–281. Springer, Heidelberg (2007)MATH Ding, J., Wolf, C., Yang, B.-Y.: \(\ell \)-invertible cycles for multivariate quadratic (MQ) public key cryptography. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 266–281. Springer, Heidelberg (2007)MATH
18.
go back to reference Ding, J., Yang, B.-Y., Chen, C.-H.O., Chen, M.-S., Cheng, C.-M.: New differential-algebraic attacks and reparametrization of rainbow. In: Bellovin, S.M., Gennaro, R., Keromytis, A.D., Yung, M. (eds.) ACNS 2008. LNCS, vol. 5037, pp. 242–257. Springer, Heidelberg (2008) Ding, J., Yang, B.-Y., Chen, C.-H.O., Chen, M.-S., Cheng, C.-M.: New differential-algebraic attacks and reparametrization of rainbow. In: Bellovin, S.M., Gennaro, R., Keromytis, A.D., Yung, M. (eds.) ACNS 2008. LNCS, vol. 5037, pp. 242–257. Springer, Heidelberg (2008)
19.
go back to reference Faugére, J.-C.: A new efficient algorithm for computing Gröbner basis (\(F_4\)). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)MathSciNetMATH Faugére, J.-C.: A new efficient algorithm for computing Gröbner basis (\(F_4\)). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)MathSciNetMATH
20.
go back to reference Faugére, J.-C.: A new efficient algorithm for computing Gröbner basis without to zero (\(F_5\)). In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 75–83 (2002) Faugére, J.-C.: A new efficient algorithm for computing Gröbner basis without to zero (\(F_5\)). In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 75–83 (2002)
21.
go back to reference Fouque, P.-A., Macario-Rat, G., Perret, L., Stern, J.: Total break of the \(\ell \)-IC signature scheme. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 1–17. Springer, Heidelberg (2008) Fouque, P.-A., Macario-Rat, G., Perret, L., Stern, J.: Total break of the \(\ell \)-IC signature scheme. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 1–17. Springer, Heidelberg (2008)
22.
go back to reference Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000) Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)
23.
go back to reference Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster point multiplication on elliptic curves with efficient endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001) Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster point multiplication on elliptic curves with efficient endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)
24.
go back to reference Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. J. Crypt. 24(3), 446–469 (2011)MathSciNetMATH Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. J. Crypt. 24(3), 446–469 (2011)MathSciNetMATH
25.
go back to reference Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998) Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)
26.
go back to reference Kipnis, A., Patarin, J., Goubin, L.: Unbalanced oil and vinegar signature schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999) Kipnis, A., Patarin, J., Goubin, L.: Unbalanced oil and vinegar signature schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)
27.
go back to reference Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999) Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999)
28.
go back to reference Kipnis, A., Shamir, A.: Cryptanalysis of the oil and vinegar signature scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998) Kipnis, A., Shamir, A.: Cryptanalysis of the oil and vinegar signature scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998)
29.
go back to reference Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988) Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)
30.
go back to reference Moh, T.-T.: A fast public key system with signature ans master key functions. In: CrypTEC 1999, pp. 63–69 (1999) Moh, T.-T.: A fast public key system with signature ans master key functions. In: CrypTEC 1999, pp. 63–69 (1999)
31.
go back to reference Moh, T.-T.: A public key system with signature and master key functions. Commun. Algebra 27(5), 2207–2222 (1999)MathSciNetMATH Moh, T.-T.: A public key system with signature and master key functions. Commun. Algebra 27(5), 2207–2222 (1999)MathSciNetMATH
32.
go back to reference Möller, B.: Algorithms for multi-exponentiation. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 165–180. Springer, Heidelberg (2001) Möller, B.: Algorithms for multi-exponentiation. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 165–180. Springer, Heidelberg (2001)
33.
go back to reference Patarin, J.: Cryptanalysis of the matsumoto and imai public key scheme of Eurocrypt ’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995) Patarin, J.: Cryptanalysis of the matsumoto and imai public key scheme of Eurocrypt ’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)
34.
go back to reference Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996) Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)
35.
go back to reference Patarin, J., Goubin, L., Courtois, N.T.: \(C_-+^*\) and HM: variations around two schemes of T. Matsumoto and H. Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998) Patarin, J., Goubin, L., Courtois, N.T.: \(C_-+^*\) and HM: variations around two schemes of T. Matsumoto and H. Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998)
36.
go back to reference Patarin, J., Courtois, N.T., Goubin, L.: FLASH, a fast multivariate signature algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, p. 298. Springer, Heidelberg (2001) Patarin, J., Courtois, N.T., Goubin, L.: FLASH, a fast multivariate signature algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, p. 298. Springer, Heidelberg (2001)
37.
go back to reference Petzoldt, A., Bulygin, S., Buchmann, J.: Selecting parameters for the rainbow signature scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 218–240. Springer, Heidelberg (2010) Petzoldt, A., Bulygin, S., Buchmann, J.: Selecting parameters for the rainbow signature scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 218–240. Springer, Heidelberg (2010)
38.
go back to reference Petzoldt, A., Bulygin, S., Buchmann, J.: CyclicRainbow – a multivariate signature scheme with a partially cyclic public key. In: Gong, G., Gupta, K.C. (eds.) INDOCRYPT 2010. LNCS, vol. 6498, pp. 33–48. Springer, Heidelberg (2010) Petzoldt, A., Bulygin, S., Buchmann, J.: CyclicRainbow – a multivariate signature scheme with a partially cyclic public key. In: Gong, G., Gupta, K.C. (eds.) INDOCRYPT 2010. LNCS, vol. 6498, pp. 33–48. Springer, Heidelberg (2010)
39.
go back to reference Porras, J., Baena, J., Ding, J.: ZHFE, a new multivariate public key encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 229–245. Springer, Heidelberg (2014) Porras, J., Baena, J., Ding, J.: ZHFE, a new multivariate public key encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 229–245. Springer, Heidelberg (2014)
40.
go back to reference Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., Kachisa, E.J.: On the final exponentiation for calculating pairings on ordinary elliptic curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)MATH Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., Kachisa, E.J.: On the final exponentiation for calculating pairings on ordinary elliptic curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)MATH
41.
go back to reference Tao, C., Diene, A., Tang, S., Ding, J.: Simple matrix scheme for encryption. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 231–242. Springer, Heidelberg (2013) Tao, C., Diene, A., Tang, S., Ding, J.: Simple matrix scheme for encryption. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 231–242. Springer, Heidelberg (2013)
42.
go back to reference Tao, C., Xiang, H., Petzoldt, A., Ding, J.: Simple matrix - a multivariate public key cryptosystem (MPKC) for encryption. Finite Fields Appl. 35, 352–368 (2015)MathSciNetMATH Tao, C., Xiang, H., Petzoldt, A., Ding, J.: Simple matrix - a multivariate public key cryptosystem (MPKC) for encryption. Finite Fields Appl. 35, 352–368 (2015)MathSciNetMATH
43.
go back to reference Thomae, E., Wolf, C.: Roots of square: cryptanalysis of double-layer square and square+. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 83–97. Springer, Heidelberg (2011) Thomae, E., Wolf, C.: Roots of square: cryptanalysis of double-layer square and square+. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 83–97. Springer, Heidelberg (2011)
46.
go back to reference Yang, B.-Y., Chen, J.-M.: All in the XL family: theory and practice. In: Park, C., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–86. Springer, Heidelberg (2005) Yang, B.-Y., Chen, J.-M.: All in the XL family: theory and practice. In: Park, C., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–86. Springer, Heidelberg (2005)
Metadata
Title
A Multivariate Encryption Scheme with Rainbow
Authors
Takanori Yasuda
Kouichi Sakurai
Copyright Year
2016
Publisher
Springer International Publishing
DOI
https://doi.org/10.1007/978-3-319-29814-6_19

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