Abstract
In 2013, Tao et al. introduced the ABC Simple Matrix Scheme for Encryption, a multivariate public key encryption scheme. The scheme boasts great efficiency in encryption and decryption, though it suffers from very large public keys. It was quickly noted that the original proposal, utilizing square matrices, suffered from a very bad decryption failure rate. As a consequence, the designers later published updated parameters, replacing the square matrices with rectangular matrices and altering other parameters to avoid the cryptanalysis of the original scheme presented in 2014 by Moody et al.
In this work we show that making the matrices rectangular, while decreasing the decryption failure rate, actually, and ironically, diminishes security. We show that the combinatorial rank methods employed in the original attack of Moody et al. can be enhanced by the same added degrees of freedom that reduce the decryption failure rate. Moreover, and quite interestingly, if the decryption failure rate is still reasonably high, as exhibited by the proposed parameters, we are able to mount a reaction attack to further enhance the combinatorial rank methods. To our knowledge this is the first instance of a reaction attack creating a significant advantage in this context.
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References
Cabarcas, D., Smith-Tone, D., Verbel, J.A.: Key recovery attack for ZHFE. In: Lange, T., Takagi, T. [13], pp. 289–308 (2017)
Cartor, R., Smith-Tone, D.: An updated security analysis of PFLASH. In: Lange, T., Takagi, T. [13], pp. 241–254 (2017)
Cartor, R., Smith-Tone, D.: EFLASH: a new multivariate encryption scheme. In: Cid, C., Jacobson Jr., M. (eds.) Selected Areas in Cryptography - SAC 2018. LNCS, vol. 11349, pp. 281–299. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-10970-7_13
Chen, M.-S., Yang, B.-Y., Smith-Tone, D.: Pflash - secure asymmetric signatures on smart cards. In: Lightweight Cryptography Workshop 2015 (2015). http://csrc.nist.gov/groups/ST/lwc-workshop2015/papers/session3-smith-tone-paper.pdf
Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005). https://doi.org/10.1007/11496137_12
Ding, J.: A new variant of the matsumoto-imai cryptosystem through perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 305–318. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24632-9_22
Ding, J., Petzoldt, A., Wang, L.: The cubic Simple Matrix encryption scheme. In: Mosca, M. [18], pp. 76–87 (2014)
Dubois, V., Fouque, P.-A., Stern, J.: Cryptanalysis of SFLASH with slightly modified parameters. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 264–275. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72540-4_15
Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_3
Fouque, P.-A., Granboulan, L., Stern, J.: Differential cryptanalysis for multivariate schemes. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 341–353. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_20
Ikematsu, Y., Perlner, R., Smith-Tone, D., Takagi, T., Vates, J.: HFERP - a new multivariate encryption scheme. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 396–416. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_19
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). https://doi.org/10.1007/3-540-48910-X_15
Lange, T., Takagi, T. (eds.): PQCrypto 2017. LNCS, vol. 10346. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6
Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Barstow, D., et al. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-45961-8_39
Moody, D., Perlner, R., Smith-Tone, D.: An asymptotically optimal structural attack on the ABC multivariate encryption scheme. In: Mosca, M. [18], pp. 180–196 (2014)
Moody, D., Perlner, R., Smith-Tone, D.: Key recovery attack on the cubic ABC Simple Matrix multivariate encryption scheme. In: Avanzi, R., Heys, H. (eds.) SAC 2016. LNCS, vol. 10532, pp. 543–558. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-69453-5_29
Moody, D., Perlner, R., Smith-Tone, D.: Improved attacks for characteristic-2 parameters of the cubic ABC Simple Matrix encryption scheme. In: Lange, T., Takagi, T. [13], pp. 255–271 (2017)
Mosca, M. (ed.): PQCrypto 2014. LNCS, vol. 8772. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11659-4
Patarin, J.: The oil and vinegar signature scheme. In: Dagstuhl Workshop on Cryptography, September 1997 (1997)
Patarin, J., Courtois, N., Goubin, L.: FLASH, a fast multivariate signature algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 298–307. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45353-9_22
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). https://doi.org/10.1007/3-540-44750-4_20
Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_4
Patarin, J., Courtois, N., Goubin, L.: QUARTZ, 128-bit long digital signatures. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 282–297. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45353-9_21
Perlner, R., Petzoldt, A., Smith-Tone, D.: Total break of the SRP encryption scheme. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 355–373. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_18
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). https://doi.org/10.1007/BFb0055733
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci. Stat. Comp. 26, 1484 (1997)
Smith-Tone, D., Verbel, J.: A key recovery attack for the extension field cancellation encryption scheme. In: concurrent submission to PQCrypto 2020 (2020)
Szepieniec, A., Ding, J., Preneel, B.: Extension field cancellation: a new central trapdoor for multivariate quadratic systems. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 182–196. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_12
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). https://doi.org/10.1007/978-3-642-38616-9_16
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)
The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.7) (2019). https://www.sagemath.org
Wolf, C., Braeken, A., Preneel, B.: On the security of stepwise triangular systems. Des. Codes Cryptogr. 40(3), 285–302 (2006)
Yang, B.-Y., Chen, J.-M.: Building secure tame-like multivariate public-key cryptosystems: the new TTS. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 518–531. Springer, Heidelberg (2005). https://doi.org/10.1007/11506157_43
Yasuda, T., Sakurai, K.: A multivariate encryption scheme with rainbow. In: Qing, S., Okamoto, E., Kim, K., Liu, D. (eds.) ICICS 2015. LNCS, vol. 9543, pp. 236–251. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29814-6_19
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Apon, D., Moody, D., Perlner, R., Smith-Tone, D., Verbel, J. (2020). Combinatorial Rank Attacks Against the Rectangular Simple Matrix Encryption Scheme. In: Ding, J., Tillich, JP. (eds) Post-Quantum Cryptography. PQCrypto 2020. Lecture Notes in Computer Science(), vol 12100. Springer, Cham. https://doi.org/10.1007/978-3-030-44223-1_17
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