Abstract
We consider some attacks on multi-prime RSA (MPRSA) with a modulus N = p 1 p 2…p r (r ≥ 3). It is believed that the small private exponent attack on the MPRSA is less effective than that on RSA (see Hinek et al.’s work at SAC 2003), which means that one can use a smaller private exponent in the MPRSA than that in the original RSA. However, our attacks show that private exponents which are significantly beyond Hinek’s bound may be insecure when the prime difference Δ (Δ = p r − p 1 = N γ, 0 < γ < 1/r, suppose p 1 < p 2 < ⋯ < p r ) is small. By exploring the relation between φ(N) and its upper bound, our proposed small private exponent attack can make full use of the benefit brought by small prime difference. It is shown that the MPRSA is insecure when \(\delta<1-\sqrt{1+\gamma-2/r}\), where δ is the exponential of the private exponent d with base N, i.e., d = N δ. This result is a perfect extension of the best known small private exponent attack. We also present a Fermat-like factoring attack on the MPRSA which can directly factor the modulus N when \(\Delta<N^{1/r^2}\). These results surpass those of Bahig et al. (ICICS 2012) and the attacks are experimentally proved effective in practice.
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References
ANSI X9.31-1998, Digital signatures using reversible public key cryptography for the financial services industry (rDSA), American National Standards Institute (1998)
Bahig, H.M., Bhery, A., Nassr, D.I.: Cryptanalysis of multi-prime RSA with small prime difference. In: Chim, T.W., Yuen, T.H. (eds.) ICICS 2012. LNCS, vol. 7618, pp. 33–44. Springer, Heidelberg (2012)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N 0.292. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 1–11. Springer, Heidelberg (1999)
Boneh, D., Shacham, H.: Fast variants of RSA. CryptoBytes 5(1), 1–9 (2002)
Ciet, M., Koeune, F., Laguillaumie, F., Quisquater, J.-J.: Short private exponent attacks on fast variants of RSA. UCL Crypto Group Technical Report Series CG-2002/4, University Catholique de Louvain (2002)
Collins, T., Hopkins, D., Langford, S., Sabin, M.: Public key cryptographic apparatus and method. US patent #5, 848, 149 (1997)
Compaq Computer Corperation: Cryptography using Compaq multiprime technology in a parallel processing environment (2000)
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. Journal of Cryptology 10, 233–260 (1997)
Herrmann, M., May, A.: Maximizing small root bounds by linearization and applications to small secret exponent RSA. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 53–69. Springer, Heidelberg (2010)
Hinek, M.J., Low, M.K., Teske, E.: On some attacks on multiprime RSA. In: Nyberg, K., Heys, H.M. (eds.) SAC 2002. LNCS, vol. 2595, pp. 385–404. Springer, Heidelberg (2003)
Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)
Kühnel, M.: RSA vulnerabilities with small prime difference. In: Armknecht, F., Lucks, S. (eds.) WEWoRC 2011. LNCS, vol. 7242, pp. 122–136. Springer, Heidelberg (2012)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 513–534 (1982)
May, A.: Secret exponent attacks on RSA-type schemes with moduli N = p r q. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 218–230. Springer, Heidelberg (2004)
National Institute of Standards and Technology: Digital signature standard, FIPS Publication 186-3 (2009), http://www.nist.gov/cmvp
Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21, 120–126 (1978)
RSA Laboratories: Public Key Cryptography Standards PKCS #1 v2.1: RSA cryptography standard (2001)
Shoup, V.: NTL number theory C++ library, http://www.shoup.net/ntl
de Weger, B.: Cryptanalysis of RSA with small prime difference. Applicable Algebra in Engineering, Communication and Computing 13, 17–28 (2002)
Wiener, M.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36, 553–558 (1990)
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Zhang, H., Takagi, T. (2013). Attacks on Multi-Prime RSA with Small Prime Difference. In: Boyd, C., Simpson, L. (eds) Information Security and Privacy. ACISP 2013. Lecture Notes in Computer Science, vol 7959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39059-3_4
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DOI: https://doi.org/10.1007/978-3-642-39059-3_4
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