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New number-theoretic cryptographic primitives. (English) Zbl 1465.94059

It is known that factoring unbalanced integers, e.g. integers of the form \(n = {p^r}q\), \(p\), \(q\)-primes is easy if \(r\) is large, but for small values of \(r\) the task seems to be as hard as in the general case. Thus, a new family of one-way functions and signature schemes based on this fact is introduced, e.g. for signing a document signer generates multiple moduli \({n_i} = p_i^2{q_i}\) using secret primes \({p_i}\), \({q_i}\). The signature is a bounded-size prime whose Jacobi symbols with respect to the \({n_i}\)s match the message digest.

MSC:

94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11A15 Power residues, reciprocity
11R18 Cyclotomic extensions

Software:

ESIGN
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Bach and J. Shallit, Algorithmic Number Theory. Vol. 1: Efficient Algorithms, MIT Press, Cambridge, 1996. · Zbl 0873.11070
[2] M. Bellare and P. Rogaway, Random oracles are practical: A paradigm for designing efficient protocols, ACM Conference on Computer and Communications Security, ACM Press, New York (1993), 62-73.
[3] D. Boneh, G. Durfee and N. Howgrave-Graham, Factoring N = p^rq for large r, Advances in Cryptology—CRYPTO ’99, Lecture Notes in Comput. Sci. 1666, Springer, Berlin (1999), 326-337. · Zbl 1007.11077
[4] P. C. Caranay and R. Scheidler, An efficient seventh power residue symbol algorithm, Int. J. Number Theory 6 (2010), no. 8, 1831-1853. · Zbl 1242.11079
[5] H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993. · Zbl 0786.11071
[6] I. B. Damgård, On the randomness of Legendre and Jacobi sequences, Advances in Cryptology—CRYPTO’88, Lecture Notes in Comput. Sci. 403, Springer, Berlin (1990), 163-172. · Zbl 0719.94013
[7] I. B. Damgård and G. S. Frandsen, Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers, J. Symbolic Comput. 39 (2005), no. 6, 643-652. · Zbl 1156.11346
[8] H. Davenport, On the distribution of quadratic residues (mod p), J. Lond. Math. Soc. 6 (1931), no. 1, 49-54.
[9] H. Davenport, On the distribution of quadratic residues (mod p). II, J. Lond. Math. Soc. 8 (1933), no. 1, 46-52.
[10] W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory IT-22 (1976), no. 6, 644-654. · Zbl 0435.94018
[11] C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem. Applications in Computing, Coding, Cryptography, World Scientific, River Edge, 1996. · Zbl 0907.11002
[12] A. Fiat and A. Shamir, How to prove yourself: Practical solutions to identification and signature problems, Advances in Cryptology—CRYPTO’86, Lecture Notes in Comput. Sci. 263, Springer, Berlin (1987), 186-194. · Zbl 0636.94012
[13] A. Fujioka, T. Okamoto and S. Miyaguchi, ESIGN: An efficient digital signature implementation for smart cards, Advances in Cryptology—EUROCRYPT’91, Lecture Notes in Comput. Sci. 547, Springer, Berlin (1991), 446-457. · Zbl 0825.94184
[14] O. Goldreich, Foundations of Cryptography. Basic Tools, Cambridge University, Cambridge, 2001. · Zbl 1007.94016
[15] S. Goldwasser, S. Micali and R. L. Rivest, A digital signature scheme secure against adaptive chosen-message attacks. Special issue on cryptography, SIAM J. Comput. 17 1988, no. 2, 281-308. · Zbl 0644.94012
[16] L. Goubin, C. Mauduit and A. Sárközy, Construction of large families of pseudorandom binary sequences, J. Number Theory 106 (2004), no. 1, 56-69. · Zbl 1049.11089
[17] L. Granboulan, How to repair ESIGN, Security in Communication Networks—SCN 2002, Lecture Notes in Comput. Sci. 2576, Springer, Berlin (2003), 234-240. · Zbl 1022.68537
[18] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Grad. Texts in Math. 84, Springer, New York, 1990. · Zbl 0712.11001
[19] J. Katz, Digital Signatures, Springer, New York, 2010. · Zbl 1202.94002
[20] F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exp. Math. 13 (1995), no. 5, 385-416. · Zbl 0843.11046
[21] A. K. Lenstra, Unbelievable security (Matching AES security using public key systems), Advances in Cryptology—ASIACRYPT 2001, Lecture Notes in Comput. Sci. 2248, Springer, Berlin (2001), 67-86. · Zbl 1062.94550
[22] A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515-534. · Zbl 0488.12001
[23] A. K. Lenstra and E. Verheul, Selecting cryptographic key sizes, J. Cryptology 14 (2001), no. 4, 255-293. · Zbl 1006.94020
[24] H. W. Lenstra, Jr., Euclid’s algorithm in cyclotomic fields, J. Lond. Math. Soc. (2) 10 (1975), no. 4, 457-465. · Zbl 0313.12001
[25] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649-673. · Zbl 0629.10006
[26] H. W. Lenstra, Jr., The number field sieve: An annotated bibliography, The Development of the Number Field Sieve, Lecture Notes in Math. 1554, Springer, Berlin (1993), 1-3. · Zbl 0806.11063
[27] N. Manohar and B. Fisch, Factoring n = p^2q, Final project report CS359C, Stanford University, 2017.
[28] A. May, Secret exponent attacks on RSA-type schemes with moduli N = p^rq, Public Key Cryptography—PKC 2004, Lecture Notes in Comput. Sci. 2947, Springer, Berlin (2004), 218-230. · Zbl 1198.94113
[29] A. Menezes, M. Qu, D. Stinson and Y. Wang, Evaluation of security level of cryptography: ESIGN signature scheme, External Evaluation Report ex-1053-2000, CRYPTREC, 2001.
[30] T. Okamoto, E. Fujisaki and H. Morita, TSH-ESIGN: Efficient digital signature scheme using trisection size hash, Submission to IEEE P1363a, November 1998. [Online; accessed 7-February-2019].
[31] T. Okamoto and A. Shibaishi, A fast signature scheme based on quadratic inequalities, 1985 IEEE Symposium on Security and Privacy, IEEE Press, Piscataway (1985), 123-133.
[32] T. Okamoto and S. Uchiyama, A new public-key cryptosystem as secure as factoring, Advances in Cryptology—EUROCRYPT’98, Lecture Notes in Comput. Sci. 1403, Springer, Berlin (1998), 308-318. · Zbl 0919.94028
[33] R. Peralta, On the distribution of quadratic residues and nonresidues modulo a prime number, Math. Comp. 58 (1992), no. 197, 433-440. · Zbl 0745.11057
[34] R. Peralta and E. Okamoto, Faster factoring of integers of a special form, IEICE Trans. Fundam. Electron. Comm. Comp. Sci. E79 (1996), no. A4, 489-493.
[35] R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), no. 2, 120-126. · Zbl 0368.94005
[36] A. Sárközy and C. L. Stewart, On pseudorandomness in families of sequences derived from the Legendre symbol, Period. Math. Hungar. 54 (2007), no. 2, 163-173. · Zbl 1136.11049
[37] H. Sato, T. Takagi, S. Tezuka and K. Takaragi, Generalized powering functions and their application to digital signatures, Advances in Cryptology—ASIACRYPT 2003, Lecture Notes in Comput. Sci. 2894, Springer, Berlin (2003), 434-451. · Zbl 1205.11130
[38] R. Scheidler and H. C. Williams, A public-key cryptosystem utilizing cyclotomic fields, Des. Codes Cryptogr. 6 (1995), no. 2, 117-131. · Zbl 0829.94005
[39] K. Schmidt-Samoa, A new Rabin-type trapdoor permutation equivalent to factoring, Electron. Notes Theor. Comput. Sci. 157 (2006), no. 3, 79-94. · Zbl 1294.94076
[40] K. Schmidt-Samoa and T. Takagi, Paillier’s cryptosystem modulo p^2q and its applications to trapdoor commitment schemes, Progress in Cryptology—Mycrypt 2005, Lecture Notes in Comput. Sci. 3715, Springer, Berlin (2005), 296-313. · Zbl 1126.94340
[41] C. P. Schnorr, Efficient signature generation by smart cards, J. Cryptology 4 (1991), no. 3, 161-174. · Zbl 0743.68058
[42] J. Stern, D. Pointcheval, J. Malone-Lee and N. P. Smart, Flaws in applying proof methodologies to signature schemes, Advances in cryptology—CRYPTO 2002, Lecture Notes in Comput. Sci. 2442, Springer, Berlin (2002), 93-110. · Zbl 1026.94550
[43] T. Takagi, Fast RSA-type cryptosystem modulo p^kq., Advances in Cryptology—CRYPTO’98, Lecture Notes in Comput. Sci. 1462, Springer, Berlin (1998), 318-326. · Zbl 0931.94041
[44] L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts Math. 83, Springer, New York, 1997. · Zbl 0966.11047
[45] A. Weilert, Fast computation of the biquadratic residue symbol, J. Number Theory 96 (2002), no. 1, 133-151. · Zbl 1043.11003
[46] H. C. Williams, An M^3 public-key encryption scheme, Advances in Cryptology—CRYPTO’85, Lecture Notes in Comput. Sci. 218, Springer, Berlin (1986), 358-368. · Zbl 0612.94010
[47] BlueKrypt, Cryptographic key length recommendations, 2018.
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