Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti

CiteScore 2018: 1.41

SCImago Journal Rank (SJR) 2018: 0.342
Source Normalized Impact per Paper (SNIP) 2018: 1.076

Mathematical Citation Quotient (MCQ) 2018: 0.75

See all formats and pricing
More options …
Ahead of print


New number-theoretic cryptographic primitives

Éric Brier / Houda Ferradi / Marc Joye / David Naccache
Published Online: 2019-11-09 | DOI: https://doi.org/10.1515/jmc-2019-0035


This paper introduces new prq-based one-way functions and companion signature schemes. The new signature schemes are interesting because they do not belong to the two common design blueprints, which are the inversion of a trapdoor permutation and the Fiat–Shamir transform. In the basic signature scheme, the signer generates multiple RSA-like moduli ni=pi2qi and keeps their factors secret. The signature is a bounded-size prime whose Jacobi symbols with respect to the ni’s match the message digest. The generalized signature schemes replace the Jacobi symbol with higher-power residue symbols. Given of their very unique design, the proposed signature schemes seem to be overlooked “missing species” in the corpus of known signature algorithms.

Keywords: number theory; one-way functions; digital signatures; cryptographic primitives

MSC 2010: 94A60; 11T71; 11A15; 11R18


  • [1]

    E. Bach and J. Shallit, Algorithmic Number Theory. Vol. 1: Efficient Algorithms, MIT Press, Cambridge, 1996. Google Scholar

  • [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. Google Scholar

  • [3]

    D. Boneh, G. Durfee and N. Howgrave-Graham, Factoring N=prq for large r, Advances in Cryptology—CRYPTO ’99, Lecture Notes in Comput. Sci. 1666, Springer, Berlin (1999), 326–337. Google Scholar

  • [4]

    P. C. Caranay and R. Scheidler, An efficient seventh power residue symbol algorithm, Int. J. Number Theory 6 (2010), no. 8, 1831–1853. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993. Google Scholar

  • [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. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [8]

    H. Davenport, On the distribution of quadratic residues (mod p), J. Lond. Math. Soc. 6 (1931), no. 1, 49–54. Google Scholar

  • [9]

    H. Davenport, On the distribution of quadratic residues (mod p). II, J. Lond. Math. Soc. 8 (1933), no. 1, 46–52. Google Scholar

  • [10]

    W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory IT-22 (1976), no. 6, 644–654. Google Scholar

  • [11]

    C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem. Applications in Computing, Coding, Cryptography, World Scientific, River Edge, 1996. Google Scholar

  • [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. Google Scholar

  • [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. Google Scholar

  • [14]

    O. Goldreich, Foundations of Cryptography. Basic Tools, Cambridge University, Cambridge, 2001. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [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. CrossrefGoogle Scholar

  • [17]

    L. Granboulan, How to repair ESIGN, Security in Communication Networks—SCN 2002, Lecture Notes in Comput. Sci. 2576, Springer, Berlin (2003), 234–240. Google Scholar

  • [18]

    K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Grad. Texts in Math. 84, Springer, New York, 1990. Google Scholar

  • [19]

    J. Katz, Digital Signatures, Springer, New York, 2010. Google Scholar

  • [20]

    F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exp. Math. 13 (1995), no. 5, 385–416. Google Scholar

  • [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. Google Scholar

  • [22]

    A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. CrossrefGoogle Scholar

  • [23]

    A. K. Lenstra and E. Verheul, Selecting cryptographic key sizes, J. Cryptology 14 (2001), no. 4, 255–293. CrossrefGoogle Scholar

  • [24]

    H. W. Lenstra, Jr., Euclid’s algorithm in cyclotomic fields, J. Lond. Math. Soc. (2) 10 (1975), no. 4, 457–465. Google Scholar

  • [25]

    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [27]

    N. Manohar and B. Fisch, Factoring n=p2q, Final project report CS359C, Stanford University, 2017. Google Scholar

  • [28]

    A. May, Secret exponent attacks on RSA-type schemes with moduli N=prq, Public Key Cryptography—PKC 2004, Lecture Notes in Comput. Sci. 2947, Springer, Berlin (2004), 218–230. Google Scholar

  • [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. Google Scholar

  • [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. Google Scholar

  • [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. Google Scholar

  • [33]

    R. Peralta, On the distribution of quadratic residues and nonresidues modulo a prime number, Math. Comp. 58 (1992), no. 197, 433–440. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [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. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [38]

    R. Scheidler and H. C. Williams, A public-key cryptosystem utilizing cyclotomic fields, Des. Codes Cryptogr. 6 (1995), no. 2, 117–131. CrossrefGoogle Scholar

  • [39]

    K. Schmidt-Samoa, A new Rabin-type trapdoor permutation equivalent to factoring, Electron. Notes Theor. Comput. Sci. 157 (2006), no. 3, 79–94. CrossrefGoogle Scholar

  • [40]

    K. Schmidt-Samoa and T. Takagi, Paillier’s cryptosystem modulo p2q and its applications to trapdoor commitment schemes, Progress in Cryptology—Mycrypt 2005, Lecture Notes in Comput. Sci. 3715, Springer, Berlin (2005), 296–313. Google Scholar

  • [41]

    C. P. Schnorr, Efficient signature generation by smart cards, J. Cryptology 4 (1991), no. 3, 161–174. Google Scholar

  • [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. Google Scholar

  • [43]

    T. Takagi, Fast RSA-type cryptosystem modulo pkq., Advances in Cryptology—CRYPTO’98, Lecture Notes in Comput. Sci. 1462, Springer, Berlin (1998), 318–326. Google Scholar

  • [44]

    L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts Math. 83, Springer, New York, 1997. Google Scholar

  • [45]

    A. Weilert, Fast computation of the biquadratic residue symbol, J. Number Theory 96 (2002), no. 1, 133–151. CrossrefGoogle Scholar

  • [46]

    H. C. Williams, An M3 public-key encryption scheme, Advances in Cryptology—CRYPTO’85, Lecture Notes in Comput. Sci. 218, Springer, Berlin (1986), 358–368. Google Scholar

  • [47]

    BlueKrypt, Cryptographic key length recommendations, 2018.

About the article

Received: 2019-07-18

Accepted: 2019-09-15

Published Online: 2019-11-09

Citation Information: Journal of Mathematical Cryptology, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc-2019-0035.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in