Abstract
In 1993 Boo Barkee and others have written a paper “Why you cannot even hope to use Gröbner Bases in Public Key Cryptography: an open letter to a scientist who failed and a challenge to those who have not yet failed.” Since 1994, further attempts have been made, that gave rise to several cryptosystems now known as Polly Cracker systems. None of these proposals have been successful, and while Gröbner Bases are now an established tool for cryptanalysis, the challenge of Boo Barkee still stands w.r.t. the design point of view. We outline a description of how all these attempts have failed.
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References
P. Ackermann and M. Kreuzer, Gröbner basis cryptosystems, AAECC 17 (2006), nos. 3–4, 173–194.
M. E. Alonso and M. G. Marinari, Oracle-supported drawing of the Gröbner éscalier, preprint, 2008.
J. Apel, Computational ideal theory in finitely generated extension rings, Theoret. Comput. Sci. 244 (2000), nos. 1–2, 1–33.
J. Backelin, S. Cojocaru, and V. Ufnarovski, Mathematical computations using Bergman, 2005, Lund University, Sweden, 2005. – 206 p.
W. Banks, D. Lieman, and I. Shparlinski, Cryptographic applications of sparse polynomials over finite rings, Proc. of ICISC 2000, LNCS, vol. 2015, Springer, Berlin, 2001, pp. 206–220.
F. Bao, R. H. Deng, W. Geiselmann, G. Schnorr, Steinwand R., and H. Wu, Cryptanalysis of two sparse polynomial based public key cryptosystems, Proc. of PKC 2001, LNCS, vol. 1992, Springer, Berlin, 2001, pp. 153–164.
B. Barkee, D. C. Can, J. Ecks, T. Moriarty, and R. F. Ree, Why you cannot even hope to use Gröbner bases in public key cryptography: an open letter to a scientist who failed and a challenge to those who have not yet failed, J. Symbolic Comput. 18 (1994), no. 6, 497–501.
M. Ben-Or and P. Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation, Proc. of ACM Symp. Theory Comput., ACM, New York, 1988, pp. 301–309.
A. M. Bigatti, R. La Scala, and L. Robbiano, Computing toric ideals, J. Symbolic Comput. 27 (1999), no. 4, 351–365.
O. Billet and J. Ding, Overview of cryptanalysis techniques in multivariate public key cryptography, this volume, 2009, pp. 263–283.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.
B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383.
B. Buchberger, Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional systems theory, Reidel, Dordrecht, 1985, pp. 184–232.
B. Buchberger, An algorithmical criterion for the solvability of algebraic systems of equations, London Math. Soc. LNS 251 (1998), 535–545.
B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.
S. Bulygin, Chosen-ciphertext attack on noncommutative Polly Cracker, 2005, http://arxiv.org/abs/cs/0508015v2+.
S. Bulygin and T. S. Rai, Countering chosen-ciphertext attacks against noncommutative Polly Cracker cryptosystems, 2006, talk at Special Semester on Gröbner Bases, Linz, Austria.
M. Caboara and M. Silvestri, Classification of compatible module orderings, J. Pure Appl. Algebra 142 (1999), no. 1, 13–24.
M. Caboara, F. Caruso, and C. Traverso, Gröbner bases in public key cryptography, Proc. of ISSAC 2008, to appear, 2008.
F. Caruso, P. Conti, and C. Traverso, Non-commutative factorisation and GCD with applications to public-key cryptography, 2008, Proc. of Differential Algebra and Related Computer Algebra, Le Matematiche, LXIII (1), pp. 37–39.
S. Cojocaru and V. Ufnarovski, Noncommutative Gröbner basis, Hilbert series, Anick’s resolution and BERGMAN under MS-DOS, Computer Science Journal of Moldova 3 (1995), 24–39.
P. Conti and C. Traverso, Buchberger’s algorithm and integer programming, Proc. of AAECC, LNCS, vol. 539, Springer, Berlin, 1992, pp. 130–139.
P. Conti and C. Traverso, Homomorphism attacks to non-commutative Polly Cracker, 2007, preprint.
J. H. Davenport, Factorisation of polynomials in non-commuting variables, 1991, Personal communication.
A. Dickenstein, N. Fitchas, M. Giusti, and C. Sessa, The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete Appl. Math. 33 (1991), nos. 1–3, 73–94.
R. Endsuleit, W. Geiselmann, and R. Steinwandt, Attacking a polynomial-based cryptosystem: Polly Cracker, Int. J. Inf. Secur. 1 (2002), no. 3, 143–148.
M. Fellows and N. Koblitz, Combinatorial cryptosystems galore!, Contemp. Math. 168 (1994), 51–61.
D. Grant, K. Krastev, D. Lieman, and I. Shparlinski, A public key cryptosystem based on sparse polynomials, Proc. of ICCC 1998, Springer, Berlin, 2000, pp. 114–121.
E. Green, T. Mora, and V. Ufnarovski, The non-commutative Gröbner freaks, Symbolic rewriting techniques, Progr. Comput. Sci. Appl. Logic, vol. 15, Birkhäuser, Basel, 1998, pp. 93–104.
D. Y. Grigoriev, M. Karpinski, and M. F. Singer, Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput. 19 (1990), no. 6, 1059–1063.
E. A. Hirsch, http://logic.pdmi.ras.ru/~hirsch/sat.html, 2009.
D. Hofheinz and R. Steinwandt, A “differential” attack on Polly Cracker, Proc. of ISIT 2002, 2002, pp. 211–211.
E. Kaltofen and B. M. Trager, Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators, J. Symbolic Comput. 9 (1990), no. 3, 301–320.
N. Koblitz, Algebraic aspects of cryptography, Algorithms and Computation in Mathematics, vol. 3, Springer, Berlin, 1998.
F. Levy-dit-Vehel and L. Perret, A Polly Cracker system based on satisfiability, Coding, cryptography and combinatorics, Progr. Comput. Sci. Appl. Logic, vol. 23, Birkhäuser, Basel, 2004, pp. 177–192.
L. V. Ly, Polly Two: a new algebraic polynomial-based public-key scheme, AAECC 17 (2006), nos. 3–4, 267–283.
K. Madlener and B. Reinert, Computing Gröbner bases in monoid and group rings, Proc. of ISSAC 1993, ACM, New York, 1993, pp. 254–263.
T. Matsumoto and H. Imai, Algebraic methods for constructing asymmetric cryptosystems, Proc. of AAECC, LNCS, vol. 229, Springer, Berlin, 1985, pp. 108–119.
T. Mora, A 15/01/94 communication to M.R. Fellows and N. Koblitz, 1994
F. Mora, De nugis Groebnerialium. II. Applying Macaulay’s trick in order to easily write a Gröbner basis, AAECC 13 (2003), no. 6, 437–446.
T. Mora, Solving polynomial equation systems. II, Macaulay’s paradigm and Gröbner technology, Encyclopedia of Mathematics and its Applications, vol. 99, Cambridge University Press, Cambridge, 2005.
T. Mora, Gröbner technology, this volume, 2009a, pp. 11–25.
T. Mora, Solving polynomial equation systems. III, algebraic solving and beyond, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2009b, to appear.
F. L. Pritchard, The ideal membership problem in non-commutative polynomial rings, J. Symbolic Comput. 22 (1996), no. 1, 27–48.
T. S. Rai, Infinite Gröbner bases and noncommutative Polly Cracker cryptosystems, Ph.D. thesis, Virginia Polytech. Inst. and State Univ., 2004.
B. Reinert, On Gröbner bases in monoid and group rings, Ph.D. thesis, Kaiserslautern, 1995.
B. Reinert, A systematic study of Gröbner basis methods, Ph.D. thesis, Kaiserslautern, 2003, Habilitationschrift.
R. Steinwandt, A ciphertext-only attack on Polly Two, 2006, preprint.
R. Steinwandt and W. Geiselmann, Cryptanalysis of Polly Cracker, IEEE Trans. on Inf. Th. 48 (2002), no. 11, 2990–2991.
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Levy-dit-Vehel, F., Marinari, M.G., Perret, L., Traverso, C. (2009). A Survey on Polly Cracker Systems. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_16
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