Sparse polynomial approximation in finite fields

We consider a polynomial analogue of the <italic>hidden number problem</italic> which has recently been introduced by Boneh and Venkatesan. Namely we consider the <italic>sparse polynomial approximation</italic> problem of recovering an unknown polynomial <italic>f(X) \in \F_p[X]</italic> with at most $m$ non-zero terms from approximate values of <italic>f(t)</italic> at polynomially many points <italic>t \in \F_p</italic> selected uniformly at random. The case of a polynomial <italic>f(X) = α X<italic> corresponds to the hidden number problem. The above problem is related to the <italic>noisy polynomial interpolation</italic> problem and to the <italic>sparse polynomial interpolation</italic> problem which have recently been considered in the literature. Our results are based on a combination of some number theory tools such as bounds of exponential sums and the number of solutions of congruences with the lattice reduction technique.

[1]  Amin Shokrollahi,et al.  List Decoding of Algebraic-Geometric Codes , 1999, IEEE Trans. Inf. Theory.

[2]  C. P. Schnorr,et al.  A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms , 1987, Theor. Comput. Sci..

[3]  Igor E. Shparlinski,et al.  The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces , 2003, Des. Codes Cryptogr..

[4]  Ronitt Rubinfeld,et al.  Reconstructing Algebraic Functions from Mixed Data , 1998, SIAM J. Comput..

[5]  Rudolf Lide,et al.  Finite fields , 1983 .

[6]  Dan Boneh,et al.  Rounding in lattices and its cryptographic applications , 1997, SODA '97.

[7]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[8]  Arjen K. Lenstra,et al.  Key Improvements to XTR , 2000, ASIACRYPT.

[9]  Ronitt Rubinfeld,et al.  Learning Polynomials with Queries: The Highly Noisy Case , 2000, SIAM J. Discret. Math..

[10]  R. Kannan.  ALGORITHMIC GEOMETRY OF NUMBERS , 1987 .

[11]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[12]  Igor E. Shparlinski,et al.  The Insecurity of the Digital Signature Algorithm with Partially Known Nonces , 2002, Journal of Cryptology.

[13]  B. David Saunders,et al.  Sparse Polynomial Interpolation in Nonstandard Bases , 1995, SIAM J. Comput..

[14]  Igor E. Shparlinski,et al.  Security of most significant bits of gx2 , 2002, Inf. Process. Lett..

[15]  Igor E. Shparlinski,et al.  On the Generalised Hidden Number Problem and Bit Security of XTR , 2001, AAECC.

[16]  Richard Zippel,et al.  Effective polynomial computation , 1993, The Kluwer international series in engineering and computer science.

[17]  Nigel P. Smart,et al.  Lattice Attacks on Digital Signature Schemes , 2001, Des. Codes Cryptogr..

[18]  Dan Boneh,et al.  Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes , 1996, CRYPTO.

[19]  Madhu Sudan,et al.  Improved Low-Degree Testing and its Applications , 1997, STOC '97.

[20]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[21]  Daniele Micciancio,et al.  On the hardness of the shortest vector problem , 1998 .

[22]  Igor E. Shparlinski,et al.  On the Security of Diffie-Hellman Bits , 2000, Electron. Colloquium Comput. Complex..

[23]  R. Canetti,et al.  On the statistical properties of Diffie-Hellman distributions , 2000 .

[24]  Claus-Peter Schnorr,et al.  Local Randomness in Polynomial Random Number and Random Function Generators , 1993, SIAM J. Comput..

[25]  Frédéric Magniez,et al.  Exact and Approximate Testing/Correcting of Algebraic Functions: A Survey , 2000, Theoretical Aspects of Computer Science.

[26]  Ravi Kumar,et al.  A sieve algorithm for the shortest lattice vector problem , 2001, STOC '01.

[27]  Igor E. Shparlinski,et al.  The Insecurity of Nyberg-Rueppel and Other DSA-Like Signature Schemes with Partially Known Nonces , 2001, CaLC.

[28]  Ravi Kannan,et al.  Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..

[29]  Phong Q. Nguyen,et al.  Noisy Polynomial Interpolation and Noisy Chinese Remaindering , 2000, EUROCRYPT.

[30]  Marek Karpinski,et al.  Computational Complexity of Sparse Rational Interpolation , 1994, SIAM J. Comput..

[31]  Igor E. Shparlinski,et al.  Security of the most significant bits of the Shamir message passing scheme , 2000, Math. Comput..

[32]  Hal Wasserman,et al.  Reconstructing randomly sampled multivariate polynomials from highly noisy data , 1998, SODA '98.

[33]  Igor E. Shparlinski,et al.  On The Correlation Of Binary M-sequences , 1999, Des. Codes Cryptogr..

[34]  Marek Karpinski,et al.  On Zero-Testing and Interpolation of k-Sparse Multivariate Polynomials Over Finite Fields , 1991, Theor. Comput. Sci..

[35]  André Weil,et al.  Basic number theory , 1967 .

[36]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometric codes , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[37]  Jacques Stern,et al.  Lattice Reduction in Cryptology: An Update , 2000, ANTS.

[38]  Phong Q. Nguyen.  The Dark Side of the Hidden Number Problem: Lattice Attacks on DSA , 2001 .

[39]  Arjen K. Lenstra,et al.  The XTR Public Key System , 2000, CRYPTO.