An improved worst-case to average-case connection for lattice problems

We improve a connection of the worst-case complexity and the average-case complexity of some well-known lattice problems. This fascinating connection was first discovered by Ajtai (1995). We improve the exponent of this connection from 8 to 3.5+/spl epsiv/.

[1]  DyerMartin,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991 .

[2]  Jörg M. Wills,et al.  Handbook of Convex Geometry , 1993 .

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

[4]  Oded Goldreich,et al.  Public-Key Cryptosystems from Lattice Reduction Problems , 1996, CRYPTO.

[5]  Miklós Ajtai,et al.  Generating Hard Instances of Lattice Problems , 1996, Electron. Colloquium Comput. Complex..

[6]  K. Ball Cube slicing in ⁿ , 1986 .

[7]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[8]  Miklós Simonovits,et al.  The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[9]  C. Hermite Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. , 1850 .

[10]  Cynthia Dwork,et al.  A public-key cryptosystem with worst-case/average-case equivalence , 1997, STOC '97.

[11]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

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

[13]  A. Odlyzko,et al.  Disproof of the Mertens conjecture. , 1984 .

[14]  G. L. Dirichlet Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. , 1850 .

[15]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[16]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[17]  Jeffrey C. Lagarias,et al.  Solving low density subset sum problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[18]  Jacques Stern,et al.  The hardness of approximate optima in lattices, codes, and systems of linear equations , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[19]  Martin E. Dyer,et al.  A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies , 1989, STOC.

[20]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[21]  Kenneth J. Giuliani Factoring Polynomials with Rational Coeecients , 1998 .

[22]  Oded Goldreich,et al.  Collision-Free Hashing from Lattice Problems , 1996, Electron. Colloquium Comput. Complex..

[23]  Jeffrey C. Lagarias The computational complexity of simultaneous Diophantine approximation problems , 1982, FOCS 1982.

[24]  Miklós Simonovits,et al.  Isoperimetric problems for convex bodies and a localization lemma , 1995, Discret. Comput. Geom..