On the complexity of computing short linearly independent vectors and short bases in a lattice

Motivated by Ajtai’s worst-case to average-case reduction for lattice problems, we study the complexity of computing short linearly independent vectors (short basis) in a lattice. We show that approximating the length of a shortest set of linearly independent vectors (shortest basis) within any constant factor is NP-hard. Under the assumption that problems in NP cannot be solved in DTIME(n p”‘y’og(n)) we show that no polynomial time algorithm can approximate the length of a shortest set of linearly independent vectors (shortest basis) within a factor of 2’“g’-‘(“), E > 0 arbitrary, but fixed. Finally, we obtain results on the limits of non-approximability for computing short linearly independent vectors (short basis). Our strongest result in this direction states that under reasonable complexity-theoretic assumptions, approximating the length of a shortest set of linearly independent vectors (shortest basis) within a factor of n/a is not NP-hard.

[1]  Martin Henk.  Note on Shortest and Nearest Lattice Vectors , 1997, Inf. Process. Lett..

[2]  Guy Kindler,et al.  Approximating CVP to Within Almost-Polynomial Factors is NP-Hard , 2003, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

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

[4]  Carsten Lund,et al.  Hardness of approximations , 1996 .

[5]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[6]  Jin-Yi Cai,et al.  A Relation of Primal-Dual Lattices and the Complexity of Shortest Lattice Vector Problem , 1998, Theor. Comput. Sci..

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

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

[9]  Miklós Ajtai,et al.  The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract) , 1998, STOC '98.

[10]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[11]  Jeffrey C. Lagarias,et al.  Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice , 1990, Comb..

[12]  Jin-Yi Cai.  A new transference theorem and applications to Ajtai's connection factor , 1998, Electron. Colloquium Comput. Complex..

[13]  Oded Goldreich,et al.  On the limits of non-approximability of lattice problems , 1998, STOC '98.

[14]  P. Cohn.  Symmetric Bilinear Forms , 1973 .

[15]  Jin-Yi Cai,et al.  An improved worst-case to average-case connection for lattice problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[16]  Miklós Ajtai,et al.  Generating hard instances of lattice problems (extended abstract) , 1996, STOC '96.

[17]  M. Ajtai.  The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract) , 1998, STOC '98.

[18]  W. Banaszczyk.  New bounds in some transference theorems in the geometry of numbers , 1993 .

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

[20]  Jean-Pierre Seifert,et al.  Approximating Shortest Lattice Vectors is Not Harder Than Approximating Closest Lattice Vectors , 1999, Electron. Colloquium Comput. Complex..

[21]  Daniele Micciancio,et al.  The shortest vector in a lattice is hard to approximate to within some constant , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[22]  Jacques Stern,et al.  The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations , 1997, J. Comput. Syst. Sci..

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

[24]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .