On the limits of non-approximability of lattice problems

Wo show simple constantround interactive proof systems for pcoblcms capturing the appcoximabiity, to within a factor of &, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector pcoblom @VP), These interactive proofs are for the “coNP direction”; that is, we give an interactive protocol showing that a vector is “far” from the lattice (for CVP), and an interactive protocol showing that the shortest-lattice-vector la Ulong” (for SVP). Furthermore, these interactive proof systems ace Honest-Verifier Perfect Zero-Knowledge. We conclude that approximating CVP (resp., SVP) within a factor of fi is in NP n codM. Thus, it seems unlikely that approximating these problems to within a fi factor is NP-hard, Pccviously, for the CVP (resp., SVP) problem, La&arias et, ol,, H&tad and Banaszczyk showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in NP n con/p. On the other hand, Arora et. al. showed that the gap problem corresponding to approximating CVP within 2’0so*ooo n is quasi-NP-hard.

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

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

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

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

[5]  Yacov Yacobi,et al.  The Complexity of Promise Problems with Applications to Public-Key Cryptography , 1984, Inf. Control..

[6]  Johan Håstad,et al.  Dual vectors and lower bounds for the nearest lattice point problem , 1988, Comb..

[7]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[8]  Lance Fortnow,et al.  The Complexity of Perfect Zero-Knowledge , 1987, Proceeding Structure in Complexity Theory.

[9]  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.

[10]  Johan Håstad,et al.  Some optimal inapproximability results , 1997, STOC '97.

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

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

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

[14]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

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

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

[17]  Miklós Simonovits,et al.  Random walks and an O*(n5) volume algorithm for convex bodies , 1997, Random Struct. Algorithms.

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

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

[20]  Silvio Micali,et al.  Probabilistic Encryption , 1984, J. Comput. Syst. Sci..

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

[22]  Shafi Goldwasser,et al.  Private coins versus public coins in interactive proof systems , 1986, STOC '86.

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

[24]  Oded Goldreich,et al.  Modern Cryptography, Probabilistic Proofs and Pseudorandomness , 1998, Algorithms and Combinatorics.

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

[26]  D. Micciancio.  On the Inapproximability of the Shortest Vector in a Lattice Within Some Constant Factor , 1998 .

[27]  Tatsuaki Okamoto.  On Relationships between Statistical Zero-Knowledge Proofs , 2000, J. Comput. Syst. Sci..

[28]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

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

[30]  Uri Zwick,et al.  A 7/8-approximation algorithm for MAX 3SAT? , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

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

[32]  Donald E. Knuth,et al.  The Art of Computer Programming, Volumes 1-3 Boxed Set , 1998 .

[33]  Oded Goldreich,et al.  On Completeness and Soundness in Interactive Proof Systems , 1989, Adv. Comput. Res..

[34]  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.

[35]  Amit Sahai,et al.  A complete promise problem for statistical zero-knowledge , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[36]  Uriel Feige.  A threshold of ln n for approximating set cover (preliminary version) , 1996, STOC '96.

[37]  Cheng-Shang Chang.  Calculus , 2020, Bicycle or Unicycle?.

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

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

[40]  Shafi Goldmer.  An Eflcient Probabilistic Public-Key Encryption Scheme Which Hides All Partial Information , 1985 .

[41]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[42]  Gilles Brassard,et al.  Relativized cryptography , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

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

[44]  Donald E. Knuth,et al.  The Art of Computer Programming, Vol. 2 , 1981 .

[45]  Oded Goldreich,et al.  On the possibility of basing Cryptography on the assumption that P ≠ NP , 1998, IACR Cryptol. ePrint Arch..

[46]  Amit Sahai,et al.  Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge , 1998, STOC '98.

[47]  M. Simonovits,et al.  Random walks and an O * ( n 5 ) volume algorithm for convex bodies , 1997 .

[48]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[49]  Manuel Blum,et al.  An Efficient Probabilistic Public-Key Encryption Scheme Which Hides All Partial Information , 1985, CRYPTO.