Collision Finding with Many Classical or Quantum Processors

In this thesis, we investigate the cost of finding collisions in a black-box function, a problem that is of fundamental importance in cryptanalysis. Inspired by the excellent performance of the heuristic rho method of collision finding, we define several new models of complexity that take into account the cost of moving information across a large space, and lay the groundwork for studying the performance of classical and quantum algorithms in these models.

[1]  Yaoyun Shi,et al.  Quantum lower bounds for the collision and the element distinctness problems , 2001, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[2]  Jean-Jacques Quisquater,et al.  How Easy is Collision Search? Application to DES (Extended Summary) , 1990, EUROCRYPT.

[3]  C. Thomborson,et al.  Area-time complexity for VLSI , 1979, STOC.

[4]  H. T. Kung,et al.  Sorting on a mesh-connected parallel computer , 1976, STOC '76.

[5]  D. Bernstein.  Cost analysis of hash collisions : will quantum computers make SHARCS obsolete? , 2009 .

[6]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[7]  Claudia Leopold,et al.  Parallel and Distributed Computing: A Survey of Models, Paradigms and Approaches , 2008 .

[8]  Ravi Montenegro,et al.  Near Optimal Bounds for Collision in Pollard Rho for Discrete Log , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[9]  Gilles Brassard,et al.  Quantum Algorithm for the Collision Problem , 2016, Encyclopedia of Algorithms.

[10]  J. Pollard.  A monte carlo method for factorization , 1975 .

[11]  Christof Zalka.  GROVER'S QUANTUM SEARCHING ALGORITHM IS OPTIMAL , 1997, quant-ph/9711070.

[12]  Andris Ambainis,et al.  Polynomial degree vs. quantum query complexity , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[13]  Ramarathnam Venkatesan,et al.  Random Cayley Digraphs and the Discrete Logarithm , 2002, ANTS.

[14]  Edlyn Teske.  On random walks for Pollard's rho method , 2001, Math. Comput..

[15]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[16]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[17]  Ramamohan Paturi,et al.  On the degree of polynomials that approximate symmetric Boolean functions (preliminary version) , 1992, STOC '92.

[18]  Ramarathnam Venkatesan,et al.  Non-degeneracy of Pollard Rho Collisions , 2008, ArXiv.

[19]  Noga Alon,et al.  Almost k-wise independence versus k-wise independence , 2003, Information Processing Letters.

[20]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[21]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[22]  Paul C. van Oorschot,et al.  Parallel Collision Search with Cryptanalytic Applications , 2013, Journal of Cryptology.

[23]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[24]  Hartmut Klauck,et al.  Quantum and classical strong direct product theorems and optimal time-space tradeoffs , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[25]  R. Bousso.  The Holographic principle , 2002, hep-th/0203101.

[26]  Mario Szegedy,et al.  All Quantum Adversary Methods Are Equivalent , 2005, ICALP.

[27]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[28]  Robert Spalek,et al.  Lower Bounds on Quantum Query Complexity , 2005, Bull. EATCS.

[29]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

[30]  Andrew Chi-Chih Yao,et al.  The entropic limitations on VLSI computations(Extended Abstract) , 1981, STOC '81.

[31]  Noam Nisan,et al.  CREW PRAMS and decision trees , 1989, STOC '89.

[32]  Andris Ambainis,et al.  Quantum search of spatial regions , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[33]  Scott Aaronson,et al.  Quantum lower bounds for the collision and the element distinctness problems , 2004, JACM.

[34]  Michael E. Saks,et al.  Quantum query complexity and semi-definite programming , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[35]  Harold Abelson,et al.  Information transfer and area-time tradeoffs for VLSI multiplication , 1980, CACM.

[36]  Shengyu Zhang,et al.  On the power of Ambainis lower bounds , 2005, Theor. Comput. Sci..

[37]  Samuel Kutin,et al.  Quantum Lower Bound for the Collision Problem with Small Range , 2005, Theory Comput..

[38]  R. Venkatesan.  APPLICATIONS OF CAYLEY GRAPHS , BILINEARITY , AND HIGHER-ORDER RESIDUES TO CRYPTOLOGY , 2004 .

[39]  Troy Lee,et al.  Negative weights make adversaries stronger , 2007, STOC '07.

[40]  Paul Benioff.  Space Searches with a Quantum Robot , 2000 .

[41]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[42]  Frédéric Magniez,et al.  Search via quantum walk , 2006, STOC '07.

[43]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[44]  Edlyn Teske,et al.  Speeding Up Pollard's Rho Method for Computing Discrete Logarithms , 1998, ANTS.

[45]  Richard Beigel,et al.  The polynomial method in circuit complexity , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[46]  Christof Zalka zalka.  Using Grover’s quantum algorithm for searching actual databases , 2000 .

[47]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[48]  Scott Aaronson,et al.  Quantum lower bound for the collision problem , 2001, STOC '02.

[49]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[50]  Frédéric Magniez,et al.  Lower bounds for randomized and quantum query complexity using Kolmogorov arguments , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[51]  H. T. Kung,et al.  Sorting on a mesh-connected parallel computer , 1977, CACM.

[52]  Andris Ambainis,et al.  A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs , 2005, STOC '06.

[53]  Yuval Peres,et al.  A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm , 2008, ANTS.

[54]  D. Boneh,et al.  Applications of Cayley graphs, bilinearity, and higher-order residues to cryptology , 2004 .

[55]  Ben Reichardt,et al.  Reflections for quantum query algorithms , 2010, SODA '11.

[56]  Andris Ambainis,et al.  Quantum lower bounds by quantum arguments , 2000, STOC '00.