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Collision Finding with Many Classical or Quantum Processors

Stacey Jeffery

Published Aug 31, 2011• 2 citations
OpenAlex

Abstract

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii 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. iii Acknowledgements I am deeply indebted to my supervisor, Dr. Michele Mosca, for introducing me to the subject of quantum information processing, and for the years of support and encouragement

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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learn-ing graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)

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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii 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. iii Acknowledgements I am deeply indebted to my supervisor, Dr. Michele Mosca, for introducing me to the subject of quantum information processing, and for the years of support and encouragement
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. G. Passante ii Quantum information processors have the ability to drastically change our world. By manipulating bits of information ruled by the laws of quantum mechanics, computational devices can perform some computations that are classically intractable. Most quantum algorithms rely on pure qubits as inputs and require entanglement throughout the compu-tation. In this thesis, we explore a model of computation that uses mixed qubits without en-tanglement known as DQC1 (deterministic quantum computation with one quantum bit), using the physical system of liquid-state Nuclear Magnetic Resonance (NMR). Throughout our research, we experimentally implement an algorithm that completely encapsulates the DQC1 model, and take a close look at the quantum nature of DQC1-states as given by the
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learn-ing graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)