Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199

A Library-Based Synthesis Methodology for Reversible Logic

In this paper, a library-based synthesis methodology for reversible circuits is proposed where a reversible specification is considered as a permutation comprising a set of cycles. To this end, a pre-synthesis optimization step is introduced to construct a reversible specification from an irreversible function. In addition, a cycle-based representation model is presented to be used as an intermediate format in the proposed synthesis methodology. The selected intermediate format serves as a focal point for all potential representation models. In order to synthesize a given function, a library containing seven building blocks is used where each building block is a cycle of length less than 6. To synthesize large cycles, we also propose a decomposition algorithm which produces all possible minimal and inequivalent factorizations for a given cycle of length greater than 5. All decompositions contain the maximum number of disjoint cycles. The generated decompositions are used in conjunction with a novel cycle assignment algorithm which is proposed based on the graph matching problem to select the best possible cycle pairs. Then, each pair is synthesized by using the available components of the library. The decomposition algorithm together with the cycle assignment method are considered as a binding method which selects a building block from the library for each cycle. Finally, a post-synthesis optimization step is introduced to optimize the synthesis results in terms of different costs.Comment: 24 pages, 8 figures, Microelectronics Journal, Elsevie

The weak password problem: chaos, criticality, and encrypted p-CAPTCHAs

Vulnerabilities related to weak passwords are a pressing global economic and security issue. We report a novel, simple, and effective approach to address the weak password problem. Building upon chaotic dynamics, criticality at phase transitions, CAPTCHA recognition, and computational round-off errors we design an algorithm that strengthens security of passwords. The core idea of our method is to split a long and secure password into two components. The first component is memorized by the user. The second component is transformed into a CAPTCHA image and then protected using evolution of a two-dimensional dynamical system close to a phase transition, in such a way that standard brute-force attacks become ineffective. We expect our approach to have wide applications for authentication and encryption technologies.Comment: 5 pages, 6 figer

Toward an Energy Efficient Language and Compiler for (Partially) Reversible Algorithms

We introduce a new programming language for expressing reversibility, Energy-Efficient Language (Eel), geared toward algorithm design and implementation. Eel is the first language to take advantage of a partially reversible computation model, where programs can be composed of both reversible and irreversible operations. In this model, irreversible operations cost energy for every bit of information created or destroyed. To handle programs of varying degrees of reversibility, Eel supports a log stack to automatically trade energy costs for space costs, and introduces many powerful control logic operators including protected conditional, general conditional, protected loops, and general loops. In this paper, we present the design and compiler for the three language levels of Eel along with an interpreter to simulate and annotate incurred energy costs of a program.Comment: 17 pages, 0 additional figures, pre-print to be published in The 8th Conference on Reversible Computing (RC2016

Quantum channels as a categorical completion

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201

Fault Models for Quantum Mechanical Switching Networks

The difference between faults and errors is that, unlike faults, errors can be corrected using control codes. In classical test and verification one develops a test set separating a correct circuit from a circuit containing any considered fault. Classical faults are modelled at the logical level by fault models that act on classical states. The stuck fault model, thought of as a lead connected to a power rail or to a ground, is most typically considered. A classical test set complete for the stuck fault model propagates both binary basis states, 0 and 1, through all nodes in a network and is known to detect many physical faults. A classical test set complete for the stuck fault model allows all circuit nodes to be completely tested and verifies the function of many gates. It is natural to ask if one may adapt any of the known classical methods to test quantum circuits. Of course, classical fault models do not capture all the logical failures found in quantum circuits. The first obstacle faced when using methods from classical test is developing a set of realistic quantum-logical fault models. Developing fault models to abstract the test problem away from the device level motivated our study. Several results are established. First, we describe typical modes of failure present in the physical design of quantum circuits. From this we develop fault models for quantum binary circuits that enable testing at the logical level. The application of these fault models is shown by adapting the classical test set generation technique known as constructing a fault table to generate quantum test sets. A test set developed using this method is shown to detect each of the considered faults.Comment: (almost) Forgotten rewrite from 200

Cellular Automata as a Model of Physical Systems

Cellular Automata (CA), as they are presented in the literature, are abstract mathematical models of computation. In this pa- per we present an alternate approach: using the CA as a model or theory of physical systems and devices. While this approach abstracts away all details of the underlying physical system, it remains faithful to the fact that there is an underlying physical reality which it describes. This imposes certain restrictions on the types of computations a CA can physically carry out, and the resources it needs to do so. In this paper we explore these and other consequences of our reformalization.Comment: To appear in the Proceedings of AUTOMATA 200