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Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 41 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
Twoway finite automata with quantum and classical states
 Los Alamos Preprint Archive
, 1999
"... We introduce 2way finite automata with quantum and classical states (2qcfa’s). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa’s; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum st ..."
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Cited by 25 (0 self)
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We introduce 2way finite automata with quantum and classical states (2qcfa’s). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa’s; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa’s are better than classical 2way automata. First, 2qcfa’s can recognize palindromes, a language that cannot be recognized by 2way deterministic or probabilistic finite automata. Second, in polynomial time 2qcfa’s can recognize {a n b n n ∈ N}, a language that can be recognized classically by a 2way probabilistic automaton but only in exponential time. 1
Recent progress and prospects for integer factorisation algorithms
 In Proc. of COCOON 2000
, 2000
"... Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In ..."
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Cited by 20 (1 self)
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Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now routine to factor 100decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We outline several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. In particular, we consider the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods. 1
Quantum typing, in
 Proceedings of the 2nd International Workshop on Quantum Programming Languages
"... The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambdacalculus for the QRAM model, following the work of P. Selinger (2003) on quantum flowcharts. We define a callbyvalue operational semantics, and we develop a type system using aff ..."
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Cited by 17 (0 self)
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The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambdacalculus for the QRAM model, following the work of P. Selinger (2003) on quantum flowcharts. We define a callbyvalue operational semantics, and we develop a type system using affine intuitionistic linear logic. The main result of this preprint is the subjectreduction of the language.
Reliability of CalderbankShorSteane codes and security of quantum key distribution
 J. Phys. A: Math. Gen
, 2004
"... Abstract. After Mayers (1996, 2001) gave a proof of the security of the Bennett ..."
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Cited by 17 (7 self)
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Abstract. After Mayers (1996, 2001) gave a proof of the security of the Bennett
Information rates achievable with algebraic codes on quantum discrete memoryless channels
 IEEE Trans. Information Theory
, 2005
"... The highest information rate at which quantum errorcorrection schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to ..."
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Cited by 15 (7 self)
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The highest information rate at which quantum errorcorrection schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, this work’s bound is actually the highest possible rate at which symplectic stabilizer codes work reliably.
Convolutional and tailbiting quantum errorcorrecting codes
 IEEE Trans. Inform. Theory
, 2007
"... Rate(n–2)/n unrestricted and CSStype quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical selforthogonal rate1/n F4linear and binary linear convolutional codes, respectively. These codes generally have higher rate a ..."
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Cited by 12 (4 self)
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Rate(n–2)/n unrestricted and CSStype quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical selforthogonal rate1/n F4linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate(n–2)/n block stabilizer codes with the same rate and errorcorrection capability and essentially the same decoding algorithms are derived from these convolutional codes via tailbiting. Index terms: Quantum errorcorrecting codes, CSStype codes, quantum convolutional codes, quantum tailbiting codes. I.
Quantum convolutional codes: fundamentals
, 2004
"... We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used in similar circumstances in classical communication. In this ..."
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Cited by 5 (0 self)
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We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used in similar circumstances in classical communication. In this article, we provide an efficient polynomial formalism for describing their stabilizer group, derive an online encoding circuit with linear gate complexity and study error propagation together with the existence of online decoding. Finally, we provide a maximum likelihood error estimation algorithm with linear classical complexity for any memoryless channel. 1
Tensor norms and the classical communication complexity of nonlocal quantum measurement
 SIAM J. Comput
, 2008
"... Nonlocality is at the heart of quantum information processing. In this paper we investigate the minimum amount of classical communication required to simulate a nonlocal quantum measurement. We derive general upper bounds, which in turn translate to systematic classical simulations of quantum commun ..."
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Cited by 5 (0 self)
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Nonlocality is at the heart of quantum information processing. In this paper we investigate the minimum amount of classical communication required to simulate a nonlocal quantum measurement. We derive general upper bounds, which in turn translate to systematic classical simulations of quantum communication protocols. As a concrete application, we prove that any quantum communication protocol with shared entanglement for computing a Boolean function can be simulated by a classical protocol whose cost does not depend on the amount of the shared entanglement. This implies that if the cost of communication is a constant, quantum and classical protocols, with shared entanglement and shared coins, respectively, compute the same class of functions. Yet another application is in the context of simulating quantum correlations using local hidden variable models augmented with classical communications. We give a constant cost, approximate simulation of quantum correlations of random variables whose domain is of a constant size but the dimension of the entanglement and the number of possible measurements may be arbitrary. Our upper bounds are expressed in terms of some tensor norms on the measurement operator. Those norms capture the nonlocality of bipartite operators in their own way and may be of independent interest and further applications.
Quantum Programs with Classical Output Streams (Extended Abstract)
 In Selinger [Sel05b
, 2005
"... We show how to model the semantics of quantum programs that give classical output during their execution. That is, in our model even nonterminating programs may have output. The modelling interprets a program as a measurement process on the machines state, with the classical output as measuremen ..."
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Cited by 5 (0 self)
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We show how to model the semantics of quantum programs that give classical output during their execution. That is, in our model even nonterminating programs may have output. The modelling interprets a program as a measurement process on the machines state, with the classical output as measurement result. The semantics presented here are fully abstract in the sense that two programs are equal if and only if they give the same outputs in any composition.