2,437 research outputs found
Fast Quantum Modular Exponentiation
We present a detailed analysis of the impact on modular exponentiation of
architectural features and possible concurrent gate execution. Various
arithmetic algorithms are evaluated for execution time, potential concurrency,
and space tradeoffs. We find that, to exponentiate an n-bit number, for storage
space 100n (twenty times the minimum 5n), we can execute modular exponentiation
two hundred to seven hundred times faster than optimized versions of the basic
algorithms, depending on architecture, for n=128. Addition on a neighbor-only
architecture is limited to O(n) time when non-neighbor architectures can reach
O(log n), demonstrating that physical characteristics of a computing device
have an important impact on both real-world running time and asymptotic
behavior. Our results will help guide experimental implementations of quantum
algorithms and devices.Comment: to appear in PRA 71(5); RevTeX, 12 pages, 12 figures; v2 revision is
substantial, with new algorithmic variants, much shorter and clearer text,
and revised equation formattin
Unification modulo a partial theory of exponentiation
Modular exponentiation is a common mathematical operation in modern
cryptography. This, along with modular multiplication at the base and exponent
levels (to different moduli) plays an important role in a large number of key
agreement protocols. In our earlier work, we gave many decidability as well as
undecidability results for multiple equational theories, involving various
properties of modular exponentiation. Here, we consider a partial subtheory
focussing only on exponentiation and multiplication operators. Two main results
are proved. The first result is positive, namely, that the unification problem
for the above theory (in which no additional property is assumed of the
multiplication operators) is decidable. The second result is negative: if we
assume that the two multiplication operators belong to two different abelian
groups, then the unification problem becomes undecidable.Comment: In Proceedings UNIF 2010, arXiv:1012.455
Quantum networks for elementary arithmetic operations
Quantum computers require quantum arithmetic. We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation. Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorising algorithm. We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorised
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