9,280 research outputs found
Fast matrix multiplication techniques based on the Adleman-Lipton model
On distributed memory electronic computers, the implementation and
association of fast parallel matrix multiplication algorithms has yielded
astounding results and insights. In this discourse, we use the tools of
molecular biology to demonstrate the theoretical encoding of Strassen's fast
matrix multiplication algorithm with DNA based on an -moduli set in the
residue number system, thereby demonstrating the viability of computational
mathematics with DNA. As a result, a general scalable implementation of this
model in the DNA computing paradigm is presented and can be generalized to the
application of \emph{all} fast matrix multiplication algorithms on a DNA
computer. We also discuss the practical capabilities and issues of this
scalable implementation. Fast methods of matrix computations with DNA are
important because they also allow for the efficient implementation of other
algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.Comment: To appear in the International Journal of Computer Engineering
Research. Minor changes made to make the preprint as similar as possible to
the published versio
Bit-level pipelined digit-serial array processors
A new architecture for high performance digit-serial vector inner product (VIP) which can be pipelined to the bit-level is introduced. The design of the digit-serial vector inner product is based on a new systematic design methodology using radix-2n arithmetic. The proposed architecture allows a high level of bit-level pipelining to increase the throughput rate with minimum initial delay and minimum area. This will give designers greater flexibility in finding the best tradeoff between hardware cost and throughput rate. It is shown that sub-digit pipelined digit-serial structure can achieve a higher throughput rate with much less area consumption than an equivalent bit-parallel structure. A twin-pipe architecture to double the throughput rate of digit-serial multipliers and consequently that of the digit-serial vector inner product is also presented. The effect of the number of pipelining levels and the twin-pipe architecture on the throughput rate and hardware cost are discussed. A two's complement digit-serial architecture which can operate on both negative and positive numbers is also presented
Toward the classification of Moufang loops of order 64
We show how to obtain all nonassociative Moufang loops of order less than 64
and 4262 nonassociative Moufang loops of order 64 in a unified way. We
conjecture that there are no other nonassociative Moufang loops of order 64.
The main idea of the computer search is to modify precisely one quarter of the
multiplication table in a certain way, previously applied to small 2-groups.Comment: 16 page
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
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