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Quantum
Information and Computation
ISSN: 1533-7146
published since 2001
|
Vol.3 No.4, July 2003 |
Shor's
discrete logarithm quantum algorithm for elliptic curves
(pp317-344)
J. Proos and Ch. Zalka
doi:
https://doi.org/10.26421/QIC3.4-3
Abstracts:
We show in some detail how to implement Shor's efficient
quantum algorithm for discrete logarithms for the particular case of
elliptic curve groups. It turns out that for this problem a smaller
quantum computer can solve problems further beyond current computing
than for integer factorisation. A 160 bit elliptic curve cryptographic
key could be broken on a quantum computer using around 1000 qubits while
factoring the security-wise equivalent 1024 bit RSA modulus would
require about 2000 qubits. In this paper we only consider elliptic
curves over GF(p) and not yet the equally important ones over GF(2^n) or
other finite fields. The main technical difficulty is to implement
Euclid's gcd algorithm to compute multiplicative inverses modulo p. As
the runtime of Euclid's algorithm depends on the input, one difficulty
encountered is the ``quantum halting problem''.
Key words: quantum
computation, discrete alogarithm, elliptic curves |
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