scholarly journals Improved Quantum Circuits for Elliptic Curve Discrete Logarithms

2020 ◽  
pp. 425-444
Author(s):  
Thomas Häner ◽  
Samuel Jaques ◽  
Michael Naehrig ◽  
Martin Roetteler ◽  
Mathias Soeken
Keyword(s):  
Elliptic Curve ◽  
Quantum Circuits ◽  
Discrete Logarithms

scholarly journals Efficient quantum circuits for binary elliptic curve arithmetic: reducing $T$-gate complexity

2013 ◽  
Vol 13 (7&8) ◽  
pp. 631-644
Author(s):  
Brittanney Amento ◽  
Martin Rotteler ◽  
Rainer Steinwalds
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Quantum Algorithms ◽  
Substantial Reduction ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Curves Over Finite Fields ◽  
Curve Representation ◽  
Modern Cryptography

Elliptic curves over finite fields ${\mathbb F}_{2^n}$ play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of $T$-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in $\mathbb F_{2^n}$ in depth $\bigO(n\log_2 n)$ using a polynomial basis representation, which may be of independent interest.

scholarly journals Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm

2017 ◽  
Vol 11 (3) ◽  
pp. 453-469 ◽  
Author(s):  
Steven D. Galbraith ◽  
◽  
Ping Wang ◽  
Fangguo Zhang ◽  
◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Discrete Logarithms ◽  
Giant Step ◽  
Step Algorithm

scholarly journals A quantum circuit to find discrete logarithms on ordinary binary elliptic curves in depth O(log^2n)

2014 ◽  
Vol 14 (9&10) ◽  
pp. 888-900
Author(s):  
Martin Rotteler ◽  
Rainer Steinwandt
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Quantum Circuit ◽  
Discrete Logarithm Problem ◽  
Quantum Circuits ◽  
Discrete Logarithms ◽  
Shor's Algorithm

Improving over an earlier construction by Kaye and Zalka \cite{KaZa04}, in \cite{MMCP09b} Maslov et al. describe an implementation of Shor's algorithm, which can solve the discrete logarithm problem on ordinary binary elliptic curves in quadratic depth $\bigO(n^2)$. In this paper we show that discrete logarithms on such curves can be found with a quantum circuit of depth $\bigO(\log^2n)$. As technical tools we introduce quantum circuits for ${\mathbb F}_{2^n}$-multiplication in depth $\bigO(\log n)$ and for ${\mathbb F}_{2^n}$-inversion in depth $\bigO(\log^2 n)$.

scholarly journals Quantum Resource Estimates for Computing Elliptic Curve Discrete Logarithms

2017 ◽  
pp. 241-270 ◽  
Author(s):  
Martin Roetteler ◽  
Michael Naehrig ◽  
Krysta M. Svore ◽  
Kristin Lauter
Keyword(s):  
Elliptic Curve ◽  
Discrete Logarithms

Attacks on Self-Certified Multi-Proxy Signature Schemes with Message Recovery

2011 ◽  
Vol 135-136 ◽  
pp. 316-320
Author(s):  
Qi Xie
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Proxy Signature ◽  
Public Key ◽  
Signature Verification ◽  
Signature Schemes ◽  
Discrete Logarithms ◽  
The Public ◽  
Message Recovery ◽  
Public Keys

Signature schemes with message recovery based on self-certified public keys can reduce the amount of communications and computations, since the signature verification, the public key authentication and the message recovery are simultaneously carried out in a single logical step. Integrating self-certified public-key systems and the message recovery signature schemes, in 2009, Wu et al. proposed two multi-proxy signatures based on the discrete logarithms over a finite field and the elliptic curve discrete logarithms. The proxy warrant revision attacks are proposed, and it will show that Wu et al.’s schemes can not resist the proxy warrant revision attacks by either the proxy group or the original signer.

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