Fast modular exponentiation and elliptic curve group operation in Maple

Author(s):  
S. Y. Yan ◽  
G. James
2005 ◽  
Vol 5 (6) ◽  
pp. 474-491
Author(s):  
P.R. Kaye

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a ``quantum attack'' of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In \cite{PZ03}, Proos and Zalka show how to implement the extended Euclidean algorithm to compute inverses in the prime field $\GF(p)$. They employ a number of optimizations to achieve a running time of $O(n^2)$, and a space-requirement of $O(n)$ qubits, where $n$ is the number of bits in the binary representation of $p$ (there are some trade-offs that they make, sacrificing a few extra qubits to reduce running-time). In practice, elliptic curve cryptosystems often use curves over the binary field $\GF(2^m)$. In this paper, I show how to implement the extended Euclidean algorithm for polynomials to compute inverses in $\GF(2^m)$. Working under the assumption that qubits will be an `expensive' resource in realistic implementations, I optimize specifically to reduce the qubit space requirement, while keeping the running-time polynomial. The implementation here differs from that in $\cite{PZ03}$ for $\GF(p)$, and we are able to take advantage of some properties of the binary field $\GF(2^m)$. I also optimize the overall qubit space requirement for computing the group operation for elliptic curves over $\GF(2^m)$ by decomposing the group operation to make it ``piecewise reversible'' (similar to what is done in \cite{PZ03} for curves over $\GF(p)$).


Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA

Author(s):  
Mohd Javed ◽  
Khaleel Ahmad ◽  
Ahmad Talha Siddiqui

WiMAX is the innovation and upgradation of 802.16 benchmarks given by IEEE. It has numerous remarkable qualities, for example, high information rate, the nature of the service, versatility, security and portability putting it heads and shoulder over the current advancements like broadband link, DSL and remote systems. Though like its competitors the concern for security remains mandatory. Since the remote medium is accessible to call, the assailants can undoubtedly get into the system, making the powerless against the client. Many modern confirmations and encryption methods have been installed into WiMAX; however, regardless it opens with up different dangers. In this paper, we proposed Elliptic curve Cryptography based on Cellular Automata (EC3A) for encryption and decryption the message for improving the WiMAX security


Author(s):  
Chen Zhang ◽  
Huakang Bian ◽  
Kenta Yamanaka ◽  
Akihiko Chiba
Keyword(s):  

Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.


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