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首页> 外文期刊>Journal of Applied Mathematics and Computing > $${au }$$ τ -adic expansions on hyperelliptic Koblitz curves]]>
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$${au }$$ τ -adic expansions on hyperelliptic Koblitz curves]]>

机译： $$ { tau} $$ τ 超细koblitz曲线上的 -Adic扩展]]]>

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摘要

In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the $${au }$$ τ -adic non-adjacent form ( $$au $$ τ -NAF). The $$au $$ τ -NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. On the other hand, Günther et al.?(Speeding up the arithmetic on koblitz curves of genus two. LNCS, vol. 2012, pp. 106–117. Springer, Heidelberg, 2001) have proposed two generalizations of $$au $$ τ -NAF for a family of hyperelliptic curves ( hyperelliptic Koblitz curves ) which have been proposed by Koblitz (J Cryptol 1(3):139–150, 1989). We call these generalizations $${au }$$ τ - adic sparse expansion , and $${au }$$ τ - NAF , respectively. To our knowledge, it is not known whether the three properties are true or not, especially, the existence must be satisfied for concrete cryptographic implementations. We provide an answer to the question. Our investigation shows that the $${au }$$ τ -adic sparse expansion has only the existence and the $${au }$$ τ -NAF has the existence and uniqueness. Our results guarantee the concrete cryptographic implementations of these generalizations.
机译：在椭圆曲线密码系统中，众所周知，Koblitz曲线承认使用$$ { tau} $$τ-adic非相邻的表单（$$ tau $$τ添加算法naf）。 $$ Tau $$τ-naf有三种属性：（1）存在，（2）唯一性，（3）汉明重的最小值。另一方面，Günther等人。 Koblitz（J Cryptol 1（3）：139-150,1989提出的一系列高度椭圆曲线（高度型Koblitz曲线）的$τ-naf。我们称这些概括为$$ { tau} $$τ - adic稀疏扩展，以及$$ { tau} $$τ-naf。据我们所知，尚不知道三个属性是否为真，特别是，必须满足具体加密实现。我们提供了这个问题的答案。我们的调查表明，$$ { tau} $$τ-adic稀疏扩展只有存在，$$ { tau} $$τ-naf具有存在和唯一性。我们的结果保证了这些概括的具体加密实现。

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