Boneh and Venkatesan proposed a problem called the \textit{hidden number problem} and they gave a polynomial time algorithm to solve it. And they showed that one can compute $g^{xy}$ from $g^{x}$ and $g^{y}$ if one has an oracle which computes roughly $\sqrt{\log p}$ most significant bits of $g^{xy}$ from given $g^{x}$ and $g^{y}$ in $\mathbb F_{p}$ by using an algorithm for solving the hidden number problem. Later, Shparlinski showed that one can compute $g^{x^{2}}$ if one can compute roughly $\sqrt{\log p}$ most significant bits of $g^{x^{2}}$ from given $g^{x}$. In this paper we extend these results by using some improvements on the hidden number problem and
the bound of exponential sums. We show that for given $g, g^{\alpha}, \ldots, g^{\alpha^{l}} \in \mathbb F_{\it p}^{*}$, computing about $\sqrt{\log p}$ most significant bits of
$g^{\frac{1}{\alpha}}$ is as hard as computing $g^{\frac{1}{\alpha}}$ itself, provided that the multiplicative order of $g$ is prime and not too small. Furthermore, we show that we can do it when $g$ has even much smaller multiplicative order in some special cases.

- Advisors
- Hahn, Sang-Geun
*researcher*; 한상근*researcher*

- Description
- 한국과학기술원 : 수리과학과,

- Publisher
- 한국과학기술원

- Issue Date
- 2011

- Identifier
- 466390/325007 / 020047188

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수리과학과, 2011.2, [ iii, 25 p. ]

- Keywords
Hidden number problem; Cryptography; Weak Diffie-Hellman problem; Weak Diffie-Hellman 문제; 숨겨진 수 문제; 암호학

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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