Analysis of Hidden Number Problem with Hidden Multiplier

Santanu Sarkar

doi:10.3934/amc.2017059

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2017, Volume 11, Issue 4: 805-811. Doi: 10.3934/amc.2017059

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Analysis of Hidden Number Problem with Hidden Multiplier

Santanu Sarkar

Indian Institute of Technology Madras, Sardar Patel Road, Chennai 600036, India

Received: August 31, 2016

Published: June 2018

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Abstract

In Crypto 1996, the Hidden Number Problem was introduced by Boneh and Venkatesan. Howgrave-Graham, Nguyen and Shparlinski (Mathematics of Computation 2003) generalized this problem and called it Hidden Number Problem with Hidden Multiplier (HNPHM). It has application in security analysis of timed-release crypto. They proposed a polynomial time algorithm to solve HNPHM. They showed that one can solve it if absolute error is less than $m^{0.20}$ for some positive integer $m$ . They improved this bound up to $m^{0.25}$ heuristically. It was also proved that one can not solve HNPHM if error is larger than $m^{0.5}$ . In this paper, we show that one can solve HNPHM in polynomial time heuristically if error is bounded by $m^{0.5}$ .

Keywords:

Hidden Number Problem with Hidden Multiplier,
lattice,
LLL algorithm,
sequence.

Mathematics Subject Classification: Primary: 11Y16; Secondary: 94A60.

Citation:

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Figure 1. Upper bound of $\Delta$ for increasing values of $n$ and $m \to \infty$

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Table 1. Experimental results for different values of $m$

$n$	$\bigg(\frac{\binom{n}{2}}{2\binom{n}{2}+2n}\bigg)\log_2 m$	$\log_2 \! m$	Experimental result	LLL Algorithm time in Seconds
	298	768	295	16.78
8	398	1024	395	24.39
	796	2048	793	48.49
	314	768	310	77.76
10	418	1024	415	108.96
	837	2048	834	244.68
	324	768	321	302.93
12	433	1024	429	406.54
	866	2048	863	930.16
	332	768	328	832.71
14	443	1024	439	1342.42
	887	2048	883	2798.76
	338	768	333	2283.69
16	451	1024	446	3138.10
	903	2048	899	8095.19

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References

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