**A New Lever Function with Adequate Indeterminacy**

*Shenghui Su and Ping Luo and Shuwang Lv and Maozhi Xu*

**Abstract: **The key transform of the REESSE1+ asymmetrical cryptosystem is Ci = (Ai * W ^ l(i)) ^ d (% M) with l(i) in Omega = {5, 7, ..., 2n + 3} for i = 1, ..., n, where l(i) is called a lever function. In this paper, the authors give a simplified key transform Ci = Ai * W ^ l(i) (% M) with a new lever function l(i) from {1, ..., n} to Omega = {+/-5, +/-6, ..., +/-(n + 4)}, where "+/-" means the selection of the "+" or "-" sign. Discuss the necessity of the new l(i), namely that a simplified private key is insecure if the new l(i) is a constant but not one-to-one function. Further, expound the sufficiency of the new l(i) from four aspects: (1) indeterminacy of the new l(i), (2) insufficient conditions for neutralizing the powers of W and W ^-1 even if Omega = {5, 6, ..., n + 4}, (3) verification by examples, and (4) running times of the continued fraction attack and W-parameter intersection attack which are the two most efficient of the probabilistic polytime attack algorithms so far. Last, the authors elaborate the relation between a lever function and a random oracle.

**Category / Keywords: **foundations / Asymmetric cryptosystem, Coprime sequence, Lever function, Continued fraction attack, Random oracle

**Date: **received 1 May 2013, last revised 31 Aug 2021

**Contact author: **idology98 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Note: **The content has no essential change.

**Version: **20210831:125059 (All versions of this report)

**Short URL: **ia.cr/2013/246

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