**A Lever Function to a New Codomain with Adequate Indeterminacy**

*Shenghui Su and Maozhi Xu and Shuwang Lu*

**Abstract: **The key transform of the REESSE1+ cryptosystem is Ci = (Ai * W ^ l(i)) ^ d (% M) with l(i) in O = {5, 7,
, 2n + 3} for i = 1,
, n, where l(i) is called a lever function. In this paper, the authors give a simplified transform Ci = Ai * W ^ l(i) (% M) and a new codomain Oħ = {+/-5, +/-6,
, +/-(n + 4)}, where +/- means the selection of the + or - sign. Discuss the necessity of l(.) to Oħ that a simplified private key is insecure if l(.) is only a fixed integer, and the sufficiency that a simplified private key is secure (namely Ci = Ai * W ^ l(i) (% M) is not faced with determinate polynomial time attack) if l(.) is a one-to-one function. The sufficiency is expounded from five aspects: indeterminacy of l(.) to Oħ, insufficiency of each of the four judgment conditions for counteraction of powers of W and W ^ -1 even if Oħ = {5, 6,
, n + 4}, verifying by examples, running times of continued fraction attack and indeterministic intersection attack most efficient now, and a relation between a lever function and a random oracle.

**Category / Keywords: **public-key cryptography / Public key cryptosystem; Coprime sequence; Lever function; Continued fraction attack; Random oracle

**Date: **received 1 May 2013, last revised 8 Oct 2013

**Contact author: **reesse at 126 com

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

**Note: **Very minor revision.

**Version: **20131008:102221 (All versions of this report)

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