International Association
for Cryptologic Research

The key transform of the REESSE1+ asymmetric cryptosystem is Ci = (Ai * W ^ l(i)) ^ d (% M) with l(i) belonging to Omg = {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 lever function l(i) from {1, ..., n} to Omg = {+/-5, ..., +/-(n + 4)}, where "+/-" means the selection of the "+" or "-" sign, and discuss the necessity and sufficiency of the new l(i), namely that a simplified private key is insecure if l(i) is only a fixed integer, and secure at present if l(i) is a one-to-one function. Further, the sufficiency of the new l(i) is expounded from four aspects: (i) indeterminacy of the new l(i), (ii) insufficient conditions for the neutralizing the powers of W and W ^ -1 even if Omg = {5, ..., n + 4}, (iii) verification by examples, and (iv) running times of continued fraction attack and W-parameter intersection attack which are two most efficient algorithms so far but not determinate polynomial time ones. Last, the authors elaborate a relation between a lever function and a random oracle.