Refuting the Pseudo Attack on the REESSE1+ Cryptosystem

We illustrate through example 1 and 2 that the condition at theorem 1 in [8] dissatisfies necessity, and the converse proposition of fact 1.1 in [8] does not hold, namely the condition Z / M – L / Ak < 1 / (2 Ak ) is not sufficient for f (i) + f (j) = f (k). Illuminate through an analysis and ex.3 that there is a logic error during deduction of fact 1.2, which causes each of fact 1.2, 1.3, 4 to be invalid. Demonstrate through ex.4 and 5 that each or the combination of qu+1 > qu ∆ at fact 4 and table 1 at fact 2.2 is not sufficient for f (i) + f (j) = f (k), property 1, 2, 3, 4, 5 each are invalid, and alg.1 based on fact 4 and alg.2 based on table 1 are disordered and wrong logically. Further, manifest through a repeated experiment and ex.5 that the data at table 2 is falsified, and the example in [8] is woven elaborately. We explain why Cx ≡ Ax W f(x) (% M) is changed to Cx ≡ (Ax W ) (% M) in REESSE1+ v2.1. To the signature fraud, we point out that [8] misunderstands the existence of T –1 and Q –1 % (M – 1), and forging of Q can be easily avoided through moving H. Therefore, the conclusion of [8] that REESSE1+ is not secure at all (which connotes that [8] can extract a related private key from any public key in REESSE1+) is fully incorrect, and as long as the parameter Ω is fitly selected, REESSE1+ with Cx ≡ Ax W f(x) (% M) is secure.