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تسجيل مستخدم جديدDesign and Analysis of the REESSE1+ Public Key Cryptosystem v2.21
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In this paper, the authors give the definitions of a coprime sequence and a lever function, and describe the five algorithms and six characteristics of a prototypal public key cryptosystem which is used for encryption and signature, and based on three new problems and one existent problem: the multivariate permutation problem (MPP), the anomalous subset product problem (ASPP), the transcendental logarithm problem (TLP), and the polynomial root finding problem (PRFP). Prove by reduction that MPP, ASPP, and TLP are computationally at least equivalent to the discrete logarithm problem (DLP) in the same prime field, and meanwhile find some evidence which inclines people to believe that the new problems are harder than DLP each, namely unsolvable in DLP subexponential time. Demonstrate the correctness of the decryption and the verification, deduce the probability of a plaintext solution being nonunique is nearly zero, and analyze the exact securities of the cryptosystem against recovering a plaintext from a ciphertext, extracting a private key from a public key or a signature, and forging a signature through known signatures, public keys, and messages on the assumption that IFP, DLP, and LSSP can be solved. Studies manifest that the running times of effectual attack tasks are greater than or equal to O(2^n) so far when n = 80, 96, 112, or 128 with lgM = 696, 864, 1030, or 1216. As viewed from utility, it should be researched further how to decrease the length of a modulus and to increase the speed of the decryption.
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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^2) 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 * D 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^f(x))^d (% 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 Omega is fitly selected, REESSE1+ with Cx = Ax * W^f(x) (% M) is secure.
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