  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.
