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**1 - 5**of**5**### A Simple Discrete System with Chaotic Behavior*

"... Abstract — We discuss the behavior of a particular discrete system, viz. Post’s system of tag with alphabet {0,1}, deletion number d = 3, and rules: 0 → 00, 1 → 1101. As initial string we consider all strings of length less than or equal to 15 as well as all ‘‘worst case’ ’ inputs of the form (100) ..."

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Abstract — We discuss the behavior of a particular discrete system, viz. Post’s system of tag with alphabet {0,1}, deletion number d = 3, and rules: 0 → 00, 1 → 1101. As initial string we consider all strings of length less than or equal to 15 as well as all ‘‘worst case’ ’ inputs of the form (100) m with 1 ≤ m ≤ 128. 1.

### The REESSE1+ Public Key Cryptosystem v 2.21 *

, 2012

"... Abstract: 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: ..."

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Abstract: 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 lg M ≈ 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.

### A Lightweight Hash Function Resisting Birthday Attack and Meet-in-the-middle Attack

"... Abstract: In this paper, to match a lightweight digital signing scheme of which the length of modulus is between 80 and 160 bits, a lightweight hash function called JUNA is proposed. It is based on the intractabilities MPP and ASPP, and regards a short message or a message digest as an input which i ..."

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Abstract: In this paper, to match a lightweight digital signing scheme of which the length of modulus is between 80 and 160 bits, a lightweight hash function called JUNA is proposed. It is based on the intractabilities MPP and ASPP, and regards a short message or a message digest as an input which is treated as only one block. The JUNA hash contains two algorithms: an initialization algorithm and a compression algorithm, and converts a string of n bits into another of m bits, where 80 ≤ m ≤ n ≤ 4096. The two algorithms are described, and their securities are analyzed from several aspects. The analysis shows that the JUNA hash is one-way, weakly collision-free, strongly collision-free along with a proof, especially resistant to birthday attack and meet-in-the-middle attack, and up to the security of O(2 m) arithmetic steps at present, while the time complexity of its compression algorithm is O(n) arithmetic steps. Moreover, the JUNA hash with short input and small computation may be used to reform a classical hash with output of n bits and security of O(2 n / 2) into a compact hash with output of n / 2 bits and equivalent security. Thus, it opens a door to convenience for utilization of lightweight digital signing schemes.