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In this paper, authors construct a new type of sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[0], a[1], ..., a[n]} and minimal extra-super increasing sequence {z[0], z[1], ..., z[n]}. Find that there always exists a fit n which makes (z[n] / z[n-1] - a[n] / a[n-1])= PHI, where PHI is the golden ratio conjugate with a finite precision in the range of computer expression. Further, derive the formula radic(5) = 2(z[n] / z[n-1] - a[n] / a[n-1]) + 1, where n corresponds to the demanded precision. Experiments demonstrate that the approach to radic(5) through a term ratio difference is more smooth and expeditious than through a Taylor power series, and convince the authors that lim(n to infinity) (z[n] / z[n-1] - a[n] / a[n-1]) = PHI holds., Comment: 6 pages

The key transform of the REESSE1+ asymmetrical cryptosystem is Ci = (Ai * W ^ l(i)) ^ d (% M) with l(i) in Omega = {5, 7, ..., 2n + 3} for i = 1, ..., n, where l(i) is called a lever function. In this paper, the authors give a simplified key transform Ci = Ai * W ^ l(i) (% M) with a new lever function l(i) from {1, ..., n} to Omega = {+/-5, +/-6, ..., +/-(n + 4)}, where "+/-" means the selection of the "+" or "-" sign. Discuss the necessity of the new l(i), namely that a simplified private key is insecure if the new l(i) is a constant but not one-to-one function. Further, expound the sufficiency of the new l(i) from four aspects: (1) indeterminacy of the new l(i), (2) insufficient conditions for neutralizing the powers of W and W ^-1 even if Omega = {5, 6, ..., n + 4}, (3) verification by examples, and (4) running times of continued fraction attack and W-parameter intersection attack which are the two most efficient algorithms of the probabilistic polytime attacks so far. Last, the authors detail the relation between a lever function and a random oracle., Comment: 13 pages

This paper proposes a general algorithm called Store-zechin for quickly computing the permanent of an arbitrary square matrix. Its key idea is storage, multiplexing, and recursion. That is, in a recursive process, some sub-terms which have already been calculated are no longer calculated, but are directly substituted with the previous calculation results. The new algorithm utilizes sufficiently computer memories and stored data to speed the computation of a permanent. The Analyses show that computating the permanent of an n * n matrix by Store-zechin requires (2^(n - 1)- 1)n multiplications and (2^(n-1))(n - 2)+ 1 additions while does (2^n - 1)n + 1 multiplications and (2^n - n)(n + 1)- 2 additions by the Ryser algorithm, and does (2^(n - 1))n + (n + 2) multiplications and (2^(n - 1))(n + 1)+ (n^2 - n -1) additions by the R-N-W algorithm. Therefore, Store-zechin is excellent more than the latter two algorithms, and has a better application prospect., Comment: Unsuitable

A new algorithm which is called Store-zechin, and utilizes stored data repetitively for calculating the permanent of an n * n matrix is proposed. The analysis manifests that the numbers of multiplications and additions taken by the new algorithm are respectively far smaller than those taken by the famous Ryser algorithm. Especially, for a 5-boson sampling task, the running time of the Store-zechin algorithm computing the correspondent permanent on ENIAC as well as TRADIC is lower than that of the sampling operation on a multiphoton boson sampling machine (shortly MPBSM), and thus MPBSM does not beat the early classical computers (despite of this, it is possible that when n gets large enough, a quantum boson sampling machine will beat a classical computer). On a computer, people can design an algorithm that exchanges space for time while on MPBSM, people can not do so, which is the greatest difference between a universal computer and MPBSM. This difference is right the reason why MPBSM may not be called a (photonic) quantum computer., Comment: 8 pages

This paper gives the definitions of an extra superincreasing sequence and an anomalous subset sum, and proposes a fast quantum-safe asymmetric cryptosystem called JUOAN2. The new cryptosystem is based on an additive multivariate permutation problem (AMPP) and an anomalous subset sum problem (ASSP) which parallel a multivariate polynomial problem and a shortest vector problem respectively, and composed of a key generator, an encryption algorithm, and a decryption algorithm. The authors analyze the security of the new cryptosystem against the Shamir minima accumulation point attack and the LLL lattice basis reduction attack, and prove it to be semantically secure (namely IND-CPA) on the assumption that AMPP and ASSP have no subexponential time solutions. Particularly, the analysis shows that the new cryptosystem has the potential to be resistant to quantum computing attack, and is especially suitable to the secret communication between two mobile terminals in maneuvering field operations under any weather. At last, an example explaining the correctness of the new cryptosystem is given., Comment: 8 Pages. arXiv admin note: text overlap with arXiv:1408.6226

Fraud (swindling money, property, or authority by fictionizing, counterfeiting, forging, or imitating things, or by feigning other persons privately) forms its threats against public security and network security. Anti-fraud is essentially the identification of a person or thing. In this paper, the authors first propose the concept of idology - a systematic and scientific study of identifications of persons and things, and give the definitions of a symmetric identity and an asymmetric identity. Discuss the converting symmetric identities (e.g., fingerprints) to asymmetric identities. Make a comparison between a symmetric identity and an asymmetric identity, and emphasize that symmetric identities cannot guard against inside jobs. Compare asymmetric RFIDs with BFIDs, and point out that a BFID is lightweight, economical, convenient, and environmentalistic, and more suitable for the anti-counterfeiting and source tracing of consumable merchandise such as foods, drugs, and cosmetics. The authors design the structure of a united verification platform for BFIDs and the composition of an identification system, and discuss the wide applications of BFIDs in public security and network security - antiterrorism and dynamic passwords for example., Comment: 16 pages, 13 figures

This paper gives the definition and property of a bit-pair shadow, and devises the three algorithms of a public key cryptoscheme called JUOAN that is based on a multivariate permutation problem and an anomalous subset product problem to which no subexponential time solutions are found so far, and regards a bit-pair as a manipulation unit. The authors demonstrate that the decryption algorithm is correct, deduce the probability that a plaintext solution is nonunique is nearly zero, analyze the security of the new cryptoscheme against extracting a private key from a public key and recovering a plaintext from a ciphertext on the assumption that an integer factorization problem, a discrete logarithm problem, and a low-density subset sum problem can be solved efficiently, and prove that the new cryptoscheme using random padding and random permutation is semantically secure. The analysis shows that the bit-pair method increases the density D of a related knapsack to a number more than 1, and decreases the modulus length lgM of the new cryptoscheme to 464, 544, or 640., Comment: 16 Pages. arXiv admin note: text overlap with arXiv:1408.5999

To examine the integrity and authenticity of an IP address efficiently and economically, this paper proposes a new non-Merkle-Damgard structural (non-MDS) hash function called JUNA that is based on a multivariate permutation problem and an anomalous subset product problem to which no subexponential time solutions are found so far. JUNA includes an initialization algorithm and a compression algorithm, and converts a short message of n bits which is regarded as only one block into a digest of m bits, where 80 <= m <= 232 and 80 <= m <= n <= 4096. The analysis and proof show that the new hash is one-way, weakly collision-free, and strongly collision-free, and its security against existent attacks such as birthday attack and meet-in-the- middle attack is to O(2 ^ m). Moreover, a detailed proof that the new hash function is resistant to the birthday attack is given. Compared with the Chaum-Heijst-Pfitzmann hash based on a discrete logarithm problem, the new hash is lightweight, and thus it opens a door to convenience for utilization of lightweight digital signing schemes., Comment: 18 Pages

It is well known that the inverse function of y = x with the derivative y' = 1 is x = y, the inverse function of y = c with the derivative y' = 0 is inexistent, and so on. Hence, on the assumption that the noninvertibility of the univariate increasing function y = f(x) with x > 0 is in direct proportion to the growth rate reflected by its derivative, the authors put forward a method of comparing difficulties in inverting two functions on a continuous or discrete interval called asymptotic granularity reduction (AGR) which integrates asymptotic analysis with logarithmic granularities, and is an extension and a complement to polynomial time (Turing) reduction (PTR). Prove by AGR that inverting y = x ^ x (mod p) is computationally harder than inverting y = g ^ x (mod p), and inverting y = g ^ (x ^ n) (mod p) is computationally equivalent to inverting y = g ^ x (mod p), which are compatible with the results from PTR. Besides, apply AGR to the comparison of inverting y = x ^ n (mod p) with y = g ^ x (mod p), y = g ^ (g1 ^ x) (mod p) with y = g ^ x (mod p), and y = x ^ n + x + 1 (mod p) with y = x ^ n (mod p) in difficulty, and observe that the results are consistent with existing facts, which further illustrates that AGR is suitable for comparison of inversion problems in difficulty. Last, prove by AGR that inverting y = (x ^ n)(g ^ x) (mod p) is computationally equivalent to inverting y = g ^ x (mod p) when PTR can not be utilized expediently. AGR with the assumption partitions the complexities of problems more detailedly, and finds out some new evidence for the security of cryptosystems., Comment: 13 pages

The authors discuss what is provable security in cryptography. Think that provable security is asymptotic, relative, and dynamic, and only a supplement to but not a replacement of exact security analysis. Because the conjecture P != NP has not been proven yet, and it is possible in terms of the two incompleteness theorems of Kurt Godel that there is some cryptosystem of which the security cannot or only ideally be proven in the random oracle model, the security of a cryptosystem is between provability and unprovability, and any academic conclusion must be checked and verified with practices or experiments as much as possible. Extra, a new approach to proof of P != NP is pointed out. Lastly, a reward is offered for the subexponential time solutions to the three REESSE1+ problems: MPP, ASPP, and TLP with n >= 80 and lg M >= 80, which may be regarded as a type of security proof by experiment., Comment: 5 pages

This paper gives the definitions of an anomalous super-increasing sequence and an anomalous subset sum separately, proves the two properties of an anomalous super-increasing sequence, and proposes the REESSE2+ public-key encryption scheme which includes the three algorithms for key generation, encryption and decryption. The paper discusses the necessity and sufficiency of the lever function for preventing the Shamir extremum attack, analyzes the security of REESSE2+ against extracting a private key from a public key through the exhaustive search, recovering a plaintext from a ciphertext plus a knapsack of high density through the L3 lattice basis reduction method, and heuristically obtaining a plaintext through the meet-in-the-middle attack or the adaptive-chosen-ciphertext attack. The authors evaluate the time complexity of REESSE2+ encryption and decryption algorithms, compare REESSE2+ with ECC and NTRU, and find that the encryption speed of REESSE2+ is ten thousand times faster than ECC and NTRU bearing the equivalent security, and the decryption speed of REESSE2+ is roughly equivalent to ECC and NTRU respectively., Comment: 11 pages, 2 tables

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., Comment: 14 pages, and 2 table

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., Comment: 21 pages