A New Lever Function with Adequate Indeterminacy

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 page

Beauty of Cryptography: the Cryptographic Sequences and the Golden Ratio

In this paper, the authors construct a new type of cryptographic sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[1], a[2], ..., a[n]} and minimal extra-super increasing sequence {z[1], z[2], ..., z[n]}. Prove that the minimal extra-super increasing sequence is the odd-positioned subsequence of the Fibonacci sequence, namely {z[1], z[2], ..., z[n], ...} = {F[1], F[3], ..., F[2n-1], ...}, which indicates that the approach to the golden ratio phi through the term difference ratio (z[n+1] - z[n]) / z[n] is more smooth and expeditious than through the term ratio (F[n+1] / F[n]). Further prove that the limit of the term ratio difference between the two cryptographic sequences equals the golden ratio conjugate PHI, namely lim (n to infinity) (z[n+1] / z[n] - a[n+1] / a[n]) = PHI, which reveals the beauty of cryptography

A New Lever Function with Adequate Indeterminacy

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, we 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)}. 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) the running time of continued fraction attack and running time of W-parameter intersection attack which are the two most efficient of the probabilistic polytime attack algorithms so far. Last, we detail the relation between a lever function and a random oracle

The REESSE1+ Public Key Cryptosystem v 2.21

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

Elemental Impurities in Pediatric Calcium Carbonate Preparations-High Throughput Quantification and Risk Assessment

Calcium carbonate which is extracted from the Earth in combination with other mineral impurities, is largely used in preparations for pediatric supplements. Elemental impurities in drug products pose toxicological concerns without therapeutic benefits. Thus, it is very urgent to assess the safety of chronic exposure to elements that may be present in trace amounts. In the present study, we developed high throughput ICP-MS method for the quantitative determination of 62 elemental impurities in high matric calcium carbonate samples and validated according to USP 233. Calcium carbonate preparations which state clearly used for child (including neonates, infants, toddlers and children) from 9 manufactures and two types of raw materials (light calcium carbonate and ground calcium carbonate) were investigated in terms of the content and variability of 62 elemental impurities. According to the results, ground calcium carbonate was more suitable to be used in pediatric preparations concerning elemental impurities. Parts of elemental impurities in CaCO3 preparations which are derived from the raw materials and the preparation process, may cause potential risks for children. These results indicate that it is necessary to establish a modern instrumental analysis method to evaluate and control elemental impurities in CaCO3 raw materials and preparations.</jats:p