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Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called I O -implications) in this paper. More specifically, we first give the definition of the 1 st type of FRSs and study some essential properties. Subsequently, the definition of the 2 nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1 st type of FRSs and the 2 nd type of FRSs is described. That is, in a fuzzy approximation space, the 2 nd type of lower approximation is contained in the 1 st type of lower approximation, while the 1 st type of upper approximation is included in the 2 nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses. [ABSTRACT FROM AUTHOR]

• Introduction of the Common Prime RSA and a brief description of its background mathematics. • An overview and explanation of previous CP-RSA cryptanalytic attacks. • Formulation of new cryptanalytic attack on CP-RSA using Lattice Basis Reduction method. • Explanation and detail description of new Cryptanalytic attack solution on CP-RSA. • CP-RSA Speed and efficiency measures w.r.t practical and asymptotic aspects. • Detail explanation and comparison results with the previous CP-RSA cryptanalytic attacks. Standard RSA cryptosystem becomes vulnerable, when private key d < N 0.292 is used inside CryptoChips of constrained devices, thus an alternate scheme is the Common Prime RSA (CP-RSA) variant, which provides cryptographic (decryption/signing) operations. In this paper, we perform a cryptanalytic attack on CP-RSA using lattice basis reduction method that is used to exploit possible vulnerabilities of RSA small private key attacks. In addition, we performed detail experiments on CP-RSA weak or overestimated bounds and compare results to the past studies. Our implemented cryptanalytic attack implicates more precise and direct method to exploit the CP-RSA existing theoretical and experimental bounds. Also, our results prove that CP-RSA is an effective approach that provides resistance against standard RSA small private key attacks. [ABSTRACT FROM AUTHOR]