A Multivariate Public Key Cryptographic Scheme

This paper presents a multivariate public key cryptographic scheme over a finite field with odd prime characteristic.The idea of embedding and layering is manifested in its construction.The security of the scheme is analyzed in detail,and this paper indicates that the scheme can withstand the up to date differential cryptanalysis.We give heuristic arguments to show that this scheme resists all known attacks.

Upper bounds of Schubert polynomials

Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of w.The Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following coefficient-wise inequality:Min_(x)≤■_(w)(x)≤Max_(w)xwhere both Min_(w)(x)and Max_(w)(x)are polynomials determined by D(w).Fink et al.(2018)found that■w_(x)equals the lower bound Min_(w)(x)if and only if w avoids twelve permutation patterns.In this paper,we show that■w_(x)reaches the upper bound Max_(w)(x)if and only if w avoids two permutation patterns 1432 and 1423.Similarly,for any given compositionα∈Z^(n)≥0,one can define a lower bound Min_(α)(x)and an upper bound Max_(α)(x)for the key polynomialκ_(α)(x).Hodges and Yong(2020)established thatκ_(α)(x)equals Min_(α)(x)if and only ifαavoids five composition patterns.We show thatκ_(α)(x)equals Max_(α)(x)if and only ifαavoids a single composition pattern(0,2).As an application,we obtain that whenαavoids(0,2),the key polynomialκ_(α)(x)is Lorentzian,partially verifying a conjecture of Huh et al.(2019).

Can the Polynomial Based Key Predistribution Scheme Be Used Many Times in One Wireless Sensor Network Key Establishment Protocol?

Key establishment is the basic step for the wireless sensor network （WSN） security. The polynomial based key predistribution scheme of Blom and Blundo et al. has been the basic ingredient for the key establishment for WSNs. It is tempting to use many random and different instances of polynomial based key predistribution scheme for various parts of the WSN to enhance the efficiency of WSN key establishment protocols. This paper indicates that it is not secured in general to use many instances of Blom-Blundo et al. polynomial based key predistribution scheme in a WSN key establishment protocol. Thus the previously constructed group-based type WSN key predistribution schemes using polynomial based key predistribution scheme are insecure. We propose new generalized Bloin-Blundo et al. key predistribution schemes. These new generalized Blom-Blundo et al. key predistribution schemes can be used many times in one WSN key establishment protocol with only a small increase of cost. The application to group-based WSN key predistribution schemes is given.