Construction of bent functions from near-bent functions

We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m−1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction.     

Efficient Biomimetic Hydroxylation Catalysis with a Bis(pyrazolyl)imidazolylmethane Copper Peroxide Complex

Bis(pyrazolyl)methane ligands are excellent components of model complexes used to investigate the activity of the enzyme tyrosinase. Combining the N donors 3‐tert‐butylpyrazole and 1‐methylimidazole results in a ligand that is capable of stabilising a (μ‐η²:η²)‐dicopper(II) core that resembles the active centre of tyrosinase. UV/Vis spectroscopy shows blueshifted UV bands in comparison to other known peroxo complexes, due to donor competition from different ligand substituents. This effect was investigated with the help of theoretical calculations, including DFT and natural transition orbital analysis. The peroxo complex acts as a catalyst capable of hydroxylating a variety of phenols by using oxygen. Catalytic conversion with the non‐biological phenolic substrate 8‐hydroxyquinoline resulted in remarkable turnover numbers. In stoichiometric reactions, substrate‐binding kinetics was observed and the intrinsic hydroxylation constant, k_ox, was determined for five phenolates. It was found to be the fastest hydroxylation model system determined so far, reaching almost biological activity. Furthermore, Hammett analysis proved the electrophilic character of the reaction. This sheds light on the subtle role of donor strength and its influence on hydroxylation activity.     

Nonlinear diffusion layers

In the practice of block cipher design, there seems to have grown a consensus about the diffusion function that designers choose linear functions with large branch numbers to achieve provable bounds against differential and linear cryptanalysis. In this paper, we propose two types of nonlinear functions as alternative diffusing components. One is based on a nonlinear code with parameters (16,256,6) which is known as a Kerdock code. The other is a general construction of nonlinear functions based on the T-functions, in particular, two automatons with modular addition operations. We show that the nonlinear functions possess good diffusion properties; specifically, the nonlinear function based on a Kerdock code has a better branch number than any linear counterparts, while the automatons achieve the same branch number as a linear near-MDS matrix. The advantage of adopting nonlinear diffusion layers in block ciphers is that, those functions provide extra confusion effect while a comparable performance in the diffusion effect is maintained. As an illustration, we show the application of the nonlinear diffusion functions in two example ciphers, where a 4-round differential characteristic with the optimal number of active Sboxes has a probability significantly lower (\(2^{16}\) and \(2^{10}\) times, respectively) than that of a similar cipher with a linear diffusion layer. As a result, it sheds light upon an alternative strategy of designing lightweight building blocks.     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

Highly Precise Corneal Surgery and All-Optical Histology with Femtosecond Lasers

We present an optimized femtosecond laser system for intrastromal tissue manipulation, application studies of commercial fs-LASIK procedures and results from the first successful demonstration of nonlinear (SHG) imaging of corneal ultrastructure and fs-laser-induced morphology modifications.