Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function f used as a cryptographic primitive, and interpreted as a multivariate polynomial over F/sub 2/, should not have low degree multiples obtained by multiplication with low degree nonzero functions. In this paper, we show that a Boolean function having low nonlinearity is (also) weak against algebraic attacks, and we extend this result to higher order nonlinearities. Next, we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their algebraic immunity. We identify that functions having low-degree subfunctions are weak in terms of algebraic immunity, and we analyze some existing constructions from this viewpoint. Further, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity /spl lceil/n/2/spl rceil/ (this construction, first presented at the 2005 Workshop on Fast Software Encryption (FSE 2005), has been the first one producing such functions). These functions are obtained through a doubly indexed recursive relation. We calculate their Hamming weights and deduce their nonlinearities; we show that they have very high algebraic degrees. We express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables. We deduce a very fast way of computing the output to these functions, given their input.     

Antibiotic resistance in the European countries

The incidence of complications was recorded for 269 consecutive neonatal endotracheal intubations after instituting a routine policy of premedication. Two hundred and fifty-three of the intubations were premedicated with a combination of atropine, fentanyl and succinylcholine, 194 were without incident, 28 required two attempts, 22 required more than two attempts, and 9 required a second attempt with a smaller endotracheal tube. All infants were successfully intubated. We conclude that a policy of routine neonatal premedication for intubation is safe, feasible, and humane.     

Hemodynamic profile in severe ARDS: results of the European Collaborative ARDS Study

OBJECTIVE: Although the acute respiratory distress syndrome (ARDS) was identified as long as 30 years ago, potential therapeutic objectives have been defined from small series rather than large trials. Moreover, relationships between ARDS and hemodynamics are unclear. The European Collaborative ARDS Study was designed to identify factors influencing the pathogenesis, severity, and prognosis of ARDS. Analysis of the hemodynamic profiles collected during this study and of their contribution to the above-mentioned facets of ARDS is the focus of the present report. DESIGN: Prospective clinical study. SETTING: 38 European intensive care units (ICUs). PATIENTS AND METHODS: We collected 2758 sets of data from 586 patients, including baseline data, data on proven or suspected causes of ARDS differentiating direct and nondirect lung injury, and data on baseline status including multiple organ dysfunction (MOD) with differentiation of primary ARDS from ARDS secondary to severe systemic disorders. Events during follow-up were also recorded, including whether the acute respiratory failure did or did not improve after 24 h (groups A and B, respectively). When available, hemodynamic data were recorded at enrollment (day 0), on days 1-3, 7, 14, and 21, and at discharge or at the time of death in the ICU. RESULTS: Although the rate of pre-existing disease and the nature and rate of complications varied widely among etiologic categories, differences in the hemodynamic profile occurred only between primary and secondary ARDS. Both at inclusion and during the course of the illness, variables that were used to investigate Va/Q mismatch [arterial oxygen tension (PaO2, arterial oxygen saturation, right-to-left shunt, and the PaO2/fractional inspired oxygen (FIO2) ratio] predicted survival. High pulmonary artery pressure (PAP) and low systemic artery pressure (SAP) were also related to the prognosis. In the logistic regression model including hemodynamic and oxygen-related variables, however, the only independent predictors of survival were the ratio of right over left ventricular stroke work (RVSW/LVSW) and the PaO2/FIO2 ratio at admission. On day 2, the best prognostic model included: age [odds ratio (OR) = 1.04, p = 0.0004], opportunistic pneumonia as the cause of ARDS (OR = 3.2, p = 0.03), existence of MOD (OR = 1.9, p = 0.03), PaO2/FIO2 (OR = 0.96, p = 0.005), and RVSW/LVSW (OR = 25, p = 0.02). A high RVSW/LVSW ratio, high systolic PAP, low diastolic SAP, and low PaO2/FIO2, and increased right atrial pressure were negative prognostic indicators during follow-up. CONCLUSION: In addition to the cause of ARDS and the early time-course of lung function, a high systolic PAP and a low diastolic SAP were strong independent indicators of survival.     

Problems and solutions from the fourth International Students’ Olympiad in Cryptography (NSUCRYPTO)

Mathematical problems and their solutions from the fourth International Students’ Olympiad in cryptography (NSUCRYPTO-2017) are presented. We consider problems related to attacks on ciphers and hash functions, cryptographic Boolean functions, linear branch numbers, addition chains, and error correction codes, among others. We discuss several open problems involving the algebraic structures of cryptographic functions, useful proof-of-work algorithms, the Boolean hidden shift problem, and quantum computing.     

On cryptographic properties of the cosets of R(1, m)

We introduce a new approach for the study of weight distributions of cosets of the Reed-Muller code of order 1. Our approach is based on the method introduced by Kasami (1968), using Pless (1963) identities. By interpreting some equations, we obtain a necessary condition for a coset to have a “high” minimum weight. Most notably, we are able to distinguish such cosets which have three weights only. We then apply our results to the problem of the nonlinearity of Boolean functions. We particularly study the links between this criterion and the propagation characteristics of a function     

New classes of almost bent and almost perfect nonlinear polynomials

New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function.     

Z2k-linear codes

We introduce a generalization to Z₂k of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes     

New secondary constructions of Bent functions

In this paper, we first present a novel secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, the algebraic degree and algebraic immunity of the constructed functions are analysed. Finally, we apply the construction using as initial functions some specific bent functions and then specify sufficient conditions for the resulting bent functions not to be contained in the completed Maiorana–McFarland class. In the second part of the paper, we present a corrigendum of “Constructions of bent–negabent functions and their relation to the completed Maiorana–McFarland Class” (IEEE Trans Inf Theory 61(3):1496–1506, 2015).