Transparency order versus confusion coefficient: a case study of NIST lightweight cryptography S-Boxes

Side-channel resistance is nowadays widely accepted as a crucial factor in deciding the security assurance level of cryptographic implementations. In most cases, non-linear components (e.g. S-Boxes) of cryptographic algorithms will be chosen as primary targets of side-channel attacks (SCAs). In order to measure side-channel resistance of S-Boxes, three theoretical metrics are proposed and they are reVisited transparency order (VTO), confusion coefficients variance (CCV), and minimum confusion coefficient (MCC), respectively. However, the practical effectiveness of these metrics remains still unclear. Taking the 4-bit and 8-bit S-Boxes used in NIST Lightweight Cryptography candidates as concrete examples, this paper takes a comprehensive study of the applicability of these metrics. First of all, we empirically investigate the relations among three metrics for targeted S-boxes, and find that CCV is almost linearly correlated with VTO, while MCC is inconsistent with the other two. Furthermore, in order to verify which metric is more effective in which scenarios, we perform simulated and practical experiments on nine 4-bit S-Boxes under the non-profiled attacks and profiled attacks, respectively. The experiments show that for quantifying side-channel resistance of S-Boxes under non-profiled attacks, VTO and CCV are more reliable while MCC fails. We also obtain an interesting observation that none of these three metrics is suitable for measuring the resistance of S-Boxes against profiled SCAs. Finally, we try to verify whether these metrics can be applied to compare the resistance of S-Boxes with different sizes. Unfortunately, all of them are invalid in this scenario.

On the confusion coefficient of Boolean functions

The notion of the confusion coefficient is a property that attempts to characterize confusion property of cryptographic algorithms against differential power analysis. In this article, we establish a relationship between the confusion coefficient and the autocorrelation function for any Boolean function and give a tight upper bound and a tight lower bound on the confusion coefficient for any (balanced) Boolean function. We also deduce some deep relationships between the sum-of-squares of the confusion coefficient and other cryptographic indicators (the sum-of-squares indicator, hamming weight, algebraic immunity and correlation immunity), respectively. Moreover, we obtain some trade-offs among the sum-of-squares of the confusion coefficient, the signal-to-noise ratio and the redefined transparency order for a Boolean function.

On the Modified Transparency Order of n , m -Functions

The concept of transparency order is introduced to measure the resistance of n , m -functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by TO ), redefined transparency order (denoted by RTO ), and modified transparency order (denoted by MTO ). In this paper, we firstly give a relationship between MTO and RTO and show that RTO is less than or equal to MTO for any n , m -functions. We also give a tight upper bound and a tight lower bound on MTO for balanced n , m -functions. Secondly, some relationships between MTO and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for n , m -functions are obtained, respectively. Finally, we give MTO and RTO for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.

The Notion of Transparency Order, Revisited

We revisit the definition of transparency order (TO) and that of modified transparency order (MTO) as well, which were proposed to measure the resistance of substitution boxes (S-boxes) against differential power analysis (DPA). We spot a definitional flaw in original TO, which is proved to significantly affect the soundness of TO. Regretfully, MTO overlooks this flaw, yet it happens to incur no bad effects on the correctness of MTO, even though the start point of this formulation is highly questionable. It is also this neglect that made MTO consider a variant of multi-bit DPA attack, which was mistakenly thought to appropriately serve as an alternative powerful attack. This implies the soundness of MTO is also more or less arguable. Therefore, we fix this definitional flaw and provide a revised definition named reVisited TO (VTO). For demonstrating validity and soundness of VTO, we present simulated and practical DPA attacks on implementations of $4\times 4$ and $8\times 8$ S-boxes. In addition, we also illustrate the soundness of VTO in masked S-boxes. Furthermore, as a concrete application of VTO, we present the distribution of VTO values of optimal affine equivalence classes of $4\times 4$ S-boxes and give some recommended guidelines on how to select $4\times 4$ S-boxes with higher DPA resistance at the identical level of implementation cost.

Poręczne obrazy, wymowne przedmioty. Retoryka rzeczowości w „L'Esprit Nouveau” Ozenfanta i Le Corbusiera

The paper is an attempt to draw the reader’s attention to visual reproduction as an element of modern artistic discourses and a medium of the mediated reception of art. An instrumental approach to reproduction as a neutral and ancillary vehicle of meaning, prevalent in the age of modernism, corresponded to the belief in its information efficacy and ability to overcome material, physical limitations. What mattered most were not the material, physical aspects of the existence and circulation of images, even though the avant-garde artists of the 1920s, using contemporary technology, were aware how important the medium’s and its distribution range’s “impact” was. L’Esprit Nouveau, a periodical edited in 1920-1925 by Amédée Ozenfant and Le Corbusier, was an example of a successful avant-garde strategy which let both editors, marginal in the field of art, achieve the status of “leaders” of the modernist movement, recognized or at least carefully watched by artists and critics abroad. Next to other factors, important was the visual aspect of the magazine, praised for many impressive, modern illustrations, often reproduced in other avant-garde publications. The author analyzes visual resources used and reproduced in L’Esprit Nouveau, referring to the postulates of “objectivism” and “thingness”, endorsed by the periodical, and considering the part that “ready-made” images, found in the daily press and commercial catalogues as well as on postcards. played in Le Corbusier’s polemical and programmatic texts. Their strongly persuasive message was often rooted in montage and quotations which stressed its heterogeneity. In terms of composition and aesthetics, the reproduced images supported the aesthetics of transparency, order, and thingness, so characteristic of L’Esprit Nouveau. The emblems of modernity emerged from the movement of anonymous images which acquired the value of symbols. Ozenfant’s and Le Corbusier’s use of images borrowed from popular culture, as well as from albums and art books, makes one consider not only their rhetorical effectiveness, but also their role in the creative process and thinking. In Le Corbusier’s artistic practice, those easily available, miniaturized images were a common instrument enhancing his visual, aesthetic approach. Such an approach, according to Georg Simmel, seems to be characteristic of the modernist attitude to the material world that consisted in subjective distance combined with the apparently opposite desire to “go back to things” by making them more concrete and closer to the senses.