The SQALE of CSIDH: sublinear Vélu quantum-resistant isogeny action with low exponents

Recent independent analyses by Bonnetain–Schrottenloher and Peikert in Eurocrypt 2020 significantly reduced the estimated quantum security of the isogeny-based commutative group action key-exchange protocol CSIDH. This paper refines the estimates of a resource-constrained quantum collimation sieve attack to give a precise quantum security to CSIDH. Furthermore, we optimize large CSIDH parameters for performance while still achieving the NIST security levels 1, 2, and 3. Finally, we provide a C-code constant-time implementation of those CSIDH large instantiations using the square-root-complexity Vélu’s formulas recently proposed by Bernstein, De Feo, Leroux and Smith.

Group Structure and Geometric Interpretation of the Embedded Scator Space

The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric. In this paper, the scator space of dimension 1+n (for n=2 and n=3) is interpreted as an intersection of some quadrics in the pseudo-Euclidean space of dimension 2n with zero signature. The scator product, nondistributive and rather counterintuitive in its original formulation, is represented as a natural commutative product in this extended space. What is more, the set of invertible embedded scators is a commutative group. This group is isomorphic to the group of all symmetries of the embedded scator space, i.e., isometries (in the space of dimension 2n) preserving the scator quadrics.

Moment Functions on Groups

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

COMPARISON OF KUMMER LOGARITHMIC TOPOLOGIES WITH CLASSICAL TOPOLOGIES

We compare the Kummer flat (resp., Kummer étale) cohomology with the flat (resp., étale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes, and Kato’s logarithmic multiplicative group. We are particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.

Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.