Combinatorial properties of multidimensional continued fractions

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions

Distributed Fiat-Shamir Transform

The recent surge of distribute technologies caused an increasing interest towards threshold signature protocols, that peaked with the recent NIST First Call for Multi-Party Threshold Schemes. Since its introduction, the Fiat-Shamir Transform has been the most popular way to design standard digital signature schemes. In this work, we translate the Fiat-Shamir Transform into a multi-party setting, building a framework that seeks to be an alternative, easier way to design threshold digital signatures. We do that by introducing the concept of threshold identification scheme and threshold sigma protocol, and showing necessary and sufficient conditions to prove the security of the threshold signature schemes derived from them. Lastly, we show a practical application of our framework providing an alternative security proof for Sparkle, a recent threshold Schnorr signature. In particular, we consider the threshold identification scheme underlying Sparkle and prove the security of the signature derived from it. We show that using our framework the effort required to prove the security of threshold signatures might be drastically lowered. In fact, instead of reducing explicitly its security to the security of a hard problem, it is enough to prove some properties of the underlying threshold sigma protocol and threshold identification scheme. Then, by applying the results that we prove in this paper it is guaranteed that the derived threshold signature is secure

Extensible Decentralized Secret Sharing and Application to Schnorr Signatures

Starting from links between coding theory and secret sharing we develop an extensible and decentralized version of Shamir Secret Sharing, that allows the addition of new users after the initial share distribution. On top of it we design a totally decentralized $(t,n)$-threshold Schnorr signature scheme that needs only $t$ users online during the key generation phase, while the others join later. Under standard assumptions we prove our scheme secure against adaptive malicious adversaries. Furthermore, we show how our security notion can be strengthen when considering a rushing adversary. Using a classical game-based argument, we prove that if there is an adversary capable of forging the scheme with non-negligible probability, then we can build a forger for the centralized Schnorr scheme with non-negligible probability

Faster coercion-resistant e-voting by encrypted sorting

Coercion-resistance is one of the most challenging security properties to achieve when designing an e-voting protocol. The JCJ voting scheme, proposed in 2005 by Juels, Catalano and Jakobsson, is one of the first voting systems where coercion-resistance was rigorously defined and achieved, making JCJ the benchmark for coercion-resistant protocols. Recently, the coercion-resistance definition proposed in JCJ has been disputed and improved by Cortier, Gaudry, and Yang. They identified a major problem, related to leakage of the number of discarded votes by revoting; and proposed CHide, a new protocol that solves the issue and satisfies a stronger security notion. In this work we present an improved version of CHide, with complexity $O(n\log n)$ instead of $O(n^2)$ in the number $n$ of received ballots, that relies on sorting encrypted ballots to make the tallying phase faster. The asymptotic complexity of our protocol is competitive with other state-of-the-art coercion-resistant voting protocols satisfying the stronger notion for coercion resistance

Cutting the GRASS: Threshold GRoup Action Signature Schemes

Group actions are fundamental mathematical tools, with a long history of use in cryptography. Indeed, the action of finite groups at the basis of the discrete logarithm problem is behind a very large portion of modern cryptographic systems. With the advent of post-quantum cryptography, however, the method for building protocols shifted towards a different paradigm, centered on the difficulty of discerning \u27noisy\u27 objects, as is the case for lattices, codes, and multivariate systems. This method yields promising results for \u27core\u27 primitives such as encryption or signature, but can be less than ideal in the case when more advanced functionalities are required. In this work, we show that isomorphism problems which stem from cryptographic group actions, can be viable building blocks for threshold signature schemes. In particular, we construct a full $N$-out-of-$N$ threshold signature scheme, and discuss the efficiency issues arising from extending it to the generic $T$-out-of-$N$ case. To give a practical outlook on our constructions, we instantiate them with the LESS and MEDS frameworks, which are two flavors of code-based cryptographic group actions. Finally, we highlight some ideas that would allow for a more efficient and compact $(T,N)$ threshold variant of LESS, whose security relies on new hardness assumptions

Optical coherence tomography of retinal and choroidal layers in patients with familial hypercholesterolaemia treated with lipoprotein apheresis

Detect and quantify morpho-functional alterations of the retina and choroid in patients affected by familial hypercholesterolemia (FH) treated with lipoprotein apheresis (LA) using optic coherence tomography (OCT) and optic coherence tomography-angriography (OCTA).Observational study.To be diagnosed: A group of 20 patients (40 eyes) being clinically and genetically diagnosed as FH and under treatment (FH-Group)", for at least 2 years, was compared to a control group of 20 healthy subjects (40 eyes), with a normal lipid profile and no ocular disease (CT-Group).Participants were studied with the slit lamp, binocular indirect fundoscopy, OCT and OCTA.Best corrected visual acuity (BVCA), spherical equivalent (SE), intraocular pressure (IOP), central macular thickness (CMT), choroidal thickness (CHT), retinal nerve fiber layer in four quadrants (RNFL (Superior = Sup; Inferior = Inf; Nasal = Nas Temporal = Temp), and the mean value across the four quadrants (RNFL G), foveal avascular zone (FAZ) and vascular density (VD).FH subjects had smaller RNFL superiorly (108 ± 19,38 μm OD/111 ± 16,56 μm OS FH-Group vs 127 ± 7,42 μm OD/129 ± 14,64 μm OS CT-Group; P  0,001 for both OD and OS) and inferiorly (108 ± 23,58 μm OD/115 ± 17,33 μm OS FH-Group vs 128 ± 18,15 μm OD/133 ± 17,38 μm OS CT-Group; P = 0,002 OD; P = 0,001 OS). G RNFL was consequently smaller (93 ± 12,94 μm OD/94 ± 10,49 μm OS FH-Group vs 101 ± 9,01 μm OD/101 ± 10,20 μm OS CT-Group; P = 0,03 OD; P = 0,02 OS). FH subjects had a larger FAZ (0,31 ± 0,08 mmEarly signs of retinal vessel damage in FH patients can be detected and quantified with OCT and OCTA

Familial hypercholesterolaemia: the influence of lipoprotein apheresis on choroid and retina. an oct and octa retrospective study

Purpose: High cholesterol levels in untreated familial hypercholesterolemia (FH) can significantly compromise retinal microvasculature. This type of dyslipidemia is commonly treated with combination drug therapy and lipoprotein apheresis (LA). We conducted a retrospective observational study by means of optical coherence tomography (OCT) and optical coherence tomography-angiography (OCTA) on patients affected by familial hypercholesterolemia (FH) treated by LA and healthy controls. Methods: A group of 20 patients (40 eyes; 8 males and 12 females; mean age±SD: 50, 05±13, 68) with FH were compared to a group of 20 matched healthy controls (CT group). Inclusion criteria: genetically-confirmed diagnosis of FH, treatment with combination drug therapy and LA for at least 2 years, spherical equivalent (SE)-3 to+ 3 diopters. Exclusion criteria: pre-existent retinal or choroidal damage (including macula