The Balkans Continued Fraction

In a previous article we gave a collection of continued fractions involving Catalan's constant. This paper provides more general formulae governing those continued fractions. Having distinguished different cases associated to regions in the plan, we nickname those continued fractions \enquote{The Balkans} as they divide into areas which are related but still different in nature. Because we do not provide formal proofs of those machine-constructed formulae we do not claim them to be theorems. Still, each and every proposed formula was extensively tested numerically

A Note on the Ramanujan Machine

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even and odd $\zeta$ function values, logarithms etc. We show that many relations detected by the Ramanujan Machine Project stem from a specific algebraic observation and show how to generate infinitely many. This provides an automated proof and/or an explanation of many of the relations listed as conjectures by the project (although not all of them)

On Squaring Modulo Mersenne Numbers

During the design of a new primitive inspired by Squash we accidentally stumbled on the observation described in this note. Let $n$ be a $k$-bit Mersenne number whose factors are unknown. Consider an $\ell$-bit secret number $x=2^{k/2}a+b$. We observe that there are parameter configurations where a chunk of the value $b^2$ is leaked even if $k<2\ell$. This observation does not endanger any known scheme and in particular not Squash

On The Practical Advantage of Committing Challenges in Zero-Knowledge Protocols

The Fiat-Shamir transform is a classical technique for turning any zero-knowledge $\Sigma$-protocol into a signature scheme. In essence, the idea underlying this transform is that deriving the challenge from the digest of the commitment suppresses simulatability and hence provides non-interactive proofs of interaction. It follows from that observation that if one wishes to preserve deniability the challenge size (per round) must be kept low. For instance in the original Fiat-Shamir protocol the authors recommend 18 bits but suggest that the challenge size can be made larger to reduce communication overhead, e.g. the value of 20 is proposed in \cite{micali}. We show that even with relatively small challenge sizes \textsl{practical} deniability can be destroyed by having the verifier artificially impose upon himself the use of slowed-down hash function or by resorting to a trusted agency proposing an on-line deniability enforcement service against the provers community\u27s will

A Conjecture From a Failed Cryptanalysis

This note describes an observation discovered during a failed cryptanalysis attempt. Let $P(x,y)$ be a bivariate polynomial with coefficients in $\mathbb{C}$. Form the $n\times n$ matrices $L(n)$ whose elements are defined by $P(i,j)$. Define the matrices M(n)=L(n)-\mbox{ID}_n. It appears that $\mu(n)=(-1)^n\det(M_n)$ is a polynomial in $n$ that we did not characterize. We provide a numerical example

Fiat-Shamir Goes Tropical

In a recent ePrint, Brown and Monico propose new attacks on the tropical signature scheme of Chen, Grigoriev and Shpilrain. This note provides a new countermeasures against those attacks. Thereby, we (temporarily?) shift the fire from the signature algorithm to redirect attacks on the key and on tropical polynomial factorization

Authenticating Medications with QR-Codes and Compact Digital Signatures

This paper describes a way to protect medications against falsification, a long-standing problem in the world. We combine several existing technologies to achieve the stated goal. The building-blocks used are inherent physical randomness generated during the packaging process, artificial vision, short digital signatures and QR-codes

Preservation of DNA Privacy During the Large Scale Detection of COVID-19

As humanity struggles to contain the global COVID-19 pandemic, privacy concerns are emerging regarding confinement, tracing and testing. The scientific debate concerning privacy of the COVID-19 tracing efforts has been intense, especially focusing on the choice between centralised and decentralised tracing apps. The privacy concerns regarding COVID-19 testing, however, have not received as much attention even though the privacy at stake is arguably even higher. COVID-19 tests require the collection of samples. Those samples possibly contain viral material but inevitably also human DNA. Patient DNA is not necessary for the test but it is technically impossible to avoid collecting it. The unlawful preservation, or misuse, of such samples at a massive scale may hence disclose patient DNA information with far-reaching privacy consequences. Inspired by the cryptographic concept of "Indistinguishability under Chosen Plaintext Attack", this paper poses the blueprint of novel types of tests allowing to detect viral presence without leaving persisting traces of the patient's DNA. Authors are listed in alphabetical order.Comment: 10 pages, 1 figur