Cryptanalysis of the system based on word problems using logarithmic signatures

Rapid development and advances of quantum computers are contributing to the development of public key cryptosystems based on mathematically complex or difficult problems, as the threat of using quantum algorithms to hack modern traditional cryptosystems is becoming much more real every day. It should be noted that the classical mathematically complex problems of factorization of integers and discrete logarithms are no longer considered complex for quantum calculations. Dozens of cryptosystems were considered and proposed on various complex problems of group theory in the 2000s. One of such complex problems is the problem of the word. One of the first implementations of the cryptosystem based on the word problem was proposed by Magliveras using logarithmic signatures for finite permutation groups and further proposed by Lempken et al. for asymmetric cryptography with random covers. The innovation of this idea is to extend the difficult problem of the word to a large number of groups. The article summarizes the known results of cryptanalysis of the basic structures of the cryptosystem and defines recommendations for ways to improve the cryptographic properties of structures and the use of non-commutative groups as basic structures.

CONSTRUCTING AND IMPLEMENTING SOME PERFECT VERIFIABLE SECRET SHARING SCHEMES

Most of the existing public key cryptosystems are potentially vulnerable to cryptographic attacks as they rely on the problems of discrete logarithm and factorization of integers. There is now a need for algorithms that will resist attacks on quantum computers. The article describes the implementation of Shamir’s post-quantum secret sharing scheme using long arithmetic that can be applied in modern cryptographic modules. The implementation of the Pedersen – Shamir scheme is described, which allows preserving the property of the perfection of the Shamir scheme by introducing testability. The article presents graphs reflecting the influence of the verifiability property in the Shamir secret sharing scheme on the speed of its operation.

Integer factoring and compositeness witnesses

We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp $\begin{array}{} \displaystyle \left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right) \end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored.

FACTORIZATION OF INTEGERS WITH MULTI-PATH OPTICAL INTERFERENCE

We introduce a new factorization algorithm based on the optical computation by multi-path interference of the periodicity of a "factoring" function given by exponential sums at continuous arguments. We demonstrate that this algorithm allows, in principle, the prime number decomposition of several large numbers by exploiting a remarking rescaling property of this periodic function. Such a function is recorded by measuring optical interferograms with a multi-path Michelson interferometer, a polychromatic light source and a spectrometer. The information about factors is encoded in the location of the inteferogram maxima.

Labeled Factorization of Integers

The labeled factorizations of a positive integer $n$ are obtained as a completion of the set of ordered factorizations of $n$. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of $n$. Our results include explicit enumeration formulas and some combinatorial identities. It is proved that labeled factorizations of $n$ are equinumerous with the systems of complementing subsets of $\{0,1,\dots,n-1\}$. We also give a new combinatorial interpretation of a class of generalized Stirling numbers.