Dima Grigoriev

Vincent Noel, Dima Grigoriev, Sergei Vakulenko et al.

Piecewise smooth hybrid systems, involving continuous and discrete variables, are suitable models for describing the multiscale regulatory machinery of the biological cells. In hybrid models, the discrete variables can switch on and off some molecular interactions, simulating cell progression through a series of functioning modes. The advancement through the cell cycle is the archetype of such an organized sequence of events. We present an approach, inspired from tropical geometry ideas, allowing to reduce, hybridize and analyse cell cycle models consisting of polynomial or rational ordinary differential equations.

Mariya Bessonov, Dima Grigoriev, Vladimir Shpilrain

In this short note, we address a common misconception at the interface of probability theory and public-key cryptography.

Dima Grigoriev, Vladimir Shpilrain

A blockchain is redactable if a private key holder (e.g. a central authority) can change any single block without violating integrity of the whole blockchain, but no other party can do that. In this paper, we offer a simple method of constructing redactable blockchains inspired by the ideas underlying the well-known RSA encryption scheme. Notably, our method can be used in conjunction with any reasonable hash function that is used to build a blockchain. Public immutability of a blockchain in our construction is based on the computational hardness of the RSA problem and not on properties of the underlying hash function. Corruption resistance is based on the computational hardness of the discrete logarithm problem.

Rostislav Grigorchuk, Dima Grigoriev

We suggest several automaton groups as key-agreement platforms for Anshl-Anshel-Goldfeld metascheme, they include Grigorchuk and universal Grigorchuk groups, Hanoi 3-Towers group, Basilica group and a subgroup of the affine group with the unsolvable conjugacy problem

Dima Grigoriev, Vladimir Shpilrain

We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.

Mariya Bessonov, Dima Grigoriev, Vladimir Shpilrain

We offer a probabilistic solution of Yao's millionaires' problem that gives correct answer with probability (slightly) less than 1 but on the positive side, this solution does not use any one-way functions.

Dima Grigoriev, Vladimir Shpilrain

We use various laws of classical physics to offer several solutions of Yao's millionaires' problem without using any one-way functions. We also describe several informationally secure public key encryption protocols, i.e., protocols secure against passive computationally unbounded adversary. This introduces a new paradigm of decoy-based cryptography, as opposed to "traditional" complexity-based cryptography. In particular, our protocols do not employ any one-way functions.

Dima Grigoriev, Vladimir Shpilrain

We show that some problems in information security can be solved without using one-way functions. The latter are usually regarded as a central concept of cryptography, but the very existence of one-way functions depends on difficult conjectures in complexity theory, most notably on the notorious "$P \ne NP$" conjecture. In this paper, we suggest protocols for secure computation of the sum, product, and some other functions, without using any one-way functions. A new input that we offer here is that, in contrast with other proposals, we conceal "intermediate results" of a computation. For example, when we compute the sum of $k$ numbers, only the final result is known to the parties; partial sums are not known to anybody. Other applications of our method include voting/rating over insecure channels and a rather elegant and efficient solution of Yao's "millionaires' problem". Then, while it is fairly obvious that a secure (bit) commitment between two parties is impossible without a one-way function, we show that it is possible if the number of parties is at least 3. We also show how our (bit) commitment scheme for 3 parties can be used to arrange an unconditionally secure (bit) commitment between just two parties if they use a "dummy" (e.g., a computer) as the third party. We explain how our concept of a "dummy" is different from a well-known concept of a "trusted third party". We also suggest a protocol, without using a one-way function, for "mental poker", i.e., a fair card dealing (and playing) over distance. We also propose a secret sharing scheme where an advantage over Shamir's and other known secret sharing schemes is that nobody, including the dealer, ends up knowing the shares owned by any particular player. It should be mentioned that computational cost of our protocols is negligible to the point that all of them can be executed without a computer.

Dima Grigoriev, Vladimir Shpilrain

We employ tropical algebras as platforms for several cryptographic schemes that would be vulnerable to linear algebra attacks were they based on "usual" algebras as platforms.