An efficient and secure RSA--like cryptosystem exploiting R\'edei rational functions over conics

We define an isomorphism between the group of points of a conic and the set of integers modulo a prime equipped with a non-standard product. This product can be efficiently evaluated through the use of R\'edei rational functions. We then exploit the isomorphism to construct a novel RSA-like scheme. We compare our scheme with classic RSA and with RSA-like schemes based on the cubic or conic equation. The decryption operation of the proposed scheme turns to be two times faster than RSA, and involves the lowest number of modular inversions with respect to other RSA-like schemes based on curves. Our solution offers the same security as RSA in a one-to-one communication and more security in broadcast applications.Comment: 18 pages, 1 figur

Unconditionally secure key distribution in higher dimensions by depolarization

This paper presents a prepare-and-measure scheme using N-dimensional quantum particles as information carriers where N is a prime power. One of the key ingredients used to resist eavesdropping in this scheme is to depolarize all Pauli errors introduced to the quantum information carriers. Using the Shor-Preskill-type argument, we prove that this scheme is unconditionally secure against all attacks allowed by the laws of quantum physics. For N = 2n > 2, each information carrier can be replaced by n entangled qubits. In this case, there is a family of eavesdropping attacks on which no unentangled-qubit-based prepare-and-measure (PM) quantum key distribution scheme known to date can generate a provably secure key. In contrast, under the same family of attacks, our entangled-qubit-based scheme remains secure whenever 2n ≥ 4. This demonstrates the advantage of using entangled particles as information carriers and of using depolarization of Pauli errors to combat eavesdropping attacks more drastic than those that can be handled by unentangled-qubit-based prepare-and-measure schemes. © 2005 IEEE.published_or_final_versio

Regular complete permutation polynomials over quadratic extension fields

Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $\tau_1, \tau_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $\sigma_M$ is an invertible linear map over $\mathbb{F}_{q^2}$ and $\sigma=\tau_1\circ\sigma_M\circ\tau_2$. In this paper, we prove that, for suitable $\tau_1, \tau_2$ and $\sigma_M$, the map $\sigma$ could be $r$-regular complete permutation polynomials over quadratic extension fields.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:2212.1286

Periodic representations and rational approximations of square roots

In this paper the properties of R\'edei rational functions are used to derive rational approximations for square roots and both Newton and Pad\'e approximations are given as particular cases. As a consequence, such approximations can be derived directly by power matrices. Moreover, R\'edei rational functions are introduced as convergents of particular periodic continued fractions and are applied for approximating square roots in the field of p-adic numbers and to study periodic representations. Using the results over the real numbers, we show how to construct periodic continued fractions and approximations of square roots which are simultaneously valid in the real and in the p-adic field