Post-quantum encryption algorithms of high-degree 3-variable polynomial congruences: BS cryptosystems and BS key generation

• URL: http://arxiv.org/abs/2409.03758v1
• Date: Wed, 14 Aug 2024 14:19:46 GMT
• Title: Post-quantum encryption algorithms of high-degree 3-variable polynomial congruences: BS cryptosystems and BS key generation
• Authors: Nicholas J. Daras,
• Abstract summary: We will construct post-quantum encryption algorithms based on three-variable Beal-Schur congruence. We will apply this result to generate simple and secure post-quantum encryption algorithms.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We will construct post-quantum encryption algorithms based on three-variable polynomial Beal-Schur congruence. After giving a proof of Beal's conjecture and citing some applications of it to selected cases where the discrete logarithm and some of its generalizations are unsolvable problems, we will investigate the formulation and validity of an appropriate version of the Beal's conjecture on finite fields of integers. In contrast to the infinite case, we will show that the corresponding Beal-Schur congruence equation $x^{p}+y^{q}\equiv z^{r} (mod \mathcal{N})$ has non-trivial solutions into the finite field $\mathbb{Z}_{\mathcal{N}} $, for all sufficiently large primes $\mathcal{N}$ that do not divide the product $xyz$, under certain mutual divisibility conditions of the exponents $p$, $q$ and $r$. We will apply this result to generate the so-called BS cryptosystems, i.e., simple and secure post-quantum encryption algorithms based on the Beal-Schur congruence equation, as well as new cryptographic key generation methods, whose post-quantum algorithmic encryption security relies on having an infinite number of options for the parameters $p$, $q$, $r$, $\mathcal{N}$.

Related papers

• Quantum One-Wayness of the Single-Round Sponge with Invertible Permutations [49.1574468325115]
  Sponge hashing is a widely used class of cryptographic hash algorithms. Intrepid permutations have so far remained a fundamental open problem. We show that finding zero-pairs in a random $2n$-bit permutation requires at least $Omega (2n/2)$ many queries.
  arXiv  Detail & Related papers  (2024-03-07T18:46:58Z)
• A one-query lower bound for unitary synthesis and breaking quantum cryptography [7.705803563459633]
  The Unitary Synthesis Problem asks whether any $n$qubit unitary $U$ can be implemented by an efficient quantum $A$ augmented with an oracle that computes an arbitrary Boolean function $f$. In this work, we prove unitary synthesis as an efficient challenger-ad game, which enables proving lower bounds by analyzing the maximum success probability of an adversary $Af$.
  arXiv  Detail & Related papers  (2023-10-13T05:39:42Z)
• Efficient Quantum State Synthesis with One Query [0.0]
  We present a time analogue quantum algorithm making a single query (in superposition) to a classical oracle. We prove that every $n$-qubit state can be constructed to within 0.01 error by an $On/n)$-size circuit over an appropriate finite gate set.
  arXiv  Detail & Related papers  (2023-06-02T17:49:35Z)
• Homomorphic Encryption of the k=2 Bernstein-Vazirani Algorithm [0.4511923587827301]
  We find an application of this scheme to quantum homomorphic encryption (QHE) which is an important cryptographic technology useful for delegated quantum computation. We develop QHE schemes with perfect security, $mathcalF$-homomorphism, no interaction between server and client, and quasi-compactness bounded by $O(M)$ where M is the number of gates $T$ in the circuit.
  arXiv  Detail & Related papers  (2023-03-30T14:49:15Z)
• An Efficient Quantum Decoder for Prime-Power Fields [1.0878040851638]
  We show that for $q = pm$ where $p$ is small relative to the code block-size $n$, there is a quantum algorithm that solves the problem in time. On the other hand, classical algorithms can efficiently solve the problem only for much smaller inverse factors.
  arXiv  Detail & Related papers  (2022-10-20T19:35:50Z)
• Quantum Depth in the Random Oracle Model [57.663890114335736]
  We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
  arXiv  Detail & Related papers  (2022-10-12T17:54:02Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Matrix Discrepancy from Quantum Communication [13.782852293291494]
  We develop a novel connection between discrepancy minimization and (quantum) communication complexity. We show that for every collection of symmetric $n times n$ $A_1,ldots,A_n$ with $|A_i| leq 1$ and $|A_i|_F leq n1/4$ there exist signs $x in pm 1n such that the maximum eigenvalue of $sum_i leq n x_i A_i$ is at most
  arXiv  Detail & Related papers  (2021-10-19T16:51:11Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Quantum copy-protection of compute-and-compare programs in the quantum random oracle model [48.94443749859216]
  We introduce a quantum copy-protection scheme for a class of evasive functions known as " compute-and-compare programs" We prove that our scheme achieves non-trivial security against fully malicious adversaries in the quantum random oracle model (QROM) As a complementary result, we show that the same scheme fulfils a weaker notion of software protection, called "secure software leasing"
  arXiv  Detail & Related papers  (2020-09-29T08:41:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.