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Scalar Notation01:28

Scalar Notation

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Scalar notation is a useful method for simplifying calculations involving vectors. When vectors are added or subtracted, their components can be added or subtracted separately using scalar notation. For instance, force, a vector quantity, can be broken down into its x and y components, called rectangular components, and then the magnitude and direction of these components can be determined using trigonometric functions.
Consider a man pulling a rope from a hook in the northeast direction. The...
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Couples: Scalar and Vector Formulation01:21

Couples: Scalar and Vector Formulation

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One might wonder how the captain of a large ship can navigate through the ocean with just a turn of the steering wheel. The answer lies in the concept of two parallel forces that are equal in magnitude and opposite sense, creating a couple moment.
A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment....
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Singularity Functions for Shear01:26

Singularity Functions for Shear

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In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous  variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the...
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Bulk Modulus01:21

Bulk Modulus

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The bulk modulus is a scientific term used to describe a material's resistance to uniform compression. It is the proportionality constant that links a change in pressure to the resulting relative volume change.
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Cartesian Vector Notation01:28

Cartesian Vector Notation

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Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
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Cartesian Form for Vector Formulation01:26

Cartesian Form for Vector Formulation

736
The Cartesian form for vector formulation is a process to calculate  the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.
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Related Experiment Video

Updated: Sep 11, 2025

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform
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A Post-Quantum Public-Key Signcryption Scheme over Scalar Integers Based on a Modified LWE Structure.

Mostefa Kara1, Mohammad Hammoudeh2, Abdullah Alamri3

  • 1Interdisciplinary Research Center for Intelligent Secure Systems, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

Sensors (Basel, Switzerland)
|August 14, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new post-quantum public-key encryption and signature (PQES) scheme using simple scalar integers. This efficient method offers enhanced security against quantum attacks for devices with limited resources.

Keywords:
LWEcryptosystemlight IoTpost-quantumscalar integersigncryption

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Area of Science:

  • Cryptography
  • Computer Science
  • Information Security

Background:

  • Current post-quantum cryptography often separates encryption and digital signatures, increasing complexity.
  • Existing signcryption schemes typically use complex mathematical structures like lattices or polynomials.
  • The need for efficient and implementable quantum-resistant cryptographic solutions is growing.

Purpose of the Study:

  • To present a novel post-quantum public-key encryption and signature (PQES) scheme.
  • To develop a scheme based solely on scalar integer operations for simplicity and efficiency.
  • To enhance the implementability of quantum-resistant cryptography on constrained devices.

Main Methods:

  • Developed a PQES scheme using scalar integers modulo n for keys, ciphertext, and operations.
  • Adapted modified learning-with-errors (LWE) assumptions to a scalar setting.
  • Evaluated security and performance, including signcryption and unsigncryption times.

Main Results:

  • The proposed PQES scheme operates entirely on scalar integers, reducing computational and memory overhead.
  • Achieved signcryption time of 0.0007s and unsigncryption time of 0.0011s.
  • Demonstrated strong security properties resistant to quantum attacks.

Conclusions:

  • The novel scalar-integer-based PQES scheme offers a practical balance between efficiency and quantum resistance.
  • The simplified structure enhances implementability on resource-constrained devices.
  • This approach provides a viable alternative to complex lattice-based or ring-based post-quantum cryptographic schemes.