Quadratic residues

A Comparative Analysis of Classical Cryptography versus Quantum Cryptography for Web of Medical Things (WoMT)

Published in Aboul Ella Hassanien, Nilanjan Dey, Surekha Borra, Medical Big Data and Internet of Medical Things, 2018

Let's say there exists an integer a∈Zp* then ‘a’ is called as a quadratic residue (QR) modulo ‘p’, if x2 ≡ a(mod p) for x∈Zp*. Otherwise, ‘a’ is called as a quadratic non-residue (QNR) modulo ‘p’. QRp denotes a set of quadratic residues modulo ‘p’ and QNRp denotes a set of quadratic non-residues modulo ‘p’. For example, QR7 denotes the set of all quadratic residues modulo 7. Then QR7 = {12, 22, 32,42, 52, 62} (mod 7) = {1, 2, 4}. The obtained quadratic residues are most widely used in modern cryptographic algorithms to enhance security.

*Advanced Topics in Public-Key Encryption

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Published in Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography, 2020

Jonathan Katz, Yehuda Lindell

In a group G, an element y∈G is a quadratic residue if there exists an x∈G with x2 = y. In this case, we call x a square root of y. An element that is not a quadratic residue is called a quadratic non-residue. In an abelian group, the set of quadratic residues forms a subgroup.

ElGamal-type encryption for optimal dynamic quantizer in encrypted control systems

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Published in SICE Journal of Control, Measurement, and System Integration, 2021

Kaoru Teranishi, Kiminao Kogiso

An integer a is called as a quadratic residue modulo m if there exists an integer b such that . We use Gauss's notation, i.e. we use if a is a quadratic residue modulo m; otherwise, we use .

Quadratic residues

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A Comparative Analysis of Classical Cryptography versus Quantum Cryptography for Web of Medical Things (WoMT)

*Advanced Topics in Public-Key Encryption

ElGamal-type encryption for optimal dynamic quantizer in encrypted control systems