Some Properties of the Computation of the Modular Inverse with Applications in Cryptography

Bufalo, Michele; Bufalo, Daniele; Orlando, Giuseppe

doi:10.3390/computation11040070

Open AccessArticle

Some Properties of the Computation of the Modular Inverse with Applications in Cryptography

by

Michele Bufalo

¹,

Daniele Bufalo

² and

Giuseppe Orlando

^3,4,*

¹

Department of Methods and Models for Economics, Territory and Finance, Università degli Studi di Roma “La Sapienza”, Via del Castro Laurenziano 9, 00185 Rome, Italy

²

Department of Informatics, Università degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy

³

Department of Mathematics, Università degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy

⁴

Department of Economics, HSE University, Soyuza Pechatnikov Street 16, 190121 St. Petersburg, Russia

^*

Author to whom correspondence should be addressed.

Computation 2023, 11(4), 70; https://doi.org/10.3390/computation11040070

Submission received: 10 February 2023 / Revised: 21 March 2023 / Accepted: 21 March 2023 / Published: 27 March 2023

(This article belongs to the Section Computational Engineering)

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Abstract

In the field of cryptography, many algorithms rely on the computation of modular multiplicative inverses to ensure the security of their systems. In this study, we build upon our previous research by introducing a novel sequence,

{(z_{j})}_{j \geq 0}

, that can calculate the modular inverse of a given pair of integers

(a, n)

, i.e.,

a^{- 1}; m o d, n

. The computational complexity of this approach is

O (a)

, which is more efficient than the traditional Euler’s phi function method,

O (n, \ln, n)

. Furthermore, we investigate the properties of the sequence

{(z_{j})}_{j \geq 0}

and demonstrate that all solutions of the problem belong to a specific set,

I

, that only contains the minimum values of

{(z_{j})}_{j \geq 0}

. This results in a reduction of the computational complexity of our method, especially when

a \sim n

and it also opens new opportunities for discovering closed-form solutions for the modular inverse.

Keywords: extended-Euclid algorithm; RSA algorithm; modular multiplicative inverse; public-key cryptography

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MDPI and ACS Style

Bufalo, M.; Bufalo, D.; Orlando, G. Some Properties of the Computation of the Modular Inverse with Applications in Cryptography. Computation 2023, 11, 70. https://doi.org/10.3390/computation11040070

AMA Style

Bufalo M, Bufalo D, Orlando G. Some Properties of the Computation of the Modular Inverse with Applications in Cryptography. Computation. 2023; 11(4):70. https://doi.org/10.3390/computation11040070

Chicago/Turabian Style

Bufalo, Michele, Daniele Bufalo, and Giuseppe Orlando. 2023. "Some Properties of the Computation of the Modular Inverse with Applications in Cryptography" Computation 11, no. 4: 70. https://doi.org/10.3390/computation11040070

APA Style

Bufalo, M., Bufalo, D., & Orlando, G. (2023). Some Properties of the Computation of the Modular Inverse with Applications in Cryptography. Computation, 11(4), 70. https://doi.org/10.3390/computation11040070

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Some Properties of the Computation of the Modular Inverse with Applications in Cryptography

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