Color image encryption by piecewise function and elliptic curve over the Galois field $ {G}{F}\left({2}^{{n}}\right) $

Hafeez Ur Rehman; Mohammad Mazyad Hazzazi; Tariq Shah; Amer Aljaedi; Zaid Bassfar; Hafeez Ur Rehman; Mohammad Mazyad Hazzazi; Tariq Shah; Amer Aljaedi; Zaid Bassfar

doi:10.3934/math.2024278

AIMS Mathematics

2024, Volume 9, Issue 3: 5722-5745. doi: 10.3934/math.2024278

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Research article

Color image encryption by piecewise function and elliptic curve over the Galois field $ {G}{F}\left({2}^{{n}}\right) $

1.
Department of Mathematics, Quaid-i-Azam University Islamabad, Islamabad 45320, Pakistan
2.
Department of Mathematics, College of Science, King Khalid University, Abha 61421, Saudi Arabia
3.
College of Computing and Information Technology, University of Tabuk, Tabuk 71491, Saudi Arabia
4.
Department of Information Technology, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 20 August 2023 Revised: 13 January 2024 Accepted: 16 January 2024 Published: 30 January 2024
MSC : 68P25, 81P94, 94A60

Elliptic curve (EC) cryptography supplies an efficient, secure, and lightweight method for executing computer cryptographic protocols. Its widespread use in various applications, including secure communications, digital signatures, and key agreement protocols, highlights its importance in modern computing. Moreover, EC-based image encryption is gaining popularity in cryptography as it offers strong protection with a relatively smaller key size than other famous cryptosystems. Inspired by this, we proposed a novel image encryption scheme that leverages ECs over a binary extension field (BEF). This approach also reduces computational workload using EC over BEF instead of large primes. Also, BEF can represent large numbers in a compact form, which is helpful in applications that require efficient data storage and transmission. Our scheme involves three main steps. Initially, we utilize points of an EC over a BEF and a piecewise function to mask the plain image. Next, to introduce a high level of confusion in the plain text, we create a substitution box (S-box) based on the EC and operation of BEF of order 256, which is then used to permute the pixels of the masked image. Finally, we generate pseudo-random numbers (PRNs) using EC coordinates and BEF characteristics to create diffusion in the image and obtain a cipher image. In addition, we accomplished computational experiments demonstrating that our proposed cryptosystem provides excellent security against linear, differential, and statistical attacks compared to existing cryptosystems.
- elliptic curve cryptography,
- Galois field,
- image encryption,
- S-box
Citation: Hafeez Ur Rehman, Mohammad Mazyad Hazzazi, Tariq Shah, Amer Aljaedi, Zaid Bassfar. Color image encryption by piecewise function and elliptic curve over the Galois field $ {G}{F}\left({2}^{{n}}\right) $[J]. AIMS Mathematics, 2024, 9(3): 5722-5745. doi: 10.3934/math.2024278

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Abstract

Elliptic curve (EC) cryptography supplies an efficient, secure, and lightweight method for executing computer cryptographic protocols. Its widespread use in various applications, including secure communications, digital signatures, and key agreement protocols, highlights its importance in modern computing. Moreover, EC-based image encryption is gaining popularity in cryptography as it offers strong protection with a relatively smaller key size than other famous cryptosystems. Inspired by this, we proposed a novel image encryption scheme that leverages ECs over a binary extension field (BEF). This approach also reduces computational workload using EC over BEF instead of large primes. Also, BEF can represent large numbers in a compact form, which is helpful in applications that require efficient data storage and transmission. Our scheme involves three main steps. Initially, we utilize points of an EC over a BEF and a piecewise function to mask the plain image. Next, to introduce a high level of confusion in the plain text, we create a substitution box (S-box) based on the EC and operation of BEF of order 256, which is then used to permute the pixels of the masked image. Finally, we generate pseudo-random numbers (PRNs) using EC coordinates and BEF characteristics to create diffusion in the image and obtain a cipher image. In addition, we accomplished computational experiments demonstrating that our proposed cryptosystem provides excellent security against linear, differential, and statistical attacks compared to existing cryptosystems.

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