An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA
Due to its very desirable properties, Chebyshev polynomials are often used in the design of public key cryptographic systems. This paper discretizes the Chebyshev mapping, generalizes the properties of Chebyshev polynomials, and proposes an improved public key encryption algorithm based on Chebyshev...
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MDPI AG
2024-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/3/263 |
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| author | Chunfu Zhang Yanchun Liang Adriano Tavares Lidong Wang Tiago Gomes Sandro Pinto |
| author_facet | Chunfu Zhang Yanchun Liang Adriano Tavares Lidong Wang Tiago Gomes Sandro Pinto |
| author_sort | Chunfu Zhang |
| collection | DOAJ |
| description | Due to its very desirable properties, Chebyshev polynomials are often used in the design of public key cryptographic systems. This paper discretizes the Chebyshev mapping, generalizes the properties of Chebyshev polynomials, and proposes an improved public key encryption algorithm based on Chebyshev chaotic mapping and RSA, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>R</mi><mi>P</mi><mi>K</mi><mi>C</mi><mo>−</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. This algorithm introduces alternative multiplication coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>, the selection of which is determined by the size of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mi>N</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mi>N</mi></mrow></semantics></math></inline-formula>, and the specific value selection rules are shared secrets among participants, overcoming the shortcomings of previous schemes. In the key generation and encryption/decryption stages, more complex intermediate processes are used to achieve higher algorithm complexity, making the algorithm more robust against ordinary attacks. The algorithm is also compared with other RSA-based algorithms to demonstrate its effectiveness in terms of performance and security. |
| first_indexed | 2024-04-24T17:47:58Z |
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| language | English |
| last_indexed | 2024-04-24T17:47:58Z |
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| spelling | doaj.art-19386891d0ab4584bdf6a10b317bd9ef2024-03-27T14:05:20ZengMDPI AGSymmetry2073-89942024-02-0116326310.3390/sym16030263An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSAChunfu Zhang0Yanchun Liang1Adriano Tavares2Lidong Wang3Tiago Gomes4Sandro Pinto5School of Statistics and Data Science, Zhuhai College of Science and Technology, Zhuhai 519041, ChinaSchool of Computer Science, Zhuhai College of Science and Technology, Zhuhai 519041, ChinaDepartment of Industrial Electronics, University of Minho, 4800-058 Guimaraes, PortugalSchool of Statistics and Data Science, Zhuhai College of Science and Technology, Zhuhai 519041, ChinaDepartment of Industrial Electronics, University of Minho, 4800-058 Guimaraes, PortugalDepartment of Industrial Electronics, University of Minho, 4800-058 Guimaraes, PortugalDue to its very desirable properties, Chebyshev polynomials are often used in the design of public key cryptographic systems. This paper discretizes the Chebyshev mapping, generalizes the properties of Chebyshev polynomials, and proposes an improved public key encryption algorithm based on Chebyshev chaotic mapping and RSA, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>R</mi><mi>P</mi><mi>K</mi><mi>C</mi><mo>−</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. This algorithm introduces alternative multiplication coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>, the selection of which is determined by the size of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mi>N</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mi>N</mi></mrow></semantics></math></inline-formula>, and the specific value selection rules are shared secrets among participants, overcoming the shortcomings of previous schemes. In the key generation and encryption/decryption stages, more complex intermediate processes are used to achieve higher algorithm complexity, making the algorithm more robust against ordinary attacks. The algorithm is also compared with other RSA-based algorithms to demonstrate its effectiveness in terms of performance and security.https://www.mdpi.com/2073-8994/16/3/263public-key cryptosystemChebyshev polynomialsRSAalternative multiplication coefficientssemi-group property |
| spellingShingle | Chunfu Zhang Yanchun Liang Adriano Tavares Lidong Wang Tiago Gomes Sandro Pinto An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA Symmetry public-key cryptosystem Chebyshev polynomials RSA alternative multiplication coefficients semi-group property |
| title | An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA |
| title_full | An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA |
| title_fullStr | An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA |
| title_full_unstemmed | An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA |
| title_short | An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA |
| title_sort | improved public key cryptographic algorithm based on chebyshev polynomials and rsa |
| topic | public-key cryptosystem Chebyshev polynomials RSA alternative multiplication coefficients semi-group property |
| url | https://www.mdpi.com/2073-8994/16/3/263 |
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