Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three
In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>32</mn><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | AppliedMath |
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| Online Access: | https://www.mdpi.com/2673-9909/4/1/20 |
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| author | Lucian Trifina Daniela Tărniceriu Ana-Mirela Rotopănescu |
| author_facet | Lucian Trifina Daniela Tărniceriu Ana-Mirela Rotopănescu |
| author_sort | Lucian Trifina |
| collection | DOAJ |
| description | In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>32</mn><msub><mi>k</mi><mi>L</mi></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi>L</mi></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ψ</mo></semantics></math></inline-formula> is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients’ conditions. With the fourth- and third-degree coefficients of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula>, respectively, we prove that the inverse PP is (I) a 4-PP when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> or when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and (II) a 5-PP when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-04-24T18:36:28Z |
| format | Article |
| id | doaj.art-3b648644ada14a7dbb034c28bc0ec281 |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-04-24T18:36:28Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-3b648644ada14a7dbb034c28bc0ec2812024-03-27T13:18:51ZengMDPI AGAppliedMath2673-99092024-03-014138339310.3390/appliedmath4010020Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than ThreeLucian Trifina0Daniela Tărniceriu1Ana-Mirela Rotopănescu2Department of Telecommunications and Information Technologies, “Gheorghe Asachi” Technical University, 700506 Iasi, RomaniaDepartment of Telecommunications and Information Technologies, “Gheorghe Asachi” Technical University, 700506 Iasi, RomaniaDepartment of Telecommunications and Information Technologies, “Gheorghe Asachi” Technical University, 700506 Iasi, RomaniaIn this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>32</mn><msub><mi>k</mi><mi>L</mi></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi>L</mi></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ψ</mo></semantics></math></inline-formula> is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients’ conditions. With the fourth- and third-degree coefficients of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>Ψ</mo></mrow></semantics></math></inline-formula>, respectively, we prove that the inverse PP is (I) a 4-PP when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> or when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and (II) a 5-PP when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>4</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mrow><mn>3</mn><mo>,</mo><mi>f</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2673-9909/4/1/20permutation polynomial (PP)fourth-degree PPinverse PP |
| spellingShingle | Lucian Trifina Daniela Tărniceriu Ana-Mirela Rotopănescu Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three AppliedMath permutation polynomial (PP) fourth-degree PP inverse PP |
| title | Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three |
| title_full | Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three |
| title_fullStr | Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three |
| title_full_unstemmed | Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three |
| title_short | Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three |
| title_sort | inverses for fourth degree permutation polynomials modulo 32ψ or 96ψ with ψ as a product of different prime numbers greater than three |
| topic | permutation polynomial (PP) fourth-degree PP inverse PP |
| url | https://www.mdpi.com/2673-9909/4/1/20 |
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