A Note on APN Permutations and Their Derivatives
Prior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&...
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| description | Prior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≥</mo><mn>8</mn></mrow></semantics></math></inline-formula> remains a significant challenge. Understanding and determining more properties of these functions is a crucial approach to discovering them. In this note, we study the properties of vectorial Boolean functions based on the weights of the first-order and second-order derivatives of their components. We show that a function is an APN permutation if and only if the sum of the squares of the weights of the first-order derivatives of its components is exactly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Additionally, we determined that the sum of the weights of the second-order derivatives of the components of any vectorial Boolean function is at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. This bound is achieved if and only if a function is APN. |
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| spelling | doaj-art-f91c557f2372401c9bdfa0cfbb2ec2b42025-08-20T02:04:54ZengMDPI AGMathematics2227-73902024-11-011222347710.3390/math12223477A Note on APN Permutations and Their DerivativesAugustine Musukwa0Department of Mathematics and Statistics, Mzuzu University, P/Bag 201, Mzuzu 2, MalawiPrior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≥</mo><mn>8</mn></mrow></semantics></math></inline-formula> remains a significant challenge. Understanding and determining more properties of these functions is a crucial approach to discovering them. In this note, we study the properties of vectorial Boolean functions based on the weights of the first-order and second-order derivatives of their components. We show that a function is an APN permutation if and only if the sum of the squares of the weights of the first-order derivatives of its components is exactly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Additionally, we determined that the sum of the weights of the second-order derivatives of the components of any vectorial Boolean function is at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. This bound is achieved if and only if a function is APN.https://www.mdpi.com/2227-7390/12/22/3477Boolean functionsweightfirst-order derivativessecond-order derivativesAPN functionsAPN permutations |
| spellingShingle | Augustine Musukwa A Note on APN Permutations and Their Derivatives Mathematics Boolean functions weight first-order derivatives second-order derivatives APN functions APN permutations |
| title | A Note on APN Permutations and Their Derivatives |
| title_full | A Note on APN Permutations and Their Derivatives |
| title_fullStr | A Note on APN Permutations and Their Derivatives |
| title_full_unstemmed | A Note on APN Permutations and Their Derivatives |
| title_short | A Note on APN Permutations and Their Derivatives |
| title_sort | note on apn permutations and their derivatives |
| topic | Boolean functions weight first-order derivatives second-order derivatives APN functions APN permutations |
| url | https://www.mdpi.com/2227-7390/12/22/3477 |
| work_keys_str_mv | AT augustinemusukwa anoteonapnpermutationsandtheirderivatives AT augustinemusukwa noteonapnpermutationsandtheirderivatives |