Resiliency and Nonlinearity Profiles of Some Cryptographic Functions
Boolean functions are important in terms of their cryptographic and combinatorial properties for different kinds of cryptosystems. The nonlinearity and resiliency of cryptographic functions are crucial criteria with respect to protection of ciphers from affine approximation and correlation attacks....
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/23/4473 |
| _version_ | 1797462698205642752 |
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| author | Deep Singh Amit Paul Neerendra Kumar Veronika Stoffová Chaman Verma |
| author_facet | Deep Singh Amit Paul Neerendra Kumar Veronika Stoffová Chaman Verma |
| author_sort | Deep Singh |
| collection | DOAJ |
| description | Boolean functions are important in terms of their cryptographic and combinatorial properties for different kinds of cryptosystems. The nonlinearity and resiliency of cryptographic functions are crucial criteria with respect to protection of ciphers from affine approximation and correlation attacks. In this article, some constructions of disjoint spectra Boolean that function by concatenating the functions on a lesser number of variables are provided. The nonlinearity and resiliency profiles of the constructed functions are obtained. From the profiles of the constructed functions, it is observed that the nonlinearity of these functions is greater than or equal to the nonlinearity of some existing functions. Furthermore, in the security analysis of cryptosystems, 4th order nonlinearity of Boolean functions play a crucial role. It provides protection against various higher order approximation attacks. The lower bounds on 4th order nonlinearity of some classes of Boolean functions having degree 5 are provided. The lower bounds of two classes of functions have form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><msubsup><mi>r</mi><mn>1</mn><mi>n</mi></msubsup><mrow><mo>(</mo><mi>λ</mi><msup><mi>x</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi mathvariant="double-struck">F</mi><msup><mn>2</mn><mi>n</mi></msup></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><msubsup><mi mathvariant="double-struck">F</mi><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><mo>*</mo></msubsup><mo>,</mo></mrow></semantics></math></inline-formula> where (i) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><msup><mn>2</mn><mi>i</mi></msup><mo>+</mo><msup><mn>2</mn><mi>j</mi></msup><mo>+</mo><msup><mn>2</mn><mi>k</mi></msup><mo>+</mo><msup><mn>2</mn><mo>ℓ</mo></msup><mo>+</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mo>ℓ</mo></mrow></semantics></math></inline-formula> are integers such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>></mo><mi>k</mi><mo>></mo><mo>ℓ</mo><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>2</mn><mi>i</mi></mrow></semantics></math></inline-formula>, and (ii) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><msup><mn>2</mn><mrow><mn>4</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mrow><mn>3</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mrow><mn>2</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mo>ℓ</mo></msup><mo>+</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℓ</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is an integer with property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>gcd</mi><mo>(</mo><mo>ℓ</mo><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>8</mn></mrow></semantics></math></inline-formula> are provided. The obtained lower bounds are compared with some existing results available in the literature. |
| first_indexed | 2024-03-09T17:40:19Z |
| format | Article |
| id | doaj.art-100a3df88e314641a1a6ea79f28e5283 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T17:40:19Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-100a3df88e314641a1a6ea79f28e52832023-11-24T11:34:09ZengMDPI AGMathematics2227-73902022-11-011023447310.3390/math10234473Resiliency and Nonlinearity Profiles of Some Cryptographic FunctionsDeep Singh0Amit Paul1Neerendra Kumar2Veronika Stoffová3Chaman Verma4Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151401, IndiaDepartment of Mathematics, Guru Nanak Dev University, Amritsar 143005, IndiaDepartment of Computer Science and IT, Central University of Jammu, Jammu 181143, IndiaDepartment of Mathematics and Computer Science, Trnava University, 91843 Trnava, SlovakiaDepartment of Media and Educational Informatics, Faculty of Informatics, Eötvös Loránd University, 1053 Budapest, HungaryBoolean functions are important in terms of their cryptographic and combinatorial properties for different kinds of cryptosystems. The nonlinearity and resiliency of cryptographic functions are crucial criteria with respect to protection of ciphers from affine approximation and correlation attacks. In this article, some constructions of disjoint spectra Boolean that function by concatenating the functions on a lesser number of variables are provided. The nonlinearity and resiliency profiles of the constructed functions are obtained. From the profiles of the constructed functions, it is observed that the nonlinearity of these functions is greater than or equal to the nonlinearity of some existing functions. Furthermore, in the security analysis of cryptosystems, 4th order nonlinearity of Boolean functions play a crucial role. It provides protection against various higher order approximation attacks. The lower bounds on 4th order nonlinearity of some classes of Boolean functions having degree 5 are provided. The lower bounds of two classes of functions have form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><msubsup><mi>r</mi><mn>1</mn><mi>n</mi></msubsup><mrow><mo>(</mo><mi>λ</mi><msup><mi>x</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi mathvariant="double-struck">F</mi><msup><mn>2</mn><mi>n</mi></msup></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><msubsup><mi mathvariant="double-struck">F</mi><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><mo>*</mo></msubsup><mo>,</mo></mrow></semantics></math></inline-formula> where (i) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><msup><mn>2</mn><mi>i</mi></msup><mo>+</mo><msup><mn>2</mn><mi>j</mi></msup><mo>+</mo><msup><mn>2</mn><mi>k</mi></msup><mo>+</mo><msup><mn>2</mn><mo>ℓ</mo></msup><mo>+</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mo>ℓ</mo></mrow></semantics></math></inline-formula> are integers such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>></mo><mi>k</mi><mo>></mo><mo>ℓ</mo><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>2</mn><mi>i</mi></mrow></semantics></math></inline-formula>, and (ii) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><msup><mn>2</mn><mrow><mn>4</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mrow><mn>3</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mrow><mn>2</mn><mo>ℓ</mo></mrow></msup><mo>+</mo><msup><mn>2</mn><mo>ℓ</mo></msup><mo>+</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℓ</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is an integer with property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>gcd</mi><mo>(</mo><mo>ℓ</mo><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>8</mn></mrow></semantics></math></inline-formula> are provided. The obtained lower bounds are compared with some existing results available in the literature.https://www.mdpi.com/2227-7390/10/23/4473affine approximation attackBoolean functionsdisjoint spectra functionshigher order nonlinearitiesresiliencyWalsh–Hadamard transform (WHT) |
| spellingShingle | Deep Singh Amit Paul Neerendra Kumar Veronika Stoffová Chaman Verma Resiliency and Nonlinearity Profiles of Some Cryptographic Functions Mathematics affine approximation attack Boolean functions disjoint spectra functions higher order nonlinearities resiliency Walsh–Hadamard transform (WHT) |
| title | Resiliency and Nonlinearity Profiles of Some Cryptographic Functions |
| title_full | Resiliency and Nonlinearity Profiles of Some Cryptographic Functions |
| title_fullStr | Resiliency and Nonlinearity Profiles of Some Cryptographic Functions |
| title_full_unstemmed | Resiliency and Nonlinearity Profiles of Some Cryptographic Functions |
| title_short | Resiliency and Nonlinearity Profiles of Some Cryptographic Functions |
| title_sort | resiliency and nonlinearity profiles of some cryptographic functions |
| topic | affine approximation attack Boolean functions disjoint spectra functions higher order nonlinearities resiliency Walsh–Hadamard transform (WHT) |
| url | https://www.mdpi.com/2227-7390/10/23/4473 |
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