Revisiting linearly extended discrete functions
The authors introduced a new family of cryptographic schemes in a previous research article, which includes many practical encryption schemes, such as the Feistel family. Given a finite field of order qq, any n>m≥0n\gt m\ge 0, the authors described a new way to extend discrete functions with doma...
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De Gruyter
2024-12-01
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| Series: | Journal of Mathematical Cryptology |
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| Online Access: | https://doi.org/10.1515/jmc-2024-0010 |
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| author | Gravel Claude Panario Daniel |
| author_facet | Gravel Claude Panario Daniel |
| author_sort | Gravel Claude |
| collection | DOAJ |
| description | The authors introduced a new family of cryptographic schemes in a previous research article, which includes many practical encryption schemes, such as the Feistel family. Given a finite field of order qq, any n>m≥0n\gt m\ge 0, the authors described a new way to extend discrete functions with domain size qm{q}^{m} and range size qn−m{q}^{n-m} to a permutation over qn{q}^{n} elements using theory from linear error correcting codes. The authors previously showed that the knowledge about the differentials and correlations of the resulting permutation reduces solely to those of the extended discrete function. We show how the perfect secrecy of extended nonlinear functions transfers to the family of bijective linear extensions. We investigate how the concrete security of the family of nonlinear functions relates to the family of permutations obtained by such a type of linear extension. We also explore how the interplay between the entropy and the total variation distance (near-perfect secrecy with unbounded adversary) affects the mixing rate (number of iterations of the feedback linear extensions) with respect to the uniform distribution of the permutations over qn{q}^{n} elements. We give a new proof that a distribution close to the uniform distribution has a large entropy. |
| format | Article |
| id | doaj-art-55fb642de4414294b9dba720462348bc |
| institution | OA Journals |
| issn | 1862-2984 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of Mathematical Cryptology |
| spelling | doaj-art-55fb642de4414294b9dba720462348bc2025-08-20T01:56:28ZengDe GruyterJournal of Mathematical Cryptology1862-29842024-12-01181235005171510.1515/jmc-2024-0010Revisiting linearly extended discrete functionsGravel Claude0Panario Daniel1Department of Computer Science, Toronto Metropolitan University, Toronto, CanadaSchool of Mathematics and Statistics, Carleton University, Ottawa, CanadaThe authors introduced a new family of cryptographic schemes in a previous research article, which includes many practical encryption schemes, such as the Feistel family. Given a finite field of order qq, any n>m≥0n\gt m\ge 0, the authors described a new way to extend discrete functions with domain size qm{q}^{m} and range size qn−m{q}^{n-m} to a permutation over qn{q}^{n} elements using theory from linear error correcting codes. The authors previously showed that the knowledge about the differentials and correlations of the resulting permutation reduces solely to those of the extended discrete function. We show how the perfect secrecy of extended nonlinear functions transfers to the family of bijective linear extensions. We investigate how the concrete security of the family of nonlinear functions relates to the family of permutations obtained by such a type of linear extension. We also explore how the interplay between the entropy and the total variation distance (near-perfect secrecy with unbounded adversary) affects the mixing rate (number of iterations of the feedback linear extensions) with respect to the uniform distribution of the permutations over qn{q}^{n} elements. We give a new proof that a distribution close to the uniform distribution has a large entropy.https://doi.org/10.1515/jmc-2024-0010pseudo-random objectsfinite fieldlinear algebramixing rateentropytotal variation distance94a6037a25 |
| spellingShingle | Gravel Claude Panario Daniel Revisiting linearly extended discrete functions Journal of Mathematical Cryptology pseudo-random objects finite field linear algebra mixing rate entropy total variation distance 94a60 37a25 |
| title | Revisiting linearly extended discrete functions |
| title_full | Revisiting linearly extended discrete functions |
| title_fullStr | Revisiting linearly extended discrete functions |
| title_full_unstemmed | Revisiting linearly extended discrete functions |
| title_short | Revisiting linearly extended discrete functions |
| title_sort | revisiting linearly extended discrete functions |
| topic | pseudo-random objects finite field linear algebra mixing rate entropy total variation distance 94a60 37a25 |
| url | https://doi.org/10.1515/jmc-2024-0010 |
| work_keys_str_mv | AT gravelclaude revisitinglinearlyextendeddiscretefunctions AT panariodaniel revisitinglinearlyextendeddiscretefunctions |