Secret Sharing, Zero Sum Sets, and Hamming Codes
A <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-secret sharing scheme is a met...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1644 |
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| author | Selda Çalkavur Patrick Solé |
| author_facet | Selda Çalkavur Patrick Solé |
| author_sort | Selda Çalkavur |
| collection | DOAJ |
| description | A <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-secret sharing scheme is a method of distribution of information among <i>n</i> participants such that any <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> of them can reconstruct the secret but any <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> cannot. A ramp secret sharing scheme is a relaxation of that protocol that allows that some <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-coalitions could reconstruct the secret. In this work, we explore some ramp secret sharing schemes based on quotients of polynomial rings. The security analysis depends on the distribution of zero-sum sets in abelian groups. We characterize all finite commutative rings for which the sum of all elements is zero, a result of independent interest. When the quotient is a finite field, we are led to study the weight distribution of a coset of shortened Hamming codes. |
| first_indexed | 2024-03-10T16:06:24Z |
| format | Article |
| id | doaj.art-2f474c88107a47b4867df7acbea352d9 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:06:24Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2f474c88107a47b4867df7acbea352d92023-11-20T14:50:47ZengMDPI AGMathematics2227-73902020-09-01810164410.3390/math8101644Secret Sharing, Zero Sum Sets, and Hamming CodesSelda Çalkavur0Patrick Solé1Mathematics Department, Kocaeli University, 41380 Kocaeli, TurkeyI2M, Aix Marseille University, Centrale Marseilles, CNRS, 163 Avenue de Luminy, 13009 Marseille, FranceA <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-secret sharing scheme is a method of distribution of information among <i>n</i> participants such that any <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> of them can reconstruct the secret but any <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> cannot. A ramp secret sharing scheme is a relaxation of that protocol that allows that some <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-coalitions could reconstruct the secret. In this work, we explore some ramp secret sharing schemes based on quotients of polynomial rings. The security analysis depends on the distribution of zero-sum sets in abelian groups. We characterize all finite commutative rings for which the sum of all elements is zero, a result of independent interest. When the quotient is a finite field, we are led to study the weight distribution of a coset of shortened Hamming codes.https://www.mdpi.com/2227-7390/8/10/1644secret sharingthreshold schemeramp schemepolynomial residue ringzero-sum set |
| spellingShingle | Selda Çalkavur Patrick Solé Secret Sharing, Zero Sum Sets, and Hamming Codes Mathematics secret sharing threshold scheme ramp scheme polynomial residue ring zero-sum set |
| title | Secret Sharing, Zero Sum Sets, and Hamming Codes |
| title_full | Secret Sharing, Zero Sum Sets, and Hamming Codes |
| title_fullStr | Secret Sharing, Zero Sum Sets, and Hamming Codes |
| title_full_unstemmed | Secret Sharing, Zero Sum Sets, and Hamming Codes |
| title_short | Secret Sharing, Zero Sum Sets, and Hamming Codes |
| title_sort | secret sharing zero sum sets and hamming codes |
| topic | secret sharing threshold scheme ramp scheme polynomial residue ring zero-sum set |
| url | https://www.mdpi.com/2227-7390/8/10/1644 |
| work_keys_str_mv | AT seldacalkavur secretsharingzerosumsetsandhammingcodes AT patricksole secretsharingzerosumsetsandhammingcodes |