The least primitive roots mod p
Let p>1p\gt 1 be a large prime number, and let ε>0\varepsilon \gt 0 be a small number. The established unconditional upper bounds of the least primitive root u≠±1,v2u\ne \pm 1,{v}^{2} in the prime finite field Fp{{\mathbb{F}}}_{p} have exponential magnitudes u≪p1⁄4+εu\ll {p}^{1/4+\varepsilon }...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-04-01
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| Series: | Journal of Mathematical Cryptology |
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| Online Access: | https://doi.org/10.1515/jmc-2024-0017 |
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| _version_ | 1850265823237636096 |
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| author | Carella Nelson |
| author_facet | Carella Nelson |
| author_sort | Carella Nelson |
| collection | DOAJ |
| description | Let p>1p\gt 1 be a large prime number, and let ε>0\varepsilon \gt 0 be a small number. The established unconditional upper bounds of the least primitive root u≠±1,v2u\ne \pm 1,{v}^{2} in the prime finite field Fp{{\mathbb{F}}}_{p} have exponential magnitudes u≪p1⁄4+εu\ll {p}^{1/4+\varepsilon }. This note contributes a new result to the literature. It proves that the upper bound of the least primitive root has polynomial magnitude u≤(logp)1+εu\le {\left(\log p)}^{1+\varepsilon } unconditionally. |
| format | Article |
| id | doaj-art-cfa734a5a4254484a8dd048f349653da |
| institution | OA Journals |
| issn | 1862-2984 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of Mathematical Cryptology |
| spelling | doaj-art-cfa734a5a4254484a8dd048f349653da2025-08-20T01:54:19ZengDe GruyterJournal of Mathematical Cryptology1862-29842025-04-011911799210.1515/jmc-2024-0017The least primitive roots mod pCarella Nelson0Department of Mathematics, Fordham University and CUNY, Bronx, NY 10458, New York, United States of AmericaLet p>1p\gt 1 be a large prime number, and let ε>0\varepsilon \gt 0 be a small number. The established unconditional upper bounds of the least primitive root u≠±1,v2u\ne \pm 1,{v}^{2} in the prime finite field Fp{{\mathbb{F}}}_{p} have exponential magnitudes u≪p1⁄4+εu\ll {p}^{1/4+\varepsilon }. This note contributes a new result to the literature. It proves that the upper bound of the least primitive root has polynomial magnitude u≤(logp)1+εu\le {\left(\log p)}^{1+\varepsilon } unconditionally.https://doi.org/10.1515/jmc-2024-0017primitive rootleast primitive rootfinite fieldcryptographic algorithmcomplexity theory11a0711n0511n32 |
| spellingShingle | Carella Nelson The least primitive roots mod p Journal of Mathematical Cryptology primitive root least primitive root finite field cryptographic algorithm complexity theory 11a07 11n05 11n32 |
| title | The least primitive roots mod p |
| title_full | The least primitive roots mod p |
| title_fullStr | The least primitive roots mod p |
| title_full_unstemmed | The least primitive roots mod p |
| title_short | The least primitive roots mod p |
| title_sort | least primitive roots mod p |
| topic | primitive root least primitive root finite field cryptographic algorithm complexity theory 11a07 11n05 11n32 |
| url | https://doi.org/10.1515/jmc-2024-0017 |
| work_keys_str_mv | AT carellanelson theleastprimitiverootsmodp AT carellanelson leastprimitiverootsmodp |