On some patterns of TNAF for scalar multiplication over Koblitz curve
A τ-adic non-adjacent form (TNAF) of an element α of the ring Z(τ) is an expansion whereby the digits are generated by iteratively dividing α by τ, allowing the remainders of -1,0 or 1. The application of TNAF as a multiplier of scalar multiplication (SM) on the Koblitz curve plays a key role in Ell...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Published: |
Faculty of Science, University of Malaya
2022
|
| Online Access: | http://psasir.upm.edu.my/id/eprint/102385/ |
| _version_ | 1848863789211975680 |
|---|---|
| author | Yunos, Faridah Rosli, Rosimah Muslim, Norliana |
| author_facet | Yunos, Faridah Rosli, Rosimah Muslim, Norliana |
| author_sort | Yunos, Faridah |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | A τ-adic non-adjacent form (TNAF) of an element α of the ring Z(τ) is an expansion whereby the digits are generated by iteratively dividing α by τ, allowing the remainders of -1,0 or 1. The application of TNAF as a multiplier of scalar multiplication (SM) on the Koblitz curve plays a key role in Elliptical Curve Cryptography (ECC). There are several patterns of TNAF (α) expansion in the form of {equation presented} and 8k1+8k2that have been produced in prior work in the literature. However, the construction of their properties based upon pyramid number formulas such as Nichomacus's theorem and Faulhaber's formula remains to be rather complex. In this work, we derive such types of TNAF in a more concise manner by applying the power of Frobenius map (τm) based on v-simplex and arithmetic sequences. |
| first_indexed | 2025-11-15T13:38:30Z |
| format | Article |
| id | upm-102385 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-15T13:38:30Z |
| publishDate | 2022 |
| publisher | Faculty of Science, University of Malaya |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1023852023-07-10T02:10:58Z http://psasir.upm.edu.my/id/eprint/102385/ On some patterns of TNAF for scalar multiplication over Koblitz curve Yunos, Faridah Rosli, Rosimah Muslim, Norliana A τ-adic non-adjacent form (TNAF) of an element α of the ring Z(τ) is an expansion whereby the digits are generated by iteratively dividing α by τ, allowing the remainders of -1,0 or 1. The application of TNAF as a multiplier of scalar multiplication (SM) on the Koblitz curve plays a key role in Elliptical Curve Cryptography (ECC). There are several patterns of TNAF (α) expansion in the form of {equation presented} and 8k1+8k2that have been produced in prior work in the literature. However, the construction of their properties based upon pyramid number formulas such as Nichomacus's theorem and Faulhaber's formula remains to be rather complex. In this work, we derive such types of TNAF in a more concise manner by applying the power of Frobenius map (τm) based on v-simplex and arithmetic sequences. Faculty of Science, University of Malaya 2022-09-30 Article PeerReviewed Yunos, Faridah and Rosli, Rosimah and Muslim, Norliana (2022) On some patterns of TNAF for scalar multiplication over Koblitz curve. Malaysian Journal of Science, 41 (1 spec.). pp. 9-16. ISSN 1394-3065 https://mjs.um.edu.my/index.php/MJS/article/view/34829 10.22452/mjs.sp2022no1.2 |
| spellingShingle | Yunos, Faridah Rosli, Rosimah Muslim, Norliana On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title | On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title_full | On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title_fullStr | On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title_full_unstemmed | On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title_short | On some patterns of TNAF for scalar multiplication over Koblitz curve |
| title_sort | on some patterns of tnaf for scalar multiplication over koblitz curve |
| url | http://psasir.upm.edu.my/id/eprint/102385/ http://psasir.upm.edu.my/id/eprint/102385/ http://psasir.upm.edu.my/id/eprint/102385/ |