Maximum and minimum norms for τ-NAF expansion on Koblitz Curve
Background/Objectives: The scalar multiplication in Elliptic Curve Cryptosystem (ECC) is the dominant operation of computing integer multiple for an integer n and a point P on elliptic curve. In 1997, Solinas4 introduced the τ-adic non-adjacent form (τ-NAF) expansion of an element n of ring Z(τ) on...
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| Format: | Article |
| Language: | English |
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Indian Society for Education and Environment
2016
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| Online Access: | http://psasir.upm.edu.my/id/eprint/54761/ http://psasir.upm.edu.my/id/eprint/54761/1/Maximum%20and%20minimum%20norms%20for%20%CF%84-NAF%20expansion%20on%20Koblitz%20Curve.pdf |
| _version_ | 1848852623908667392 |
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| author | Ali, Nur Adawiah Yunos, Faridah |
| author_facet | Ali, Nur Adawiah Yunos, Faridah |
| author_sort | Ali, Nur Adawiah |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | Background/Objectives: The scalar multiplication in Elliptic Curve Cryptosystem (ECC) is the dominant operation of computing integer multiple for an integer n and a point P on elliptic curve. In 1997, Solinas4 introduced the τ-adic non-adjacent form (τ-NAF) expansion of an element n of ring Z(τ) on Koblitz Curve. However in 2000, Solinas estimated the length of τ-NAF expansion by using maximum and minimum norms that obtained by direct evaluation method. In 2014, Yunos et al.9 introduced the formula of norm for every τ-NAF to improve this method. However, a lot of combination of norm should be considered when length of expansion is more than 15. So, the objective of this paper is to built the formulas to calculate the number of maximum and minimum norms for τ-NAF occurring among of all elements in Z(τ). Application/Improvement: With these formulas, we can estimate the length of τ-NAF expansion more accurately. |
| first_indexed | 2025-11-15T10:41:02Z |
| format | Article |
| id | upm-54761 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T10:41:02Z |
| publishDate | 2016 |
| publisher | Indian Society for Education and Environment |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-547612018-04-23T04:59:26Z http://psasir.upm.edu.my/id/eprint/54761/ Maximum and minimum norms for τ-NAF expansion on Koblitz Curve Ali, Nur Adawiah Yunos, Faridah Background/Objectives: The scalar multiplication in Elliptic Curve Cryptosystem (ECC) is the dominant operation of computing integer multiple for an integer n and a point P on elliptic curve. In 1997, Solinas4 introduced the τ-adic non-adjacent form (τ-NAF) expansion of an element n of ring Z(τ) on Koblitz Curve. However in 2000, Solinas estimated the length of τ-NAF expansion by using maximum and minimum norms that obtained by direct evaluation method. In 2014, Yunos et al.9 introduced the formula of norm for every τ-NAF to improve this method. However, a lot of combination of norm should be considered when length of expansion is more than 15. So, the objective of this paper is to built the formulas to calculate the number of maximum and minimum norms for τ-NAF occurring among of all elements in Z(τ). Application/Improvement: With these formulas, we can estimate the length of τ-NAF expansion more accurately. Indian Society for Education and Environment 2016-07 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/54761/1/Maximum%20and%20minimum%20norms%20for%20%CF%84-NAF%20expansion%20on%20Koblitz%20Curve.pdf Ali, Nur Adawiah and Yunos, Faridah (2016) Maximum and minimum norms for τ-NAF expansion on Koblitz Curve. Indian Journal of Science and Technology, 9 (28). pp. 1-8. ISSN 0974-6846; ESSN: 0974-5645 http://www.indjst.org/index.php/indjst/article/view/97358 10.17485/ijst/2016/v9i28/97358 |
| spellingShingle | Ali, Nur Adawiah Yunos, Faridah Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title | Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title_full | Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title_fullStr | Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title_full_unstemmed | Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title_short | Maximum and minimum norms for τ-NAF expansion on Koblitz Curve |
| title_sort | maximum and minimum norms for τ-naf expansion on koblitz curve |
| url | http://psasir.upm.edu.my/id/eprint/54761/ http://psasir.upm.edu.my/id/eprint/54761/ http://psasir.upm.edu.my/id/eprint/54761/ http://psasir.upm.edu.my/id/eprint/54761/1/Maximum%20and%20minimum%20norms%20for%20%CF%84-NAF%20expansion%20on%20Koblitz%20Curve.pdf |