Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves
In ECC, scalar multiplication is the dominant operation, namely computing nP from a point P on an elliptic curve where the multiplier n is an integer, defined as the point resulting from adding P+P+⋯+P, n times. The τ-NAF proposed by Solinas, is one of the most efficient algorithms to compute scalar...
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| Format: | Article |
| Language: | English |
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Institute for Mathematical Research,Universiti Putra Malaysia
2015
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| Online Access: | http://psasir.upm.edu.my/id/eprint/35102/ http://psasir.upm.edu.my/id/eprint/35102/1/Pseudo%20t-Adic%20non%20adjacent%20form%20for%20scalar%20multiplication%20on%20Koblitz%20curves.pdf |
| _version_ | 1848847960620662784 |
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| author | Yunos, Faridah Mohd Atan, Kamel Ariffin Kamel Ariffin, Muhammad Rezal Md. Said, Mohamad Rushdan |
| author_facet | Yunos, Faridah Mohd Atan, Kamel Ariffin Kamel Ariffin, Muhammad Rezal Md. Said, Mohamad Rushdan |
| author_sort | Yunos, Faridah |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | In ECC, scalar multiplication is the dominant operation, namely computing nP from a point P on an elliptic curve where the multiplier n is an integer, defined as the point resulting from adding P+P+⋯+P, n times. The τ-NAF proposed by Solinas, is one of the most efficient algorithms to compute scalar multiplications on Koblitz curves. In this paper, we introduced an equivalent multiplier to τ-NAF namely pseudo TNAF. It is based on the idea of transforming the τ-NAF expression to a reduced τ-NAF that has been done by some researchers. It can eliminate the elliptic doublings in scalar multiplication method, and double the number of elliptic additions. We provide the formula for obtaining a total of lattice points in Voronoi region of modulo r+st where r+sτ an element of ring Z(τ). This helps us to find all the multipliers n that based on τ-NAF. We also discuss the estimation of operational costs when using pseudo TNAF as a multiplier of scalar multiplication. |
| first_indexed | 2025-11-15T09:26:54Z |
| format | Article |
| id | upm-35102 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T09:26:54Z |
| publishDate | 2015 |
| publisher | Institute for Mathematical Research,Universiti Putra Malaysia |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-351022019-01-23T01:33:23Z http://psasir.upm.edu.my/id/eprint/35102/ Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves Yunos, Faridah Mohd Atan, Kamel Ariffin Kamel Ariffin, Muhammad Rezal Md. Said, Mohamad Rushdan In ECC, scalar multiplication is the dominant operation, namely computing nP from a point P on an elliptic curve where the multiplier n is an integer, defined as the point resulting from adding P+P+⋯+P, n times. The τ-NAF proposed by Solinas, is one of the most efficient algorithms to compute scalar multiplications on Koblitz curves. In this paper, we introduced an equivalent multiplier to τ-NAF namely pseudo TNAF. It is based on the idea of transforming the τ-NAF expression to a reduced τ-NAF that has been done by some researchers. It can eliminate the elliptic doublings in scalar multiplication method, and double the number of elliptic additions. We provide the formula for obtaining a total of lattice points in Voronoi region of modulo r+st where r+sτ an element of ring Z(τ). This helps us to find all the multipliers n that based on τ-NAF. We also discuss the estimation of operational costs when using pseudo TNAF as a multiplier of scalar multiplication. Institute for Mathematical Research,Universiti Putra Malaysia 2015-06 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/35102/1/Pseudo%20t-Adic%20non%20adjacent%20form%20for%20scalar%20multiplication%20on%20Koblitz%20curves.pdf Yunos, Faridah and Mohd Atan, Kamel Ariffin and Kamel Ariffin, Muhammad Rezal and Md. Said, Mohamad Rushdan (2015) Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves. Malaysian Journal of Mathematical Sciences, 9 (spec.). pp. 71-88. ISSN 1823-8343 http://einspem.upm.edu.my/journal/fullpaper/vol9s/5.%20Faridah%20Yunos.pdf |
| spellingShingle | Yunos, Faridah Mohd Atan, Kamel Ariffin Kamel Ariffin, Muhammad Rezal Md. Said, Mohamad Rushdan Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title | Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title_full | Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title_fullStr | Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title_full_unstemmed | Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title_short | Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves |
| title_sort | pseudo τ - adic non adjacent form for scalar multiplication on koblitz curves |
| url | http://psasir.upm.edu.my/id/eprint/35102/ http://psasir.upm.edu.my/id/eprint/35102/ http://psasir.upm.edu.my/id/eprint/35102/1/Pseudo%20t-Adic%20non%20adjacent%20form%20for%20scalar%20multiplication%20on%20Koblitz%20curves.pdf |