Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling
Reducing the computation time of scalar multiplication for elliptic curve cryptography is a significant challenge. This study proposes an efficient scalar multiplication method for elliptic curves over finite fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
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2025-03-01
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| author | Fu-Jung Kan Yan-Haw Chen Jeng-Jung Wang Chong-Dao Lee |
| author_facet | Fu-Jung Kan Yan-Haw Chen Jeng-Jung Wang Chong-Dao Lee |
| author_sort | Fu-Jung Kan |
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| description | Reducing the computation time of scalar multiplication for elliptic curve cryptography is a significant challenge. This study proposes an efficient scalar multiplication method for elliptic curves over finite fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>m</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The proposed method first converts the scalar into a binary number. Then, using Horner’s rule, the binary number is divided into fixed-length bit-words. Each bit-word undergoes repeating point doubling, which can be precomputed. However, repeating point doubling typically involves numerous inverse operations. To address this, significant effort has been made to develop formulas that minimize the number of inverse operations. With the proposed formula, regardless of how many times the operation is repeated, only a single inverse operation is required. Over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>m</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, the proposed method for scalar multiplication outperforms the sliding window method, which is currently regarded as the fastest available. However, the introduced formulas require more multiplications, squares, and additions. To reduce these operations, we further optimize the square operations; however, this introduces a trade-off between computation time and memory size. These challenges are key areas for future improvement. |
| format | Article |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-1f413526b0aa4a87b2bf102ba6dcae2a2025-08-20T03:43:20ZengMDPI AGMathematics2227-73902025-03-0113692410.3390/math13060924Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point DoublingFu-Jung Kan0Yan-Haw Chen1Jeng-Jung Wang2Chong-Dao Lee3Department of Electronic Engineering, I-Shou University, Kaohsiung 84001, TaiwanDepartment of Information Engineering, I-Shou University, Kaohsiung 84001, TaiwanDepartment of Information Engineering, I-Shou University, Kaohsiung 84001, TaiwanDepartment of Information Engineering, I-Shou University, Kaohsiung 84001, TaiwanReducing the computation time of scalar multiplication for elliptic curve cryptography is a significant challenge. This study proposes an efficient scalar multiplication method for elliptic curves over finite fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>m</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The proposed method first converts the scalar into a binary number. Then, using Horner’s rule, the binary number is divided into fixed-length bit-words. Each bit-word undergoes repeating point doubling, which can be precomputed. However, repeating point doubling typically involves numerous inverse operations. To address this, significant effort has been made to develop formulas that minimize the number of inverse operations. With the proposed formula, regardless of how many times the operation is repeated, only a single inverse operation is required. Over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>m</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, the proposed method for scalar multiplication outperforms the sliding window method, which is currently regarded as the fastest available. However, the introduced formulas require more multiplications, squares, and additions. To reduce these operations, we further optimize the square operations; however, this introduces a trade-off between computation time and memory size. These challenges are key areas for future improvement.https://www.mdpi.com/2227-7390/13/6/924elliptic curvescalar multiplicationinverse operationfinite field |
| spellingShingle | Fu-Jung Kan Yan-Haw Chen Jeng-Jung Wang Chong-Dao Lee Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling Mathematics elliptic curve scalar multiplication inverse operation finite field |
| title | Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling |
| title_full | Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling |
| title_fullStr | Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling |
| title_full_unstemmed | Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling |
| title_short | Efficient Scalar Multiplication of ECC Using Lookup Table and Fast Repeating Point Doubling |
| title_sort | efficient scalar multiplication of ecc using lookup table and fast repeating point doubling |
| topic | elliptic curve scalar multiplication inverse operation finite field |
| url | https://www.mdpi.com/2227-7390/13/6/924 |
| work_keys_str_mv | AT fujungkan efficientscalarmultiplicationofeccusinglookuptableandfastrepeatingpointdoubling AT yanhawchen efficientscalarmultiplicationofeccusinglookuptableandfastrepeatingpointdoubling AT jengjungwang efficientscalarmultiplicationofeccusinglookuptableandfastrepeatingpointdoubling AT chongdaolee efficientscalarmultiplicationofeccusinglookuptableandfastrepeatingpointdoubling |