**Efficient Quintuple Formulas for Elliptic Curves and Efficient Scalar Multiplication Using Multibase Number Representation**

*Pradeep Kumar Mishra and Vassil Dimitrov*

**Abstract: **In the current work we propose two efficient formulas for computing the $5$-fold ($5P$) of an elliptic curve point $P$. One formula is for curves over finite fields of even characteristic and the other is for curves over prime fields. Double base number systems (DBNS) have been gainfully exploited to compute scalar multiplication efficiently in ECC. Using the proposed point quintupling formulas one can use 2,5 and 3,5 (besides 3,5) as bases of the double base number system. In the current work we propose a scalar multiplication algorithm, which expands the scalar using three bases 2, 3 and 5 and computes the scalar multiplication very efficiently. The proposed scheme is faster than all sequential scalar multiplication algorithms reported in literature.

**Category / Keywords: **Elliptic Curve Cryptosystems, Scalar Multiplication, Quintupling, Efficient Curve Arithmetic

**Date: **received 7 Feb 2007, last revised 10 Apr 2007

**Contact author: **pradeep at math ucalgary ca

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20070410:061728 (All versions of this report)

**Short URL: **ia.cr/2007/040

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