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Fast Scalar Multiplication for Elliptic Curves over Prime Fields by Efficiently Computable Formulas
Saud Al Musa, Guangwu Xu
Public-key cryptography
This paper addresses fast scalar multiplication for elliptic curves over finite fields. In the first part of the paper, we obtain several efficiently computable formulas for basic elliptic curves arithmetic in the family of twisted Edwards curves over prime fields. Our $2Q+P$ formula saves about $2.8$ field multiplications, and our $5P$ formula saves about $4.2$ field multiplications in standard projective coordinate systems, compared to the latest existing results. In the second part of the...
Fast Scalar Multiplication for Elliptic Curves over Binary Fields by Efficiently Computable Formulas
Saud Al Musa, Guangwu Xu
Public-key cryptography
This paper considers efficient scalar multiplication of elliptic curves over binary fields with a twofold purpose. Firstly, we derive the most efficient $3P$ formula in $\lambda$-projective coordinates and $5P$ formula in both affine and $\lambda$-projective coordinates. Secondly, extensive experiments have been conducted to test various multi-base scalar multiplication methods (e.g., greedy, ternary/binary, multi-base NAF, and tree-based) by integrating our fast formulas. The experiments...
On the Enumeration of Double-Base Chains with Applications to Elliptic Curve Cryptography
Christophe Doche
The Double-Base Number System (DBNS) uses two bases, $2$ and $3$, in order to represent any integer $n$. A Double-Base Chain (DBC) is a special case of a DBNS expansion. DBCs have been introduced to speed up the scalar multiplication $[n]P$ on certain families of elliptic curves used in cryptography. In this context, our contributions are twofold. First, given integers $n$, $a$, and $b$, we outline a recursive algorithm to compute the number of different DBCs with a leading factor...
A Note on Point Multiplication on Supersingular Elliptic Curves over Ternary Fields
Kwang Ho Kim
Implementation
Recently, the supersingular elliptic curves over ternary fields are widely used in pairing based crypto-applications since they achieve the best possible ratio between security level and space requirement. We propose new algorithms for projective arithmetic on the curves, where the point tripling is field multiplication free, and point addition and point doubling requires one field multiplication less than the known best algorithms, respectively. The algorithms combined with DBNS can lead to...
Efficient Quintuple Formulas for Elliptic Curves and Efficient Scalar Multiplication Using Multibase Number Representation
Pradeep Kumar Mishra, Vassil Dimitrov
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...
Extended Double-Base Number System with applications to Elliptic Curve Cryptography
Christophe Doche, Laurent Imbert
Public-key cryptography
We investigate the impact of larger digit sets on the length of
Double-Base Number system (DBNS) expansions.
We present a new representation system called {\em extended DBNS}
whose expansions can be extremely sparse.
When compared with double-base chains, the average length of
extended DBNS expansions of integers of size in the range 200--500 bits
is approximately reduced by $20\%$ using one precomputed point, $30\%$ using two, and $38\%$ using
four.
We also discuss a new approach to...
This paper addresses fast scalar multiplication for elliptic curves over finite fields. In the first part of the paper, we obtain several efficiently computable formulas for basic elliptic curves arithmetic in the family of twisted Edwards curves over prime fields. Our $2Q+P$ formula saves about $2.8$ field multiplications, and our $5P$ formula saves about $4.2$ field multiplications in standard projective coordinate systems, compared to the latest existing results. In the second part of the...
This paper considers efficient scalar multiplication of elliptic curves over binary fields with a twofold purpose. Firstly, we derive the most efficient $3P$ formula in $\lambda$-projective coordinates and $5P$ formula in both affine and $\lambda$-projective coordinates. Secondly, extensive experiments have been conducted to test various multi-base scalar multiplication methods (e.g., greedy, ternary/binary, multi-base NAF, and tree-based) by integrating our fast formulas. The experiments...
The Double-Base Number System (DBNS) uses two bases, $2$ and $3$, in order to represent any integer $n$. A Double-Base Chain (DBC) is a special case of a DBNS expansion. DBCs have been introduced to speed up the scalar multiplication $[n]P$ on certain families of elliptic curves used in cryptography. In this context, our contributions are twofold. First, given integers $n$, $a$, and $b$, we outline a recursive algorithm to compute the number of different DBCs with a leading factor...
Recently, the supersingular elliptic curves over ternary fields are widely used in pairing based crypto-applications since they achieve the best possible ratio between security level and space requirement. We propose new algorithms for projective arithmetic on the curves, where the point tripling is field multiplication free, and point addition and point doubling requires one field multiplication less than the known best algorithms, respectively. The algorithms combined with DBNS can lead to...
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...
We investigate the impact of larger digit sets on the length of Double-Base Number system (DBNS) expansions. We present a new representation system called {\em extended DBNS} whose expansions can be extremely sparse. When compared with double-base chains, the average length of extended DBNS expansions of integers of size in the range 200--500 bits is approximately reduced by $20\%$ using one precomputed point, $30\%$ using two, and $38\%$ using four. We also discuss a new approach to...
