Malaysian Journal of Mathematical Sciences, December 2021, Vol. 15(S)
Special Topics: New Ideas in Cryptology
Power of Frobenius Endomorphism and its Performance on PseudoTNAF System
Yunos, F., Yusof, A. M., Hadani, N. H., Ariffin, M. R. K., and Sapar, S. H.
Corresponding Email: [email protected]
Received date: 15 June 2021
Accepted date: 8 October 2021
Abstract:
Let \(E\) be an elliptical curve defined over \(F_{2^m}\) and the mapping $\tau$ is a Frobenius endomorphism from the set $F_{2^m }$ to itself. The Koblitz curve is a special curve whose $\tau$ has been used to improve the calculation performance of its scalar multiplication, $nP$ where $P$ is a point on the curve $E$. Moreover, the multiplier, $n$ is $\tau$ -adic non adjacent form (TNAF) expansion where its digit is generated by the repeated division of an integer in the ring of $Z(\tau)$ by $\tau$. Previous research has found that the power of Frobenius endomorphism $\tau^m$ has some advantages in TNAF, Reduced TNAF and their equivalent i.e. pseudoTNAF expansions. In this paper, new finding of $\tau^m$ based on $v$-simplex and arithmetic sequences is provided. With this approach, the performance of converting modulo $\rho \frac{\tau^m -1}{\tau-1}$ to $r + s \tau$ an element of $Z(\tau)$ in pseudoTNAF$'$s system is enhanced.
Keywords: cryptography; field; Frobenius endomorphism; Koblitz curve; number of elliptic points; sequence of arithmetic; sequence of simplex; $\tau$-adic non adjacent
