Fast elliptic curve point multiplication based on binary and binary non-adjacent scalar form methods
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Fast elliptic curve point multiplication based on binary and binary non-adjacent scalar form methods Denis Khleborodov1
Received: 27 June 2017 / Accepted: 8 December 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract This article presents two methods for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. Two new effective algorithms have been presented: one of them is based on a binary Non-Adjacent Form of scalar representation and another one on a binary of scalar representation method. All algorithms were developed based on simple and composite operations with point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. Theorems concerning their computational complexity are formulated and proved for these new algorithms. In the end of this article comparative analysis of both new algorithms among themselves and previously known algorithms are represented. Keywords Elliptic curves · EoT security · IoT security · Mobile security · Scalar multiplication · ECC · Lightweight cryptography Mathematics Subject Classification (2010) 03D15
1 Introduction Elliptic Curve Cryptography (ECC), independently introduced by Miller [22] and Koblitz [17] in 1980’s, is a promising technology for new cryptographic applications.
Communicated by: Ivan Oseledets Denis Khleborodov
[email protected] 1
Mathematical Studies in Information Security Section, Lomonosov Moscow State University, Moscow, Russia
D. Khleborodov Table 1 Relation of security levels (SL) in symmetric (SC) and asymmetric (FFC, IFC, ECC) cryptosystems SL
SC
FFC (DSA, DH), bit
IFC (RSA), bit
ECC (ECDS), f bit
80
2TDEA
L = 1024, N = 160
1024
160 . . . 223
112
3TDEA
L = 2048, N = 224
2048
224 . . . 255
128
AES-128
L = 3072, N = 256
3072
256 . . . 383
192
AES-192
L = 7680, N = 384
7680
384 . . . 511
256
AES-256
L = 15360, N = 512
15360
512+
ECC can achieve high security levels as that of RSA, but with much smaller key sizes and possibly faster computation, which result in lower power consumption as well as better memory and bandwidth savings (see Table 1 [23]). The main advantage of ECC is the possibility to use short keys and to provide a high security level at the same time. Significant feature of the approach used in ECC, in comparison with other approaches used in FFC (Finite Field Cryptography) and IFC (Integers Factorization Cryptography) is the ability to provide the same security level with significantly shorter keys (Table 1). It allows reduced CPU performance, power consumption and key storage requirements for hardware platforms. Memory is not a scarce resource today, however, memory usually is limited at IoT (Internet of Things) devices, smart cards and SIM (Subscriber Identification Module) cards due to the desire to reduce linear dimensions of such devices. Elliptic curve based transformations are also widely represented in security standards and s
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