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Elliptic curve cryptography (ECC) is a typical public key cryptography technique that can ensure equivalent security with considerably smaller key sizes than Rivest-Shamir-Adleman (RSA). Hence, various implementations based on ECC are recommended for block chain and Internet of Things (IoT) devices. Because elliptic curve scalar multiplication (ECSM) is a fundamental computation in ECC, enhancing the security and efficiency of ECSM is important. ECSM specifies a scalar multiplication algorithm and elliptic curve addition formulae. Elliptic curve addition formulae on affine coordinates are compact from a memory cost perspective but weak against side channel attacks. Elliptic curve complete addition (CA) formulae can achieve secure ECSMs but are inefficient. A newly proposed secure ECSM, which uses a right-to-left (RL) scalar multiplication algorithm and (extended) affine coordinates, takes advantage of elliptic curve addition formulae on affine coordinates. However, it can only scan the input scalars from right to left. We propose new ECSMs, which can scan the input scalars from left to right (LR) based on (extended) affine coordinates. We also prove that our LR ECSMs satisfy secure generality without requiring exceptional computations. We enhance the efficiency of both LR and RL ECSMs with optimized inversion. Our LR ECSM with a memory of 12 field elements reduces that of the Montgomery ladder and Joye’s LR with CA formulae by 36.84% and that of 2-ary RL with (extended) affine coordinates by 14.29%, respectively. Our compact ECSMs are fit for applications on IoT devices and block chain, with a critical memory requirement.

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Elliptic curve cryptography (ECC) can ensure an equivalent security with much smaller key sizes. Elliptic curve scalar multiplication (ECSM) is a fundamental computation used in ECC. This paper focuses on ECSM resisting simple power attack and safe error attack of side-channel attack specifically. Elliptic curve complete addition (CA) formulae can achieve secure ECSM algorithms but are inefficient from memory and computational cost perspectives. Another secure ECSM, which uses (extended) affine, is more efficient for both memory and computational costs. However, it scans input scalars from right to left. In this paper, our developed scalar multiplication algorithms also use their extended affine, but scan from left to right (LR). We also prove the security of our LR ECSM algorithms and analyze them both theoretically and experimentally. Our new LR ECSM algorithms can reduce the amount of memory by 37.5% and reduce the computational time by more than 40% compared to Joye's regular 2-ary LR algorithm with CA formulae.