Montgomery Residue Representation Fault-Tolerant Computation in GF (2 k ) Silvana Medoˇ s ⋆ and Serdar Bozta¸ s School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V, Melbourne 3001, Australia {silvana.medos,serdar.boztas} @ems.rmit.edu.au Abstract. In this paper, we are concerned with protecting elliptic curve computation in a tamper proof device by protecting finite field compu- tation against active side channel attacks, i.e., fault attacks. We propose residue representation of the field elements for fault tolerant Montgomery residue representation multiplication algorithm, by providing fault mod- els for fault attacks, and countermeasures to some fault inducing attacks. Keywords: finite field, fault tolerant computation, fault attacks. 1 Introduction Finite field arithmetic is fundamental for Elliptic Curve Cryptography (ECC) which was proposed independently by Koblitz [13] and Miller [20] in 1985. ECC has received commercial acceptance and has been included in numerous stan- dards. Its computation relies on a very large finite field (with more than 2 160 elements). Security of ECC is based on the difficulty of the discrete logarithm problem (DLP), but it is proven that security of cryptosystems does not only de- pend on the mathematical properties. Side channel attacks provide information which reveals important and compromising details about secret data. Some of these details can be used as a new trapdoor to invert a trapdoor one-way func- tion without the secret key. This allows an adversary to break a cryptographic protocol, even if it proved to be secure in the mathematical sense. Specifically, in case of fault attacks which are active attacks, an adversary has to tamper with an attacked device in order to create faults. E.g. if an adversary can inflict some physical stress on the smartcard, he can induce faults into circuitry or memory, as a result these faults are manifested in computation as a errors. Therefore, faulty final result is computed. Moreover, if computation depends on some se- cret key, facts about secret key can be concluded. For further references please see [4], [7], [10], [12]. In this paper we are concerned with protecting elliptic curve computation in a tamper proof device by protecting finite field computation against active side channel attacks, i.e., fault attacks where an adversary induces faults into a de- vice, while it executes the correct program. Our paper is organized as follows. ⋆ The first author was supported by ARC Linkage grant LP0455324. Y. Mu, W. Susilo, and J. Seberry (Eds.): ACISP 2008, LNCS 5107, pp. 419–432, 2008. c  Springer-Verlag Berlin Heidelberg 2008