Pairings on Elliptic Curves over Finite Commutative Rings
نویسندگان
چکیده
The Weil and Tate pairings are defined for elliptic curves over fields, including finite fields. These definitions extend naturally to elliptic curves over Z/NZ , for any positive integer N , or more generally to elliptic curves over any finite commutative ring, and even the reduced Tate pairing makes sense in this more general setting. This paper discusses a number of issues which arise if one tries to develop pairing-based cryptosystems on elliptic curves over such rings. We argue that, although it may be possible to develop some cryptosystems in this setting, there are obstacles in adapting many of the main ideas in pairingbased cryptography to elliptic curves over rings. Our main results are: (i) an oracle that computes reduced Tate pairings over such rings (or even just over Z/NZ ) can be used to factorise integers; and (ii) an oracle that determines whether or not the reduced Tate pairing of two points is trivial can be used to solve the quadratic residuosity problem.
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