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In this work, we are interested in arithmetic on large prime field and their extensions of small degree. We explain why it is very interesting to use RNS arithmetic for the base field   p   when computations in   p k    have to be done, essentially thanks to lazy reduction. This is for example the case for pairing computations on ordinary curves (as MNT or BN curves). We show that using RNS can considerably decrease the number of basic operations required for a pairing computation in many popular situations.

RNS arithmetic in 𝔽 p k and application to fast pairing computation