Effective compression maps for torus-based cryptography

We give explicit parametrizations of the algebraic tori over any finite field for any prime power . Applying the construction for to a quadratic field we show that the set of -rational points of the torus is birationally equivalent to the affine part of a Singer arc in . This gives a simple, yet efficient compression and decompression algorithm from to that can be substituted in the faster implementation of CEILIDH (Granger et al., in Algorithmic number theory, pp 235-249, Springer, Berlin, 2004) achieving a theoretical 30 % speedup and that is also cheaper than the recently proposed factor- compression technique in Karabina (IEEE Trans Inf Theory 58(5):3293-3304, 2012). The compression methods here presented have a wide class of applications to public-key and pairing-based cryptography over any finite field.