Effective compression maps for torus-based cryptography

Abstract

We give explicit parametrizations of the algebraic tori $$\mathbb {T}_{n}$$Tn over any finite field $$\mathbb {F}_{q}$$Fq for any prime power $$n$$n. Applying the construction for $$n=3$$n=3 to a quadratic field $$\mathbb {F}_{q^2}$$Fq2 we show that the set of $$\mathbb {F}_q$$Fq-rational points of the torus $$\mathbb {T}_{6}$$T6 is birationally equivalent to the affine part of a Singer arc in $$\mathbb {P}^2(\mathbb {F}_{q^2})$$P2(Fq2). This gives a simple, yet efficient compression and decompression algorithm from $$\mathbb {T}_{6}(\mathbb {F}_{q})$$T6(Fq) to $$\mathbb {A}^2(\mathbb {F}_{q})$$A2(Fq) 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-$$6$$6 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.