Abstract We give a comparison of the performance of the recently proposed torus - based public key cryptosystem CEILIDH, and XTR Un - derpinning both systems is the mathematics of the two dimensional algebraic torus T6(Fp ) However, while they both attain the same discrete logarithm security and each achieve a compression factor of three for all data transmissions, the arithmetic performed in each is fundamentally di erent In its inception, the designers of CEILIDH were reluctant to claim it o ers any particular advantages over XTR other than its exact compression and decompression technique From both an algorithmic and arithmetic perspective, we develop an e cient version of CEILIDH and show that while it seems bound to be inherently slower than XTR, the di erence in performance is much smaller than what one might infer from the original description Also, thanks to CEILIDH's simple group law, it provides a greater exibility for applications, and may thus be considered a worthwhile alternative to XTR