Practical cryptography in high dimensional tori

At Crypto 2004, van Dijk and Woodruff introduced a new way of using the algebraic tori $T_n$ in cryptography, and obtained an asymptotically optimal $n/\phi(n)$ savings in bandwidth and storage for a number of cryptographic applications. However, the computational requirements of compression and decompression in their scheme were impractical, and it was left open to reduce them to a practical level. We give a new method that compresses orders of magnitude faster than the original, while also speeding up the decompression and improving on the compression factor (by a constant term). Further, we give the first efficient implementation that uses T_{30}, compare its performance to XTR, CEILIDH, and ECC, and present new applications. Our methods achieve better compression than XTR and CEILIDH for the compression of as few as two group elements. This allows us to apply our results to ElGamal encryption with a small message domain to obtain ciphertexts that are 10\% smaller than in previous schemes.