Asymptotically Optimal Communication for Torus-Based Cryptography

Abstract

We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discrete-log based public-key system achieving the optimal communication rate, partially answering the conjecture in [4]. For n the product of distinct primes, we construct efficient ElGamal signature and encryption schemes in a subgroup of \(F_{q^n}^*\) in which the number of bits exchanged is only a φ(n)/n fraction of that required in traditional schemes, while the security offered remains the same. We also present a Diffie-Hellman key exchange protocol averaging only φ(n)log2 q bits of communication per key. For the cryptographically important cases of n=30 and n=210, we transmit a 4/5 and a 24/35 fraction, respectively, of the number of bits required in XTR [14] and recent CEILIDH [24] cryptosystems.