Abstract. We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discretelog 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 ∗ qn 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. 1

Let \Gamma be a group, and let C \Gamma be the group ring of \Gamma over C . We first give a simplified and self-contained proof of Zalesskii's theorem [23] that the canonical trace on C \Gamma takes rational values on idempotents. Next, we contribute to the conjecture of idempotents by proving the following result: for a group \Gamma , denote by P \Gamma the set of primes p such that \Gamma embeds in a finite extension of a pro-p-group; if \Gamma is torsion-free and P \Gamma is infinite, then the only idempotents in C \Gamma are 0 and 1. This implies Bass' theorem [2] asserting that the conjecture of idempotents holds for torsion-free subgroups of GLn (C ) .

Abstract—This paper considers the problem of reliable communication over discrete-time channels whose impulse responses have length and exactly non-zero coefficients, and whose support and coefficients remain fixed over blocks of channel uses but change independently from block to block. Here, it is assumed that the channel’s support and coefficient realizations are both unknown, although their statistics are known. Assuming Gaussian non-zero-coefficients and noise, and focusing on the high-SNR regime, it is first shown that the ergodic noncoherent channel capacity has pre-log factor 1 for any. It is then shown that, to communicate with arbitrarily small error probability at rates in accordance with the capacity pre-log factor, it suffices to use pilot-aided orthogonal frequency-division multiplexing (OFDM) with pilots per fading block, in conjunction with an appropriate noncoherent decoder. Since the achievability result is proven using a noncoherent decoder whose complexity grows exponentially in the number of fading blocks, a simpler decoder, based on +1pilots, is also proposed. Its-achievable +1 rate is shown to have pre-log factor equal to 1 with the previously considered channel, while its achievable rate is shown to +1 have pre-log factor 1 when the support of the block-fading channel remains fixed over time. Index Terms—Bayes model averaging, compressed sensing, fading channels, noncoherent capacity, noncoherent communication, sparse channels. I.