Asymptotically Optimal Communication for Torus-Based Cryptography

  • Marten van Dijk
  • David Woodruff
Conference paper

DOI: 10.1007/978-3-540-28628-8_10

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3152)
Cite this paper as:
van Dijk M., Woodruff D. (2004) Asymptotically Optimal Communication for Torus-Based Cryptography. In: Franklin M. (eds) Advances in Cryptology – CRYPTO 2004. CRYPTO 2004. Lecture Notes in Computer Science, vol 3152. Springer, Berlin, Heidelberg


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)log2q 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Marten van Dijk
    • 1
    • 2
  • David Woodruff
    • 1
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Philips Research LaboratoriesEindhovenThe Netherlands

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