Observability of Permutations, and Stream Ciphers

Robert E. Byerly, Lance D. Drager, Jeffrey M. Lee

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We study the observability of a permutation on a finite set by a complex-valued function. The analysis is done in terms of the spectral theory of the unitary operator on functions defined by the permutation. Any function f can be written uniquely as a sum of eigenfunctions of this operator; we call these eigenfunctions the eigencomponents of f. It is shown that a function observes the permutation if and only if its eigencomponents separate points and if and only if the function has no nontrivial symmetry that preserves the dynamics. Some more technical conditions are discussed. An application to the security of stream ciphers is discussed.

Original languageEnglish
Pages (from-to)3326-3330+3350
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2003


  • Control theory
  • Dynamical systems
  • Observability
  • Permutations
  • Stream cipher


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