General Report on Tunny

22B Page 46

character of the ith impulse of the U-stream at a given ciphering position.

L is used to denote the operative character of the limitation.

(b) Some further definitions.

The following symbols are generally used in Tunny-analysis and must be defined here:
 U  = letter preceding U Ui = character preceding Ui U  = letter preceding U U  = letter following U U  = letter following U                         and so on.
 ΔU   = U + U Δ2U = Δ(ΔU) ΔnU = Δ(Δn-1U) Δ2U = U + U Δ2U = U + U                                         and so on.
 Uij = Ui + Uj = Ui + a cross
 Ui x : P(Ui = x) > ½ Ui   . : P(Ui = .) > ½ Ui x : P(Ui = x) = p where p > ½ Ui   . : P(Ui = .) = p where p > ½

(c) Two general theorems.

 Theorem I : Δ(U + V) = ΔU + ΔV (A1) Theorem II: Δ2U = Δ2U (A2) Proof    : Δ2U = Δ(ΔU) = (U + U) + (U + U) = U + U = Δ2U Theorem II is a special case of the general theorem: ΔnU = ΔnU if and only if n = 2r.  (See R5 p. 114)

22B     THE CHI-STREAM.

The chi-stream differs from a random sequence of letters in its periodicity in each impulse taken separately and in the deliberately arranged equality of dots and crosses in each impulse.