Chaos Lab

Discrete map · 1D · dim 1

PWLCM (piecewise-linear chaotic map)

A workhorse of chaos-based cryptography: simple, fast, uniform invariant density, large positive Lyapunov exponent, and parameter p easy to use as a key.

PWLCM bifurcation across p ∈ (0, 0.5). Uniform invariant density everywhere.

Equations

x_{n+1} = x_n / p              if 0 < x_n ≤ p
           = (x_n − p) / (0.5 − p)   if p < x_n ≤ 0.5
           = symmetric for x_n > 0.5

At a glance

Parametersp ∈ (0, 0.5)
Chaotic foralways
Lyapunov exponentλ = − [p log p + (0.5 − p) log(0.5 − p)] · 2
HistoryWidely used in chaos cryptography since the 1990s.

Try it

Open the interactive playground at /tools/cobweb.

See also

Quick quiz

Test yourself on pwlcm

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.PWLCM stands for:

  2. Q2.PWLCMs are popular in chaos crypto because of:

  3. Q3.The PWLCM parameter p lies in:

  4. Q4.What invariant density does PWLCM have?

  5. Q5.PWLCM is conjugate / similar to which other map?

  6. Q6.A standard pitfall when implementing PWLCM in cryptography is:

  7. Q7.Why is PWLCM faster than logistic for software encryption?

  8. Q8.PWLCM-based ciphers are most commonly used for:

0 of 8 answered