Applications · Information security
Chaos in cryptography
Chaotic dynamical systems share several useful properties with cryptographic primitives: extreme sensitivity to initial conditions (analogous to the avalanche property), broadband-like output spectra, ergodic invariant densities, and parameterised families with many free 'keys'.
Toy chaotic image cipher
step 1 (permutation): apply Arnold cat map (k rounds) to scramble pixel positions
step 2 (diffusion): XOR each pixel with a logistic-map keystream
x_{n+1} = r · x_n · (1 − x_n), byte = floor(x · 256)
key = (x_0, r, k). Reverse: regenerate keystream, XOR, then inverse cat map
(period 48 on a 64×64 grid). Drift the decrypt seed by 0.001: the picture
falls apart, classic SDIC consequence in chaos-based cryptography.Tightly bound to the master Hash Lab discussion at hash.suparnpatra.com.
r > 3.57 is chaotic; r = 4 is the canonical full chaos. Try r = 3.7 to see weaker mixing.
Overall score: 4 / 8 tests passed at α = 0.01
| Test | p-value | Verdict (α = 0.01) | What it checks |
|---|---|---|---|
| Frequency (monobit) | 0.057433 | PASS | Are 0s and 1s balanced overall? |
| Block frequency (M = 128) | 0.020173 | PASS | Is the 1-density similar in every 128-bit window? |
| Runs | 0.000000 | FAIL | Are runs of identical bits the right length? |
| Longest run (block = 8) | 0.000000 | FAIL | Max run of 1s within each 8-bit block. |
| Cumulative sums | 0.112266 | PASS | Max excursion of the ±1 walk should match a Brownian-bridge tail. |
| Serial (m = 3) | 0.000000 | FAIL | Are all 3-bit patterns equally likely? |
| Approximate entropy (m = 3) | 0.000000 | FAIL | Bandt-Pompe-style block entropy. |
| DFT spectral (first 2048 bits) | 0.239553 | PASS | Periodic structure in the bit stream? |
Toy NIST SP800-22 scoreboard
Four of the simplest tests: monobit frequency, runs, longest-run (block = 8), and approximate-entropy (block = 3). p-value < 0.01 rejects the random null. The deliberately biased source should fail multiple tests; a well-tuned chaotic PRNG should pass.
Caveat: passing these toy tests is necessary but not sufficient for cryptographic use.
Stream ciphers
A chaotic map iterated under a secret seed produces a pseudo-random sequence used as a keystream. Examples: logistic-map ciphers, PWLCM-based ciphers, multi-stage coupled-map ciphers. Strength depends critically on finite-precision quantisation, which can destroy the underlying ergodicity.
Block ciphers
Baptista's 1998 cipher (Phys. Lett. A) used the logistic map as a substitution; modern cipher designs use Hénon, Arnold cat, or coupled maps for image and video encryption with confusion-diffusion structure.
Chaos-based hash functions
Hash compression functions built from chaotic maps are a research topic: PWLCM-based hashes, Chebyshev-polynomial hashes. Suparn's thesis explores cryptographic hashing using chaos; see the dedicated chaos-hash section at hash.suparnpatra.com (Hash Lab).
S-boxes
Chaotic maps generate S-boxes for AES-like designs. Strength is measured by nonlinearity, differential uniformity, and bit-independence; chaos-generated S-boxes often score comparably to AES's once linearity is checked.
Pitfalls
Alvarez and Li (2006) list standard mistakes: parameters revealed by analytic forms, quantisation eliminating positive Lyapunov exponents, key space that looks large but has equivalents, and a lack of provable-security tradition. Statistical NIST STS tests are necessary but not sufficient.
See also
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