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'.

Key seed x₀: 0.32145

r (logistic): 3.980

Cat-map rounds: 5

Decrypt seed (try same / different): 0.32145

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.

growth rate r: 3.9990

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?

Bit length: 10,000

Seed: 0.31415

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.

Test yourself on cryptography

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