Read First – Construct a conformal map

 

Using Conformal Mapping in Block Encryption

Conformal mappings preserve angles and local structures, making them conceptually relevant to cryptography. Here’s how the conformal mapping described earlier can be adapted for block encryption.

1. Encoding Data as Complex Numbers

Block encryption works on binary data. Encode data as complex numbers by representing each block (e.g., 128 bits) as a pair of real numbers to form a complex number
z = x + iy.

2. Applying the Conformal Map

Use the conformal mapping:

f(z) = [(exp(πi·log(z) / 3) – i) / (exp(πi·log(z) / 3) + i)].

This nonlinear, invertible transformation ensures diffusion, as small changes in z produce significant changes in f(z).

3. Combining with a Key

Introduce a secret key to enhance security by modifying parameters of the conformal map. For example:

  • Use a key-dependent angle θk instead of π/3.
  • Modify the logarithmic base or scaling factor with the key:
    fk(z) = [(exp(θk·i·logk(z) / 3) – i) / (exp(θk·i·logk(z) / 3) + i)].

4. Normalization and Data Output

The output of fk(z) is a complex number in the unit disk. Rescale and encode it back to binary format for subsequent encryption rounds or final ciphertext.

5. Inverse Mapping for Decryption

Decryption involves applying the inverse map:

z = fk-1(w),

where w is the ciphertext.

Advantages of Using Conformal Mappings in Cryptography

  • Nonlinearity: Introduces strong nonlinearity, enhancing resistance to cryptanalysis.
  • Invertibility: Conformal mappings are inherently invertible, meeting encryption requirements.
  • Diffusion: Spreads small input changes across the entire block.

Challenges

  • Numerical Precision: High precision is needed for operations like logarithms and exponentials.
  • Efficiency: The transformations may introduce computational overhead.
  • Key Sensitivity: Ensuring key-modified transformations maintain cryptographic properties.

By integrating conformal maps like f(z) into cryptographic algorithms, novel encryption schemes can be developed, especially in scenarios requiring strong geometric transformations.