🔐 UNNS in Cryptology: Symbolic Fingerprints and Modular Echoes

By Ihor Chomko

Abstract:
Unbounded Nested Number Sequences (UNNS) offer a reproducible structure with symbolic echoes and integer-preserving behavior. This post explores how UNNS can be applied to cryptographic systems, including hash generation, echo-based key exchange, and modular fingerprinting. Python examples included!

🧮 UNNS Formula

Each term is computed as:

S_N(M) = (M × N) + (M / N) + (M − N) + (M + N)

This yields predictable growth and integer-preserving behavior across nests.

🔐 Cryptographic Applications

1. Symbolic Fingerprinting

Each UNNS value can serve as a symbolic fingerprint. Here's how to compute it:

def symbolic_fingerprint(N, M):
    return (M * N) + (M / N) + (M - N) + (M + N)

# Example: fingerprint for Nest=5, Modulus=12
print(symbolic_fingerprint(5, 12))

2. Hash Generation

Use the fingerprint as a seed for cryptographic hashing:

import hashlib

def hash_fingerprint(N, M):
    val = symbolic_fingerprint(N, M)
    return hashlib.sha256(str(val).encode()).hexdigest()

# Example hash
print(hash_fingerprint(5, 12))  



🔐 UNNS Hash Generator
  Nest (N):
  

  Modulus (M):

3. Echo-Based Key Exchange

Two parties agree on a nest and modulus. They validate keys via cross-nest echoes:

def echo_overlap(N1, N2, M):
    val1 = symbolic_fingerprint(N1, M)
    val2 = symbolic_fingerprint(N2, M)
    return abs(val1 - val2) < 1e-6  # Allow floating-point tolerance

# Example: check if S₅(12) ≈ S₄(12)
print(echo_overlap(5, 4, 12))  # Output: False

4. Integer-Preserving Encryption

Encrypt only values that yield integers across nests:

def is_integer_preserving(N, M):
    val = symbolic_fingerprint(N, M)
    return val.is_integer()

# Example: check if S₁(25) is integer
print(is_integer_preserving(1, 25))  # Output: True

5. Obfuscated Padding

Use noninteger values as cryptographic padding:

def generate_padding(N, M_range):
    return [symbolic_fingerprint(N, M) for M in M_range if not symbolic_fingerprint(N, M).is_integer()]

# Example: padding values for Nest=3, M=1..10
print(generate_padding(3, range(1, 11)))

📊 Visualization Ideas

Plot integer-preserving positions across nests
Visualize echo overlaps as a heatmap
Fingerprint entropy vs. modulus growth

⚖️ Advantages

Predictable symbolic structure
Modular layering and echo validation
Integer-preserving behavior across nests

🔬 Research Directions

Integrate UNNS into lattice-based cryptography
Explore echo-based zero-knowledge proofs
Design symbolic hash functions with tunable entropy

📋 Conclusion

UNNS offers a mathematically grounded framework for cryptographic design. Its symbolic fingerprints, echo overlaps, and integer-preserving behavior open new doors for modular encryption and interpretability. The next step is empirical testing and protocol integration.

Pages

2025/08/17