A Simple Computational Model for Acceptance/Rejection of Binary Sequence Generators

A simple binary model to compute the degree of balancedness in the output sequence of LFSR-combinational generators has been developed. The computational method is based exclusively on the handling of binary strings by means of logic operations. The proposed model can serve as a deterministic alternative to existing probabilistic methods for checking balancedness in binary sequence generators. The procedure here described can be devised as a first selective criterium for acceptance/rejection of this type of generators.

💡 Research Summary

The paper introduces a deterministic computational model for evaluating the balancedness of binary sequences produced by LFSR‑combinational generators. Balancedness, defined as an approximately equal number of zeros and ones over a full period, is a fundamental quality criterion for cryptographic stream ciphers and pseudo‑random number generators. Traditional verification relies on statistical test suites such as NIST SP800‑22 or Diehard, which require the generation of large sample streams, repeated testing, and probabilistic interpretation of the results. These methods are costly, non‑deterministic, and unsuitable for early‑stage design decisions where a quick accept/reject filter is needed.

The proposed model eliminates the need to actually generate the output sequence. Instead, it works directly on the binary representation of the LFSR’s state space and on the Boolean expression that defines the combinational output function. The procedure consists of four main steps:

1. State‑Space Enumeration – Given an n‑stage LFSR described by its feedback polynomial and an initial seed, all 2ⁿ possible register contents are enumerated as n‑bit binary strings. This set, denoted S₀, is the exhaustive representation of the LFSR’s trajectory over one full period.

2. Boolean Function Mapping – The combinational function f(x₁,…,x_k) is parsed into a logic‑gate network (AND, OR, XOR, NOT). For each gate, the corresponding bitwise operation is applied simultaneously to every element of the current string set. For example, if f = x₁ ⊕ (x₂ ∧ x₃), the algorithm extracts the bits at positions 1, 2, and 3 from each string in S₀, computes the AND of the latter two, then XORs the result with the first bit, producing a new set S_f of output bits.

3. Counting Ones – Because each operation preserves the length of the strings, the final set S_f contains exactly one bit per original state, representing the output of the generator for that state. Counting the number of ‘1’s across S_f yields the total number of ones that would appear in the full‑period output sequence.

4. Balancedness Decision – An acceptance threshold ε (typically 0.01, i.e., 1 %) is defined. The ratio ρ = (#ones) / (2ⁿ) is compared against the interval