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 addresses the problem of verifying the balancedness of binary sequences generated by LFSR‑combinational generators, a property essential for applications ranging from spread‑spectrum communications to cryptography. Traditional approaches rely on statistical tests (e.g., the monobit or frequency test) applied to randomly selected subsequences, which only provide probabilistic evidence and become impractical when the period of the generator is astronomically large (e.g., 2^38 bits).

To overcome these limitations, the authors propose a deterministic computational model that calculates the exact number of ‘1’s in the output sequence without generating the whole period. The core insight is that the number of ‘1’s equals the number of minterms present in the Boolean function that drives the generator when expressed in minterm (canonical) form. Consequently, the problem reduces to converting the Boolean function from its Algebraic Normal Form (ANF) to a sum of minterms and then counting the contributions of each minterm.

The conversion is formalized through a dual mapping Φ_F, which replaces each ANF monomial m_i m_j … m_k by the corresponding minterm M_α whose binary configuration α has 1’s at the positions of the true variables and 0’s where the variables appear complemented. Φ_F is an involution (Φ_F∘Φ_F = F), allowing a three‑step conversion: (1) apply Φ_F to obtain a minterm sum, (2) expand each minterm into its full ANF representation and cancel common terms, (3) apply Φ_F again to return to a minterm sum.

Each minterm M_α is represented by an L‑bit string (L being the total number of LFSR stages). The number of terms in the ANF expansion of M_α is 2^{L‑d(α)}, where d(α) is the Hamming weight of α. The authors introduce the concept of “maximum common development” (MCD) to capture the overlap between two minterms: the bitwise OR of their binary strings yields a new minterm that represents all common terms. When two minterms are XORed, the common terms cancel, which is mathematically expressed as subtracting twice the size of the MCD.

To manage the bookkeeping of added and cancelled minterms, an auxiliary function H = Σ_i s_i·M_{α_i} is defined, where s_i records the signed multiplicity of each minterm. The iterative update rule

 H_i = H_{i‑1} + M_{α_i} − 2·MCD(M_{α_i}, H_{i‑1})

ensures that each new minterm is added while any overlap with previously accumulated minterms is correctly removed. After processing all N minterms of Φ_F, the final H_N contains a list of distinct minterms β_j with associated signs s_j. The exact number of ‘1’s in the output sequence is then

 U_F = Σ_j s_j·2^{L‑d(β_j)}.

If U_F ≈ T/2 (where T = 2^L−1 is the period of a maximum‑length LFSR), the generator is considered balanced; otherwise it is rejected.

The methodology works for a single LFSR as well as for multiple LFSRs combined together. In the multi‑LFSR case, the total variable count L is the sum of the individual register lengths, and global minterms are formed by the logical AND of the constituent minterms from each register. The same MCD and H‑based algorithm applies unchanged.

The paper provides illustrative examples, such as a 3‑stage LFSR with primitive polynomial x³+x²+1, and demonstrates how the model predicts the exact number of ‘1’s for various Boolean functions, matching exhaustive simulation results. It also discusses design rules derived from the model, enabling engineers to construct generators that are guaranteed to be (quasi‑)balanced without resorting to empirical testing.

Key contributions include: (1) a rigorous mathematical link between minterm count and sequence balancedness, (2) an efficient algorithm based solely on bitwise operations (OR, AND, XOR) that avoids full sequence generation, and (3) practical design guidelines for both single‑ and multi‑LFSR combinational generators.

Limitations are acknowledged: the approach assumes maximum‑length LFSRs (primitive connection polynomials) and may become memory‑intensive for Boolean functions with a very large number of minterms, necessitating pre‑optimization. Nonetheless, the deterministic nature of the method offers a valuable first‑stage filter for generator design, complementing traditional statistical tests and providing certainty where probabilistic methods cannot.