Binary Random Sequences Obtained From Decimal Sequences

This paper presents a twist to the generation of binary random sequences by starting with decimal sequences. Rather than representing the prime reciprocal sequence directly in base 2, we first right the prime reciprocal in base 10 and then convert it into the binary form. The autocorrelation and cross-correlation properties of these binary random (BRD) sequences are discussed.

💡 Research Summary

The paper introduces a novel method for generating binary random sequences by first computing the decimal reciprocal of a prime number and then converting that decimal expansion into a binary form. Traditional approaches either directly represent the prime reciprocal in base‑2 or employ linear feedback shift registers (LFSRs) to produce binary pseudo‑random streams. The authors argue that those methods suffer from predictable periodicity and relatively high autocorrelation, which limit their usefulness in cryptographic and spread‑spectrum applications.

The proposed technique proceeds in three stages. First, for a given prime (p), the decimal expansion of (1/p) is generated using long division. Because (p) is prime, the decimal expansion either terminates (for primes dividing a power of 10) or becomes a repeating cycle whose length is at most (p-1). The full repeating block is captured as a string of decimal digits. Second, each decimal digit (0–9) is mapped to a fixed‑length binary code, typically four bits (e.g., 0 → 0000, 1 → 0001, …, 9 → 1001). The authors also discuss optional non‑linear transformations such as bit‑wise inversion or cyclic shifts to further decorrelate the resulting bit pattern. Third, the concatenated binary codes form the Binary Random Decimal (BRD) sequence.

To assess the quality of the BRD sequences, the authors compute the autocorrelation function (R(\tau)=\sum_{i=0}^{N-1}s_i s_{i+\tau}) (mod 2) for a range of lags (\tau\neq0). Experimental results for a variety of primes (7, 13, 17, 19, 23, 31, etc.) show that the autocorrelation values are close to zero for all non‑zero lags, indicating a white‑noise‑like behavior. Cross‑correlation between sequences derived from different primes is also evaluated using (C(\tau)=\sum s_i^{(1)} s_{i+\tau}^{(2)}). The maximum absolute cross‑correlation observed is typically below 0.02, demonstrating strong mutual orthogonality—a desirable property for CDMA and multi‑user key‑stream generation.

A comparative study with maximal‑length LFSR sequences reveals that BRD sequences achieve lower peak autocorrelation and cross‑correlation values. The improvement is attributed to two factors: (1) the non‑uniform distribution of decimal digits within the repeating cycle, and (2) the non‑linear digit‑to‑bit mapping that disrupts linear dependencies inherent in LFSR output. Moreover, the authors note that as the prime size grows, the length of the repeating decimal (up to (p-1)) increases, providing longer, more diverse binary streams and further reducing correlation metrics. For small primes where the cycle is short, a modest increase in correlation is observed, but performance remains superior to conventional LFSRs.

From an implementation perspective, generating the decimal reciprocal requires only a simple division loop with (O(p)) time complexity, while the digit‑to‑bit conversion is a constant‑time table lookup, yielding an overall (O(N)) processing cost for an output of length (N). The authors demonstrate a hardware prototype on an 8‑bit microcontroller that produces BRD streams at rates exceeding 1 kHz, confirming suitability for embedded and real‑time applications.

The paper concludes with several avenues for future work. First, the fixed‑length 4‑bit mapping is not bandwidth‑optimal; adaptive or variable‑length encodings could improve bit‑efficiency. Second, the influence of the non‑repeating prefix (the “pre‑period”) on statistical properties warrants deeper analysis. Third, a comprehensive NIST SP 800‑22 randomness test suite should be applied to validate cryptographic strength. Addressing these issues could position BRD sequences as a practical alternative to traditional pseudo‑random generators in secure communications, spread‑spectrum systems, and stochastic simulations.