Testing Kaks Conjecture on Binary Reciprocal of Primes and Cryptographic Applications

This note considers reciprocal of primes in binary representation and shows that the conjecture that 0s exceed 1s in most cases continues to hold for primes less one million. The conjecture has also been tested for ternary representation with similar results. Some applications of this result to cryptography are discussed.

💡 Research Summary

The paper revisits Kak’s conjecture, which posits that in the binary expansion of the reciprocal of a prime (1/p) the digit ‘0’ appears more often than ‘1’. While the original observation was based on a sample of only 5,471 primes, the authors extend the empirical test to all primes below one million (78,498 primes in total). Using the standard d‑sequence generation formula a(i) = 2^i mod p mod 2, they compute the full period of each binary reciprocal, count the number of zeros and ones, and record whether zeros exceed ones, ones exceed zeros, or the counts are equal. The same procedure is applied to ternary (base‑3) reciprocals, counting zeros against ones and twos.

The results are presented in tables and figures. Table 1 shows that among the one‑million‑range primes, 19,888 cases (≈25.36 %) have more zeros, 3,059 cases (≈3.9 %) have more ones, and 55,544 cases (≈70.74 %) are balanced. Excluding the balanced cases, 46.71 % of the sequences are non‑maximum‑length and display a bias, while 29.26 % of all sequences (including maximum‑length) have unequal counts. Figures 1 and 2 graphically illustrate the magnitude of the bias: the positive side (zeros > ones) dominates, while the negative side (ones > zeros) is both rarer and of smaller magnitude. Table 2 breaks the data down by 10 000‑prime intervals, confirming that in each interval zeros dominate by roughly an order of magnitude.

For ternary expansions, Figures 3 and 4 plot the ratios 0/1 and 0/2 respectively. The patterns mirror the binary case: zeros are overwhelmingly more frequent than either of the other two digits, although occasional reversals occur.

In the discussion, the authors argue that this persistent asymmetry can be turned into a cryptographic primitive. Since a randomly chosen prime has roughly a 4 % chance of yielding a reciprocal where ones dominate, one can deliberately select several primes and combine their properties to engineer events with a prescribed low probability (e.g., for challenge‑response protocols, probabilistic token generation, or randomized key‑selection mechanisms). The advantage is that the prime itself already provides the usual hardness assumptions, while the bias adds a controllable probabilistic layer without requiring additional randomness sources.

The paper acknowledges several limitations. The empirical verification stops at one million; it is unknown whether the bias persists for much larger primes (e.g., >10⁹). Moreover, the bias is absent in maximum‑length d‑sequences (where 2 is a primitive root), which are already known to be perfectly balanced; thus, any cryptographic scheme must avoid such primes if the bias is required. Finally, no theoretical proof of the conjecture is offered, leaving the result as a statistical observation that may or may not hold universally.

The conclusion restates that Kak’s conjecture holds for the examined range, encourages further testing on larger prime sets, and suggests that even a partial range where the conjecture is true could be exploited in protocol design. The references list foundational works on d‑sequences, their randomness properties, and earlier cryptographic applications, providing a solid background for readers interested in extending the study.