Symmetric measures of pseudorandomness for binary sequences

We compare ordinary and symmetric variants of two classical measures of pseudorandomness for binary sequences, the $2$-adic complexity and the linear complexity. In the periodic setting, we show that for binary periodic sequences constructed from the binary expansions of non-palindromic primes, the symmetric $2$-adic complexity can be strictly smaller than the ordinary $2$-adic complexity. We also give a direct proof (of the known result) that the linear complexity of a periodic binary sequence is invariant under reversal, and hence coincides with its symmetric version. In the aperiodic setting, we provide explicit families of finite binary sequences for which both the $N$th symmetric 2-adic complexity and the $N$th symmetric linear complexity are substantially smaller than their ordinary counterparts. Furthermore, we show that the expected values of the $N$th rational complexity and of the $N$th exponential linear complexity exceed those of their symmetric analogues by at least a term of order of magnitude $N$. Thus, the effect of symmetrization is clearly visible on an exponential scale. We also establish lower bounds for the expected values of the symmetric rational complexity, symmetric $2$-adic complexity, symmetric linear complexity, and symmetric exponential linear complexity.

💡 Research Summary

The paper investigates the effect of symmetrisation on two classical pseudorandomness measures for binary sequences: the 2‑adic complexity and the linear complexity. For a binary sequence S, the symmetric version of a complexity measure is defined as the minimum of the value for S and for its reversal S_rev. The authors treat both periodic and aperiodic (finite) sequences and compare ordinary and symmetric complexities in each setting.

In the periodic case the 2‑adic complexity λ(S) is given by λ(S)=log₂ |2^T−1 / gcd(2^T−1, S_T(2))|, where T is the period and S_T(x) is the generating polynomial. By constructing sequences from the binary expansions of non‑palindromic primes p, and letting q be the binary reversal of p, the authors choose a period T that is a multiple of ord_q(2) but not of ord_p(2). Consequently 2^T−1 is divisible by q but not by p, which yields λ(S)=log₂(2^T−1) while λ_sym(S)=log₂((2^T−1)/q). Hence the symmetric 2‑adic complexity can be strictly smaller, by an amount log₂ q, than the ordinary one. A table of reversible prime pairs (p,q) illustrates that many such examples exist.

For linear complexity the paper provides a direct proof that for any periodic binary sequence L(S)=L(S_rev). The proof works from the definition of a linear recurrence and shows that a recurrence for S can be transformed into one for S_rev of the same order, establishing invariance under reversal. Therefore the symmetric linear complexity coincides with the ordinary linear complexity in the periodic case, making further study of the symmetric version unnecessary for periodic sequences.

The aperiodic (finite) case is where the most striking differences appear. The authors give two explicit families of length‑N sequences. In the first family a block of k≈N/2 leading zeros is followed by an arbitrary suffix; the ordinary rational complexity Λ(S_N) equals 2^k, while the reversed sequence has Λ(S_rev_N)≈2^{N−k}, which is much smaller. Hence λ_sym(S_N)≈N/2, a substantial reduction compared with λ(S_N)≈k. The second family uses a block of leading ones, leading to a similar reduction. These constructions demonstrate that symmetric 2‑adic and rational complexities can be dramatically lower than their ordinary counterparts.

The paper then analyses expected values. Known results give E_rat(N)=2^{N/2}+O(N/ log N) and E_2‑adic(N)=N/2+O(log N). By counting sequences whose ordinary complexity is below a threshold W=2^{N/2−r(N)} (with any r(N)→∞) and using the fact that each such sequence appears at most twice in the min{Λ,Λ_rev} calculation, the authors prove lower bounds E_rat^sym(N)≥2^{N/2}−r(N) and E_2‑adic^sym(N)≥N/2−log N. Choosing r(N)=log N yields E_2‑adic^sym(N)=N/2+O(log N) and E_rat^sym(N)=2^{N/2}+O(N/ log N). Thus the expected symmetric complexities are still of the same order as the ordinary ones, but they are at least an additive term of order N smaller. The paper also provides lower bounds for the expected symmetric rational, 2‑adic, linear, and exponential linear complexities, all matching the leading term of the ordinary expectations up to lower‑order corrections.

In summary, the work shows that symmetrisation does not affect linear complexity for periodic sequences but can significantly reduce 2‑adic complexity. For finite sequences, both symmetric 2‑adic and rational complexities can be substantially smaller, and the expected values reflect a clear, quantifiable gap of order N. These findings enrich the theoretical understanding of pseudorandomness measures and suggest that symmetric versions may be more appropriate in contexts where reversal invariance is desirable, such as certain cryptographic constructions.