A New Family of Binary Sequences via Elliptic Function Fields over Finite Fields of Odd Characteristics
This work extends a known construction from even to odd characteristic, providing new binary sequences with guaranteed properties for applications in cryptography and communications.
The authors construct a new family of binary sequences using cyclic elliptic function fields over finite fields of odd characteristic, achieving bounded balance, correlation, and linear complexity. The sequences have length q+1+t, size q^{d-1}-1, balance ≤ (d+1)⌊2√q⌋+|t|+d, correlation ≤ (2d+1)⌊2√q⌋+|t|+2d, and linear complexity ≥ (q+1+2t-d-(d+1)⌊2√q⌋)/(d+d⌊2√q⌋).
Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to the cyclic elliptic function fields with odd characteristic by using the quadratic residue map $η$ instead of the trace map used therein. For any cyclic elliptic function field with $q+1+t$ rational points and any positive integer $d$ with $\gcd(d, q+1+t)=1$, we construct a new family of binary sequences of length $q+1+t$, size $q^{d-1}-1$, balance upper bounded by $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d,$ the correlation upper bounded by $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$ and the linear complexity lower bounded by $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$ where $\lfloor x\rfloor$ stands for the integer part of $x\in\mathbb{R}$.