A class of Boolean functions with four-valued Walsh spectra

In this paper, we propose an infinite class of Boolean functions with four-valued Walsh spectra. These functions have a simple trace expression of the form f(x) = trⁿ ₁ (¿^{d(2 n +1)}) + tr²ⁿ ₁(bx) for b ¿ F_{2 2n} and d satisfying d(2^l +1) = 2ⁱ(mod 2ⁿ-1) with integers I and i, where x ¿ F_{2 2n}. Their cryptographic properties, including balancedness, spectrum distribution, nonlinearity, algebraic degree and algebraic immunity, are investigated. We prove that the proposed functions have high nonlinearity, and algebraic degrees n - gcd(n, I) + 2. Our computer simulation shows these functions have optimal or suboptimal algebraic immunity.