Soonhak Kwon

Namhun Koo, Soonhak Kwon, Minwoo Ko et al.

Recently, several studies have shown that when $q\equiv3\pmod{4}$, for certain choices of $r$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\Fq$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \Fq$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \Fqmul$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$.

Namhun Koo, Soonhak Kwon, Minwoo Ko et al.

Recent studies on binomials of the form $F_r(x) = x^r(1 + χ(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in characteristic $3$, where instances of extremely low boomerang uniformity-namely $0$ or $1$-seem to arise more frequently than in other characteristics. First, we provide a systematic analysis of Almost Perfect Nonlinear (APN) power functions in characteristic $3$. We present an explicit parametrization of APN exponents arising from the construction of Zha and Wang and demonstrate through numerical results for $n \le 13$ that this generalized framework accounts for several previously known and sporadic APN instances. Building on this classification, we identify and rigorously prove two classes of binomials $F_r$ that are locally-PN and possess the minimum possible boomerang uniformity of $0$. These classes involve exponents derived from the aforementioned APN construction and the differentially 4-uniform exponent $r = 2 \cdot 3^{\frac{n-1}{2}} + 1$. Furthermore, we analyze the binomial $F_r$ with $r = 3^n - 3$, proving that it is locally-APN with boomerang uniformity $1$ when $n\ge 5$ is odd, and completely determine its boomerang spectrum through the evaluation of character sums. Our results clarify and extend existing studies on the cryptographic properties of binomials, providing a systematic characterization of several classes of binomials with very low boomerang uniformity in characteristic $3$.

Namhun Koo, Gook Hwa Cho, Byeonghwan et al.

Let F_q be a finite field with q elements with prime power q and let r>1 be an integer with $q\equiv 1 \pmod{r}$. In this paper, we present a refinement of the Cipolla-Lehmer type algorithm given by H. C. Williams, and subsequently improved by K. S. Williams and K. Hardy. For a given r-th power residue c in F_q where r is an odd prime, the algorithm of H. C. Williams determines a solution of X^r=c in $O(r^3\log q)$ multiplications in F_q, and the algorithm of K. S. Williams and K. Hardy finds a solution in $O(r^4+r^2\log q)$ multiplications in F_q. Our refinement finds a solution in $O(r^3+r^2\log q)$ multiplications in F_q. Therefore our new method is better than the previously proposed algorithms independent of the size of r, and the implementation result via SAGE shows a substantial speed-up compared with the existing algorithms.