At CRYPTO 2015, Sun et al. established the links among impossible differential, zero-correlation linear, and integral cryptanalysis over $mathbb{F}_2^{n}$ from the perspective of distinguishers. In this paper, following the definition of linear correlations over $mathbb{F}_p$ by Baignéres, Stern and Vaudenay at SAC 2007, we successfully establish comprehensive links over $mathbb{F}_p$, by reproducing the proofs and offering alternatives when necessary. Interesting and important differences between $mathbb{F}_p$ and $mathbb{F}_2^n$ are observed.
– Zero-correlation linear hulls can not lead to integral distinguishers for some cases over $mathbb{F}_p$, while this is always possible over $mathbb{F}_2^n$ proven by Sun et al..
– When the newly established links are applied to GMiMC, its impossible differential, zero-correlation linear hull and integral distinguishers can be increased by up to 3 rounds for most of the cases, and even to an arbitrary number of rounds for some special and limited cases, which only appeared in $mathbb{F}_p$. It should be noted that all these distinguishers do not invalidate GMiMC’s security claims.
The development of the theories over $mathbb{F}_p$ behind these links, and properties identified (be it similar or different) will bring clearer and easier understanding of security of primitives in this emerging $mathbb{F}_p$ field, which we believe will provide useful guides for future cryptanalysis and design.