You can't under a standard assumption known as the "Decisional NTRU Assumption". This is essentially the statement that NTRU public keys are pseudorandom. The following is definition 4.4.4 of A Decade of Lattice Cryptography. NTRU Learning Problem: For an invertible $s\in R_q^*$, and distribution $\chi$ on $R$, define $N_{s, \chi}$ to be the ...


If we know the cyclic structure of $P_1$ and $𝑃_2$, can we learn the cyclic structure of $𝑃_3$? No. Consider the case when $P_1$ is all fixed points bar a 2-cycle and $P_2$ has the same structure. $P_3$ could be the identity; it could consist of two disjoint 2-cycles and the rest fixed points; it could be a 3-cycle and the rest fixed points. We can say ...


If you are asking for examples of interesting connections than my favorite is using Genearative adverserial networks (GANs) in order to invent novel cryptographic primitives. https://arxiv.org/abs/1610.06918

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