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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 ...
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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 ...
1
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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