Simple question : I have a prime field having modulus $p$ where $p−1$ contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as its suborder/subgroup. Why are there no special cases where it’s possible to lift 2 $p$’s elements to modulus $q$ while keeping their discrete logarithm if those 2 elements lies only within the $O$’s subgroup ?
Without solving the discrete logarithm of course !
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