# 64 bit key size Diffie Hellman

I am building a cryptographic El Gamal implementation on the Cardano Blockchain for a poker game. Each hand the players generate a DH 64 bit keys and shuffle the cards together via homomorphic encryption and some non interactive zero knowledge proof. Now due to the limits of the size of a transaction the safe primes for the modulus is limited to 64 bits.

Now my question is, how secure is this encryption and how fast can one brute force this? I could not find any literature that quantified this, only that it is certainly possible in some reasonable time. A normal poker round where 5 cards are draw normally take not longer than 5 minutes so that the lower bound for the time. Is it even possible to create an bounding argument?

I found this answer on stack (1) but that did not enlighten me as I am not a cryptographer. I tried to read the linked paper by Lenstra-Verheul but it completly went over my head.

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Even if you used ECDH here, which provides much more security and performance per bit in the key size, the security strength would be about $$2^{32}$$, since most elliptic curves provide security equal to about half the number of bits in size. $$2^{32}$$ is computable on a laptop in a few seconds, so this would be trivially forgeable to virtually the entire Internet immediately. You can imagine how the finite-field version would fare in comparison.