Is it necessary for a round function $F$ in a Feistel cipher to be pseudorandom?

I stumbled across this question where the questioner asked for specific requirements for the round function $$F$$ in a Feistel network so that the construction is secure. The answer explained that a pseudorandom function in a four round Feistel cipher is sufficient for it to achieve security.

Is that also necessarily the case? In other words, can a Feistel cipher be constructed with a round function that is not a PRF?

• The Feistel circuit converts a PRF into a PRP. If not a PRF, what are you planning to have as F? Jan 13 at 16:15
• @user93353 This is exactly the question, is there an F that is not a PRF but the Feistel network still gives a PRP? Jan 13 at 16:26
• Could you explain what is not a PRF for you? See Tiny Encryption Algorithm (TEA) that uses simple round operations and it is secure with sufficient rounds. Jan 13 at 19:49
• @kelalaka I used PRF as defined by the rigorous mathematical definition. But if I understand TEA correctly, I think this is already a counter example. Thank you! Jan 16 at 10:17