Would you mind sharing what protocols you are working on that can benefit from a "lightweight cryptography"?
My impression from learning cryptography in the last 8 years is that cryptographic protocols are either exponentially secure, or any tiny weaknesses can be amplified into a complete break.
The only example of secure "intrapolynomial cryptography" [1] is Proof-of-Work, but it also requires work to be expended continuously. Not exactly a scheme for "underpowered devices".
[1] https://nakamotoinstitute.org/intrapolynomial-cryptography/