I find this new research direction interesting and likewise both for the positive results and for the negative results. The positive results offer an abstraction of ideas underlying cryptographic schemes as well as their transportation between different algebras. The negative results shed light on various issues regarding the design of cryptographic schemes. In particular, ``arithmetization'' (i.e., the existence of a matching arithmetic schemes) emerges as a new barrier (i.e., as an alternative to black-box reductions) that may explain differences among cryptographic primitives.
We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a black-box use of the underlying field, and the adversary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions.
We prove several positive and negative results in this model for various cryptographic tasks. On the positive side, we show that, under coding-related intractability assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, homomorphic encryption, and secure computation with low online complexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard (Boolean) model and the arithmetic model, and explain, in retrospect, some of the limitations of previous constructions.