Under the assumption that "secure" encryption exists, secure evaluation of functions represented by Boolean circuits is possible by Yao's garbling scheme [Y1]. My question is whether secure evaluation of functions represented by Boolean formulae is an easier problem (i.e., possible in the information theoretic setting).
(This is true, for example for secret sharing: Benaloh and Lichter [BL] gave an perfect scheme for access structures described by monotone Boolean formulae, but for monotone Boolean circuits one needs to assume secure encryption [Y2].)
[BL]: Benaloh and Lichter. Generalized secret sharing and monotone functions. Crypto’88
[Y1]: Yao. How to generate and exchange secrets. FOCS’86.
[Y2]: Yao. Protocols for secure computations. FOCS'82