We consider the symmetric encryption problem which manifests when two parties must securely transmit a message m with a short shared secret key. As we permit arbitrarily powerful adversaries, any encryption scheme must leak information about m - the mutual information between m and its ciphertext cannot be zero. Despite this, we present a family of encryption schemes which guarantee that for any message space in {0,1|ⁿ with minimum entropy n - l and for any Boolean function h: {0,1|ⁿ → {0,1|, no adversary can predict h(m) from the ciphertext of m with more than 1/n ^ω(1) advantage; this is achieved with keys of length l+ω)(logn). In general, keys of length l+s yield a bound of 2^−θ(s) on the advantage. These encryption schemes rely on no unproven assumptions and can be implemented efficiently.