How Low Can You Go? Short Structure-Preserving Signatures for Diffie-Hellman Vectors

We generalize the cryptographic notion of Order Revealing Encryption (ORE) to arbitrary functions and we present a construction that allows to determine the (partial) ordering of two vectors i.e., given E(x) and E(y) it is possible to learn whether x = y, x > y, x < y or whether x and y are incomparable. This is the first non-trivial example of a Revealing Encryption (RE) scheme with output larger than one bit, and which does not rely on cryptographic obfuscation or multilinear maps.