Cryptographic counters and applications to electronic voting

We formalize the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly. The value of the counter can only be determined by a trusted authority (or group of authorities, which may include participants themselves), and participants cannot determine any information about the increment/decrement operations performed by other parties.Previous efficient implementations of such counters have relied on fully-homomorphic encryption schemes; this is a relatively strong requirement which not all encryption schemes satisfy. We provide an alternate approach, starting with any encryption scheme homomorphic over the additive group ℤ2 (i.e., 1-bit XOR). As our main result, we show a general and efficient reduction from any such encryption scheme to a general cryptographic counter. Our main reduction does not use additional assumptions, is efficient, and gives a novel implementation of a general counter. The result can also be viewed as an efficient construction of a general n-bit cryptographic counter from any 1-bit counter which has the additional property that counters can be added securely.
As an example of the applicability of our construction, we present a cryptographic counter based on the quadratic residuosity assumption and use it to construct an efficient voting scheme which satisfies universal verifiability, privacy, and robustness.
Work done while the author was at Telcordia Technologies.