Several well-known public key encryption schemes, including those of Alekhnovich (FOCS 2003), Regev (STOC 2005), and Gentry, Peikert and Vaikuntanathan (STOC 2008), rely on the conjectured intractability of inverting noisy linear encodings. These schemes are limited in that they either require the underlying field to grow with the security parameter, or alternatively they can work over the binary field but have a low noise entropy that gives rise to sub-exponential attacks.

Motivated by the goal of efficient public key cryptography, we study the possibility of obtaining improved security over the binary field by using different noise distributions. Inspired by an abstract encryption scheme of Micciancio (PKC 2010), we consider an abstract encryption scheme that unifies all the three schemes mentioned above and allows for arbitrary choices of the underlying field and noise distributions.

Our main result establishes an unexpected connection between the power of such encryption schemes and additive combinatorics. Concretely, we show that under the ``approximate duality conjecture" from additive combinatorics (Ben-Sasson and Zewi, STOC 2011), every instance of the abstract encryption scheme over the binary field can be attacked in time 2 O ( n ) , where n is the maximum of the ciphertext size and the public key size (and where the latter excludes public randomness used for specifying the code). On the flip side, counter examples to the above conjecture (if false) may lead to candidate public key encryption schemes with improved security guarantees. We also show, using a simple argument that relies on agnostic learning of parities (Kalai, Mansour and Verbin, STOC 2008), that any such encryption scheme can be {\em unconditionally} attacked in time 2 O ( n log n ) , where n is the ciphertext size. Combining this attack with the security proof of Regev's cryptosystem, we immediately obtain an algorithm that solves the {\em learning parity with noise (LPN)} problem in time 2 O ( n log log n ) using only n 1+ samples, reproducing the result of Lyubashevsky (Random 2005) in a conceptually different way. Finally, we study the possibility of instantiating the abstract encryption scheme over constant-size rings to yield encryption schemes with no decryption error. We show that over the binary field decryption errors are inherent. On the positive side, building on the construction of matching vector families (Grolmusz, Combinatorica 2000; Efremenko, STOC 2009; Dvir, Gopalan and Yekhanin, FOCS 2010), we suggest plausible candidates for secure instances of the framework over constant-size rings that can offer perfectly correct decryption.