1. Alternative Cryptographic Primitives
For quite a while, I've been thinking is it possible to "craft" a cryptographic primitive (like ECC) suitable for the purposes of post-quantum cryptography, and today (2017-05-20) I've came to the following conclusion:
There are two classes of primitives that may be used to construct cryptographic protocol:
Groups: whose elements are equipped with addition with each other, and multiplication only with scalars.
Rings: whose elements are equipped with addition and multiplication with them selves.
The first kind would almost certainly be vulnerable to cryptanalysis by Shor's Algorithm, because they'd most likely be cyclic.
And most primitives of the second kind would be representable with a matrix, since they're probably going to be linear. And this is exactly where we're progressing with lattice-base crypto.
2. CryptoPrimitive Size-Security Efficiency and My Question
Take ECC for example, an ECDSA signature of 192-bit security has 384*2 = 768-bit signature, therefore, it has an size-security ratio of about 4:1.
However, take the currently (2017-05-20) most efficient signature scheme BLISS, instantiated using the procedure described by Rebeca Staffas, we have a 128-bit security instance (512-A) with a signature size of about 6471 bits, that's a size-security ratio of about 50:1.
Yet, BLISS is the composition of several primitives - from sampling bimodal discrete Gaussian, to using Huffman or Arithmetic Coding.
This got me wonder, and this is my question: Is there a single primitive that offers the best post-quantum size-security efficiency, or we don't get to enjoy one such in a post-quantum world anymore?