Biblio

We study the problem of generation of cycles, chains and graphs of pairing-friendly elliptic curves using in succinct non-interactive arguments for knowledge protocols in blockchain. The task to build a “stick” for existing MNT753 cycle is reduced to the factorization problem for big numbers. Together with graphs of pairing friendly elliptic curves we consider auxiliary graphs of their orders (primes or irreducible polynomials) associated to vertices and embedding degrees to edges. Numerical experiments allow us to conjecture that (except of MNT case): 1) for any fixed embedding degrees there exist only finite number of such cycles and, hence, there are no families of such cycles; 2) chains of prime order are very rare; we suppose that there are no polynomial families of such chains. It is hard to find a family of pairing friendly elliptic curves with the base field order q(x) such that ζk ∈ Q[x]/(q(x)) for k \textbackslashtextgreater 6. From other hand our examples show that we can apply Brezing-Weng construction with k=6 and D=3 iteratively to obtain chains of length 3-4. We build 1) a family of 1-chains with embedding degrees 8 and 7, where all orders are given by cyclotomic polynomials; 2) a combination of MNT cycle and near-MNT curve.

We present Ouroboros Crypsinous, the first formally analyzed privacy-preserving proof-of-stake blockchain protocol. To model its security we give a thorough treatment of private ledgers in the (G)UC setting that might be of independent interest. To prove our protocol secure against adaptive attacks, we introduce a new coin evolution technique relying on SNARKs and key-private forward secure encryption. The latter primitive-and the associated construction-can be of independent interest. We stress that existing approaches to private blockchain, such as the proof-of-work-based Zerocash are analyzed only against static corruptions.