Description
Consider the projective space P^n over a finite field F_q. A hypersurface is defined by one homogenous equation with coefficients in F_q. For d going to infinity, we show that the probability that a hypersurface of degree d is nonsingular approaches 1/\zeta_{P^n (n+1)}. This is analogous to the well-known fact that the probability that an integer is squarefree equals 1/\zeta(2) = 6/\pi^2. This is a special case of the results in Bjorn Poonen's paper ``Bertini Theorems over Finite Fields'', where he computes the probability that a given variety intersects a random hypersurface. Poonen uses the full power of algebraic geometry, whereas the special case can be proven using only elementary linear algebra and properties of finite fields.
Next sessions
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Oblivious Transfer from Zero-Knowledge Proofs (or how to achieve round-optimal quantum Oblivious Transfer without structure)
Speaker : Léo Colisson - Université Grenoble Alpes
We provide a generic construction to turn any classical Zero-Knowledge (ZK) protocol into a composable oblivious transfer (OT) protocol (the protocol itself involving quantum interactions), mostly lifting the round-complexity properties and security guarantees (plain-model/statistical security/unstructured functions…) of the ZK protocol to the resulting OT protocol. Such a construction is unlikely[…]-
Cryptography
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