Legendre PRF bounties
The Legendre PRF
The Legendre pseudorandom function is a onebit PRF defined using the Legendre symbol:
Bounties
$ 10,000
For either
 a subexponential, i.e. , classical key recovery algorithm that extracts the key using inputs chosen by the attacker^{1}
 a security proof showing the nonexistence of such an algorithm by reducing it to a wellestablished computational hardness assumption (see below)
$ 3,000 (CLAIMED)
For a classical key recovery algorithm improving on the Khovratovich ( where is the number of PRF queries needed) algorithm, using a subexponential, i.e. number of queries.^{1} ^{2}
CLAIMED: Congratulations to Ward Beullens for successfully claiming this bounty with an algorithm performing Legendre key recovery in for . A new bounty based on this performance will be announced soon, so stay tuned!
$ 1,000
For the most interesting paper on the Legendre PRF in the next year (ends 31 August 2020)^{3}
The first two bounties are for the first entry that beats the given bounds. Please send submissions to Dankrad Feist dankrad .at. ethereum .dot. org.
Computational hardness assumptions
For the reduction to a wellestablished computational hardness assumption, we consider the assumptions below which are taken from the Wikipedia page
 Integer factorization problem
 RSA problem
 Quadratic residuosity, higher residuosity and decisional composite residuosity problem
 Phihiding assumption
 Discrete logarithm, DiffieHellman and Decisional DiffieHellman in
 Lattice problems: Shortest vector and learning with errors
Concrete instances
At Devcon5, further bounties for concrete instances of the Legendre PRF were announced. For primes of size 64–148 (security levels 24–108^{4}), the following bounties are now available for recovering a Legendre key:
Prime size  Security  Prize  

64 bits  24 bits  1 ETH  CLAIMED 
74 bits  34 bits  2 ETH  CLAIMED 
84 bits  44 bits  4 ETH  
100 bits  60 bits  8 ETH  
148 bits  108 bits  16 ETH 
For each of the challenges, bits of output from the Legendre PRF are available here. To claim one of these bounties, you must find the correct key that generates the outputs.
Research papers
 Damgård, Ivan Bjerre: On The Randomness of Legendre and Jacobi Sequences (1988)
 Lorenzo Grassi, Christian Rechberger, Dragos Rotaru, Peter Scholl, Nigel P. Smart: MPCFriendly Symmetric Key Primitives (2016)
 Alexander Russell, Igor Shparlinski: Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation (2002)
 Dmitry Khovratovich: Key recovery attacks on the Legendre PRFs within the birthday bound

In all cases, probabilistic algorithms are also considered if they improve on the probabilistic versions of the known algorithms. Only classical (nonquantum) algorithms are permitted for the algorithm bounties. ↩ ↩^{2}

For this bounty, we also consider any algorithm that can distinguish a bit length output of the Legendre PRF from a random bit string with advantage ↩

A cryptographer will be appointed by the Ethereum Foundation to judge this (TBD) ↩

This was originally 44–128 bits of security, but has been reduced to 24–108 due to the Beullens algorithm. ↩