# Legendre PRF bounties

## The Legendre PRF

The Legendre pseudo-random function is a one-bit PRF $\mathbb{F}_p \rightarrow \{0,1\}$ defined using the Legendre symbol:

## Bounties

### $20,000 For either • a sub-exponential, i.e. $2^{(\log p)^c}$ for some $% $, classical key recovery algorithm that extracts the key $K$ using inputs chosen by the attacker1 • a security proof which reduces the Legendre pseudo-random function distinguishing problem to a well-known computational hardness assumption (see below) ###$ 6,000

For a classical key recovery algorithm improving on the algorithm by Kaluđerović, Kleinjung and Kostić ($O (p \log(p) \log(\log(p))/M^2)$ Legendre evaluations where $M$ is the number of PRF queries needed) algorithm by more than a polylog2 factor, using a sub-exponential, i.e. $M=2^{(\log p)^c}$ for $% $ number of queries.1 3

### \$ 2,000

For the most interesting paper on the Legendre PRF in the next year (ends 01 May 2022)4

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 well-established 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
• Phi-hiding assumption
• Discrete logarithm, Diffie-Hellman and Decisional Diffie-Hellman in $\mathbb{F}_p^{\times}$
• 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 24–1082), 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 CLAIMED
100 bits 60 bits 8 ETH
148 bits 108 bits 16 ETH

For each of the challenges, $2^{20}$ 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

1. In all cases, probabilistic algorithms are also considered if they improve on the probabilistic versions of the known algorithms. Only classical (non-quantum) algorithms are permitted for the algorithm bounties.  2

2. This was originally set as 44–128 bits of security, but has been reduced to 24–108 due to the Beullens algorithm.  2

3. For this bounty, we also consider any algorithm that can distinguish a $2^{(\log p)^c}$ bit length output of the Legendre PRF from a random bit string with advantage $>0.1$

4. This will be judged by the cryptographers of the Ethereum Foundation Cryptography Research team and cannot be appealed