PIRSA:20030088

# Explicit quantum weak coin flipping protocols with arbitrarily small bias

### APA

Arora, A. (2020). Explicit quantum weak coin flipping protocols with arbitrarily small bias. Perimeter Institute. https://pirsa.org/20030088

### MLA

Arora, Atul. Explicit quantum weak coin flipping protocols with arbitrarily small bias. Perimeter Institute, Mar. 04, 2020, https://pirsa.org/20030088

### BibTex

```          @misc{ pirsa_20030088,
doi = {10.48660/20030088},
url = {https://pirsa.org/20030088},
author = {Arora, Atul},
keywords = {Quantum Information},
language = {en},
title = {Explicit quantum weak coin flipping protocols with arbitrarily small bias},
publisher = {Perimeter Institute},
year = {2020},
month = {mar},
note = {PIRSA:20030088 see, \url{https://pirsa.org}}
}
```

Atul Arora Université Libre de Bruxelles (ULB)

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## Abstract

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. A weak coin-flipping protocol has a bias ϵ if neither player can force the outcome towards their preferred value with probability more than 1/2+ϵ. While it is known that classically ϵ=1/2, Mochon showed in 2007 [arXiv:0711.4114] that quantumly weak coin flipping can be achieved with arbitrarily small bias, i.e. ϵ(k)=1/(4k+2) for arbitrarily large k, and he proposed an explicit protocol approaching bias 1/6. So far, the best known explicit protocol is the one by Arora, Roland and Weis, with ϵ(2)=1/10 (corresponding to k=2) [STOC'19, p. 205-216]. In the current work, we present the construction of protocols approaching arbitrarily close to zero bias, i.e. ϵ(k) for arbitrarily large k. We connect the algebraic properties of Mochon's assignments---at the heart of his proof of existence---with the geometric properties of the unitaries whose existence he proved. It is this connection that allows us to find these unitaries analytically.