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Shallan recently found a proof system (see below) that enables proving that two Pedersen commitments commit to the same message (but with potentially different randomness). She employes this in her private cryptocurrency to show that two committed coins have the same value. However, soon after deployment, she receives a message from a self-proclaimed hacker. The message contains two Pedersen commitments and their openings, and a proof of message equality for these commitments. The proof is valid, but there's a twist: the openings contain different messages! How can this be? Reproduce the attack and help Shallan diagnose the problem in her system.
It is a well-known fact that 1+1=2. Recent work by Alice von Trapp (et al) suggests that under special conditions in the Swiss Alps, 1+1+1+1=1. Alice has been unable to prove this statement over the BLS12-381 scalar field. The primary difficulty appears to be the fact that 1 is not equal to 4. Alice's proving system can write a proof for every statement (x,y) where x and y are BLS12-381 scalars, and x+x+x+x=y. The proving system easily outputs a proof for the statement (1,4) showing 1+1+1+1=4, but seems unable to produce a proof for the statement (1,1) showing 1+1+1+1=1.
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