A new year's puzzle:

(We use a toy version of bitcoin in which the group order is 19. So notice that e.g. 1/7 is 7^-1 mod 19 = 11. If you want an easy way to do that, do pow(7, -1, 19) in python.)

Michael makes two Bitcoin transactions using taproot, so the signatures are Schnorr signatures. s_1 = 16 and s_2 = 18.

His transaction hashes (the H(P, R, m) values) were e_1 = 12 and e_2 = 3.

But Michael did something unusual. He used a secret nonce both times (so k_1 and k_2 are secret), but he made it known that each new nonce he uses, he will increment the value by 7.

What is Michael's private key?

#cryptography #bitcoin