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I know it is well understood that it is not a good practice to use the same nonce while generating the signatures, but I am not getting the math right.

Assume I have some UTXOs that are controlled by my private key Q. Say I have spent two of the UTXOs using nonce 'N' to generate my signature. Now the (R,S) components of the signature are public and the transactions are public so everyone has access to them.

S1 = N^(-1)*[hash(m1) + Q*R] mod p

S2 = N^(-1)*[hash(m2) + Q*R] mod p

S1 - S2 = N^(-1)*[hash(m1) - hash(m2)] mod p

Even though we know S1, S2, m1 and m2, isn't solving for N^(-1), and hence N, becomes equivalent to finding the solution to the discrete logarithm?

7

Let me rewrite your question in a different notation, where all lowercase values are integers and uppercase values are points.

  • The group generator is G (a known constant).
  • The private key is q, its corresponding public key is Q = qG.
  • The nonce is n, its corresponding point is R = nG.
  • The X coordinate of R is r.
  • The hash function is h(x).
  • A signature is (r,s), where s is computed as n-1(h(m) + qr).
  • A signature is valid iff r = x(s-1(h(m)G + rQ)).

Now for the two signatures it holds that:

  • s1 = n-1(h(m1) + qr)
  • s2 = n-1(h(m2) + qr)
  • s1 - s2 = n-1(h(m1) - h(m2))
  • n = (s1 - s2)-1(h(m1) - h(m2))

As s1 and s2 are just integers, (s1 - s2)-1 can be trivially computed using a modular inverse; there are no elliptic curve points involved here (over which this problem would be hard).

Once you know n, you can find q by rewriting the first equation:

  • ns1 = h(m1) + qr
  • ns1 - h(m1) = qr
  • q = r-1(ns1 - h(m1))
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isn't solving for N^(-1), and hence N, becomes equivalent to finding the solution to the discrete logarithm?

No, it is not. This does not require finding the discrete logarithm at all. Solving the discrete logarithm is finding the exponent to a known base. However in this problem we are trying to find the base and know what the exponent is. Furthermore, this known exponent is -1 for which finding the base of something raise to -1 is to raise the result to -1 again, i.e. taking the inverse of the inverse.

There are algorithms that exist to find the modular inverse of a number which is how N^(-1) is found in the first place. To find N, you just need to take the inverse of N^(-1) because of the identity that an inverse an inverse is the element itself.

  • Isn't the order of how the equation is written same as the public key generation? You have [hash(m1) - hash(m2)] as the base and N^(-1) as the exponent? – Ugam Kamat Mar 26 at 18:40
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    No. [hash(m1) - hash(m2)] is an integer, not a elliptic curve point. N^(-1) is also an integer. So this formula is just integer multiplication. It is not exponentiation nor is it curve point multiplication (the two things that have discrete log problems). Thus it is not solving any discrete logarithm. – Andrew Chow Mar 26 at 19:03
  • That's what I was looking for. So since N^(-1)*[hash(m1) - hash(m2)] is just integer multiplication modulo p vs curve point addition. – Ugam Kamat Mar 26 at 19:20

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