# Signing same message 2 times with ECDSA

Can multiple signatures of the same message with the same private key (different nonces) lead to a private key trace?

• Hi Topolino, please edit the question and explain what you mean with "trace". Do you mean "expose", "track" or something different? Nov 17 '21 at 21:12
• And, what is the origin of this question, what have you tried? Nov 17 '21 at 21:47
• Hint, two-equation more than one unknowns. Nov 17 '21 at 21:59
• What do you mean by “private key trace”? Do you mean allowing to recover the private key? Or do you mean revealing that the same private key was used each time? Or something else? Nov 17 '21 at 23:28
• Yes I mean recover the private key, sorry Nov 18 '21 at 18:16

From the description of signing operation in ECDSA, we see that changing the nonce $$k$$ changes $$R$$, $$x_R$$, $$y_R$$, $$r$$, $$s$$ (not $$H$$, $$e$$); thus including both components of the signature $$S=(r,s)$$.
• @Topolino: $k$ is not the private key. The private key is $d_U$. $k$ is a secret random integer in $[1,n)$, and can be called an ephemeral private key. As far as I can tell the notation in the whole of sec1v2 is consistent, and for sure things do not change arbitrarily within the section on ECDSA. Wikipedia's ECDSA article uses $d_A$ and $Q_A$ where sec1 uses $d_U$ and $Q_U$, and assimilates integers to bitstrings, but is close.