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I have a question in (textbook) RSA. Suppose we have two public keys $pk_1=(e_1,N)$ and $pk_2=(e_2,N)$ (with $\gcd(e_1,e_2) = 1$). Assume that there are $\sigma$,$\sigma'$ such that $\sigma$ and $\sigma'$ are a valid signature on $m_1$ and $m_2$ under $pk_1$ and $pk_2$ respectively ($m_1$ and $m_2$ are messages in $\mathbb{Z}_N$ ). I am trying to compute the signature $\sigma$ from $N,e_1,e_2,m_1,m_2$.

Any help would be appreciated.

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  • $\begingroup$ Are you implying $\sigma = Sig_{sk1}(m1)= Sig_{sk2}(m2)$? If so you only have to compute a classical signature either for $m1$ given $sk1$ or $m2$ given $sk2$. Note that signatures are computed with private keys, not public keys (in the case of RSA, use the exponent d and the prime factors $p,q$ of N) $\endgroup$ – Binou Jan 5 at 8:53
  • $\begingroup$ Sorry for that, i mean $\sigma = Sig_{sk1}(m1)$ and $\sigma'= Sig_{sk2}(m2)$ $\endgroup$ – user178592 Jan 5 at 9:03
  • $\begingroup$ You are signing with public keys, are you aware this is not a proper way to peeform signatures? $\endgroup$ – Binou Jan 5 at 9:12
  • $\begingroup$ I didn't mean that we sign with public keys. I mean that we sign with the private keys sk1,sk2 respectively, but these signatures are valid (under pk1,pk2). $\endgroup$ – user178592 Jan 5 at 9:22
  • $\begingroup$ Is this a homework assignment? $\endgroup$ – kelalaka Jan 5 at 10:28
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Let $d_1, d_2$ be inverses of $e_1,e_2$ respectively mod $(p-1)(q-1)$ where $N=pq$ with $p,q$ two large primes.

To sign the message $m_1$, you will compute $s_1={m_1}^{d_1}\ [N]$.

To verify the signature, provided $m_1, s_1$, one will compute ${s_1}^{e_1}\ [N]$ and check whether this equals to the message $m_1$ or not.

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  • $\begingroup$ OK I see. Thank you for this. And if now we assume that there is σ such that σ is a valid signature on m1 and m2 under pk1 and pk2 respectively (m1 and m2 are messages in $Z_N$ ). Note : This time the signature is the same and i want to compute σ via N,e1,e2,m1,m2. $\endgroup$ – user178592 Jan 5 at 9:35
  • $\begingroup$ You do exactly the same. A signature is actually a pair $s, m$, where you provide the message $m$ that has been signed in $s$. Note that in practice, this is not likely to happen. $\endgroup$ – Binou Jan 5 at 9:39

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