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In RSA signature scheme, it is a common practice to 'sign' the hash value of the message so that the length can be fit.

Question: If we have $s=m^d \bmod n$ where $d$ satisfies $ed \equiv 1 \pmod{n}$ and we also know $s$ and $m$, is this still considered secure?

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  • $\begingroup$ Moved comment to an answer. $\endgroup$ Commented Oct 30, 2014 at 14:00
  • $\begingroup$ Is the above supposed to be a signature scheme? If so, how would someone verify it? What would need to be in the public key? Hint: $n$ and $e$ are not sufficient. $\endgroup$
    – poncho
    Commented Oct 30, 2014 at 14:02
  • $\begingroup$ Yes, I'm not exactly sure what he asks, but whatever version of this without any hashing, the malleability problem stays. $\endgroup$ Commented Oct 30, 2014 at 14:03

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The main problem here is that signature is malleable. Given signatures on $m_1$ and $m_2$ allows you to construct a valid signature on $m_1m_2$ by multiplying the signatures. Therefore, the answer is no. When you hash you don't have $H(m_1m_2)=H(m_1)H(m_2)$, so hashing destroys this algebraic structure that connects messages and signatures.

EDIT: I'm assuming the question is whether you can sign without hashing when the messages to be signed are already intergers modulo $n$, i.e. you don't need to hash your messages to make them elements of this group.

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