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Here the encryption is done as follows : $$C=P^e \textrm{mod} \, n =(P_1||P_2)^e \textrm{mod} \, n.$$ Here's my scenario that the adversary with CCA2 wins.

The adversary chooses $X_1, X_2$ of the same size with $P_1, P_2$ and multiplies $C'=(X_1||X_2)^e \textrm{mod} \, n$ by $C$ to get $CC'=(P_1X_1||P_2X_2)^e \textrm{mod} \, n$. The adversary asks the decryption oracle to decrypt $CC'$ to get $(P_1X_1||P_2X_2)$. Then the adversary can get $(P_1||P_2)$ by just multiplying the inverse of $(X_1||X_2)$. Finally, the adversary can decrypt the original message as Bob does.

This is the same as CCA on textbook-RSA. I see that RSA-OAEP is IND-CCA2 secure, but why is this attack impossible?

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The attack assumes $$(P_1\mathbin\|P_2)(X_1\mathbin\|X_2)\equiv P_1X_1\mathbin\|P_2X_2\pmod n$$ which has no reason to holds for most parameters.

Note: the bit size of $P_2X_2$ has no standard definition. In the following I assume the intention is to make it the sum of the size of the arguments $P_2$ and $X_2$ expressed as fixed-size bitstrings¹.

Counterexample with 4-bit parameters $P_1$, $P_2$, $X_1$, $X_2$ all set to 1111: $P_1\mathbin\|P_2$ and $X_1\mathbin\|X_2$ are 11111111, $P_1X_1$ and $P_2X_2$ are 11100001, left hand side $(P_1\mathbin\|P_2)(X_1\mathbin\|X_2)$ is 1111111000000001, right hand side $P_1X_1\mathbin\|P_2X_2$ is 1110000111100001. They have no reason to be equal modulo $n$.


¹ Another sensible defintion would be that $P_1X_1\mathbin\|P_2X_2$ has value $P_1X_1\,2^\ell+P_2X_2$ where $\ell$ is the common bit size of bitstrings $P_2$ and $X_2$, but that does not make the equality hold either.

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  • $\begingroup$ I didn't know that it doesn't work as simple as numbers. Thank you! $\endgroup$ Commented Jul 22, 2022 at 8:19
  • $\begingroup$ @이승우 : It does not work "with numbers" either. In decimal, $99\times99$ is not $8181$. $\endgroup$
    – fgrieu
    Commented Jul 22, 2022 at 8:22
  • $\begingroup$ It is because they are not a single numbers but padded numbers, am I right? Does padding remove malleability of textbook-RSA? $\endgroup$ Commented Jul 22, 2022 at 8:35
  • $\begingroup$ @이승우 : Padding the plaintext by concatenation breaks the multiplicative property of textbook RSA: that $\operatorname{Enc}(P_1)\operatorname{Enc}(P_2)\bmod n\ =\operatorname{Enc}(P_1\,P_2\bmod m)$. Large amount of random padding seems safe for encryption, but we have no proof that it is, when we have such proof for RSAES-OAEP. However deterministic padding is always unsafe in encryption, and deterministic padding with constants sometime is unsafe for signature. $\endgroup$
    – fgrieu
    Commented Jul 22, 2022 at 9:18
  • $\begingroup$ Thank you so much! $\endgroup$ Commented Jul 22, 2022 at 13:42

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