Suppose that a block is ciphered with IDEA using CTR mode operation. To get the original message, why can't we use the cyphered rule?

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    $\begingroup$ It is not at all clear what your question is about. Can you please elaborate on the title and specify what it is that you want to know? $\endgroup$ – Ella Rose May 11 '18 at 1:53
  • $\begingroup$ Are you asking why $\operatorname{enc}_k(n) \neq \operatorname{dec}_k(n)$ for a key $k$ and counter $n$? $\endgroup$ – cypherfox May 11 '18 at 2:30
  • $\begingroup$ I read the question title to mean "Why does CTR mode not use [a block cipher's] decryption [algorithm] for message decryption?" (Even though it does use the block cipher's (block) encryption algorithm for message encryption.) $\endgroup$ – Future Security May 11 '18 at 2:46

CTR uses only a block cipher's forward algorithm, not its inverse. Both sender and recipient compute $N_? = E_K(nonce || counter)$ for each block. Then the plaintext is XORed with $N$ to get ciphertext. The recipient XORs ciphertext with $N$ to get plaintext.

$C_i = E_K(nonce || i) \oplus P_i$

$P_i = E_K(nonce || i) \oplus C_i$

Note: XOR is it's own inverse. $A \oplus B \oplus B = A$


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