how to prove that given cryptosystem is not IND-CCA secure?

Let $$m$$ and $$k$$ be positive integers ($$m$$ bit plain text) ; let $$Q$$ be a family of trapdoor one-way permutations such that $$f : \{0,1\}^k \rightarrow \{0,1\}^k$$ for all $$f$$ in $$Q$$; and let $$G : \{0,1\}^k \rightarrow \{0,1\}^m$$ be a random oracle. Let $$P = \{0,1\}^m$$ and $$C = \{0,1\}^k \times \{0,1\}^m$$, and define

$$K = \{(f, f^{(-1)},G) : f$$ in $$Q\}$$

For $$K = (f,f^{(-1)},G)$$, let $$r$$ in $$\{0,1\}^k$$ be chosen randomly, and define

encryption of $$x = (y_1,y_2) = (f(r),G(r) \oplus x)$$, where $$y_1$$ in $$\{0,1\}^k$$, $$y_2$$ in $$\{0,1\}^m$$

decryption of $$(y_1,y_2) = G(f^{(-1)}(y_1)) \oplus y_2$$

The functions $$f$$ and $$G$$ are public key; the function $$f^{(-1)}$$ is the private key

• Hint: How is IND-CCA related to NM-CCA? Can you selectively modify ciphertexts? – Squeamish Ossifrage Apr 16 at 14:39
• @SqueamishOssifrage I would like to show that cryptosystem is not semantically secure against a chosen ciphertext attack. As usually given x1,x2, a ciphertext (y1,y2) that is an encryption of xi (i=1 or i=2). Also we have access to a decryption oracle DECRYPT for this cryptosystem, which decrypts with any input except for given ciphertext, and will output the corresponding plaintext – Askhat Apr 16 at 15:16
• yes, can selectively modify ciphertexts – Askhat Apr 16 at 15:20
• Just checking the operations. Seems like encryption should be different.. $G(r) \oplus x$ isn't correct because the the output of $G$ and $x$ are in different spaces – Marc Ilunga Apr 17 at 19:36
• @MarcIlunga The plaintext $x$ is an $m$-bit string, and the range of $G$ is $m$-bit strings—what's the issue? – Squeamish Ossifrage Apr 18 at 17:18