Let's suppose I encrypt something with CTR mode and keep the key and IV secret.
Does keeping the IV secret together with the key increase strength of encryption?
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Sign up to join this communityDoes keeping the IV secret together with the key increase strength of encryption?
Very little. In particular, under a known plaintext brute-force attack, it only strengthens the key by at most about 1 bit.
Argument: given two consecutive ciphertext blocks $C_0$, $C_1$ and their matching plaintext $P_0$, $P_1$, we can rule out key $K$ if $E_K(D_K(C_0\oplus P_0)+1)\ne C_1\oplus P_1$ (where the addition is modulo $2^b$ with $b$ the block size). Block decryption $D$ is about as costly as block encryption $E$, these two operations dominate the cost, which is thus about doubled compared to a brute-force attack with known IV.
A good reason not to keep the IV secret is that it needs to be known to the receiver, thus we'd need to encrypt it, making things more complicated and harder to analyze.
The only secret is the key. CTR mode has Ind-CPA security, nothing you can gain more (Ind-CCAx,...) without integrity and authentication.
If you are setting your security on the secrecy of the IV, you are on the wrong path. Just make sure that the $(key,IV)$-pair is not repeated and read more on this post.
And, with your approach, you need an additional process to transmit the IV per message;
The former is problematic, the latter can increase the communication cost ( a new DHKE per encryption for new IV or Key altogether).
Do not complicate the processes, just stick to the standards that exist for many years designed and analyzed by serious cryptographers.