Differential Cryptanalysis of Two Joint Encryption and Error Correction Schemes

In GLOBECOM'10, Adamo et. al. proposed an interesting encryption scheme, called Error Correction-Based Cipher (ECBC), working at the physical layer. This scheme, together with its ancestor, Secret Error Correcting Code (SECC), belongs to the family of Joint Encryption and Error Correction (JEEC), which combines error correction and data encryption as one process to enable efficient implementations. In this paper, we provide rigorous investigation on the security of ECBC and SECC to unveil their cryptographic strengths under chosen-plaintext attacks. For ECBC, we found a 3-stage differential-style attack, which breaks the scheme with $O(k \times 2^{deg(f)} + 2^k)$ effort, where $deg(f)$ is the degree of the core cryptographic function $f$. For SECC, we found a similar attack of complexity $O(k \times 2^{k+1})$. Both of the attacks are significantly improved from exhaustive search, e.g., $O(2^{2k+kn+n\times2^k})$ for ECBC and $O(2^{kn+ (k+n) \times 2^k})$ for SECC. In addition, we exhibit that $f$ used in ECBC's implementation is particularly vulnerable to our attack, which allows the attacker to recover the secret generator matrix in $O(1)$. To mitigate this vulnerability, we propose a secure yet lightweight construction of $f$ achieving the maximum degree. Finally, the core part of our attack against ECBC has been implemented utilizing GPU acceleration and demonstrated on a cluster GPU instance provided by Amazon EC2. Experimental results confirm that the original implementation of ECBC scheme can be broken in (almost) constant time (${<}0.4$ second) regardless of $k$, whereas the ECBC scheme enhanced by our proposed $f$ can withstand this attack to the maximum extent.