Recovery for secret key in CTIDH-512 through Fault Injection Attack

Isogeny-based cryptography is secure in a quantum computing environment and offers relatively small key sizes compared to other post-quantum cryptographic schemes. However, the CSIDH isogeny-based cryptography scheme is vulnerable to fault injection attacks, particularly the Disorientation Attack. This paper analyzes the Disorientation Attack on CSIDH and proposes an optimized method to recover the secret key for CTIDH, an extension of CSIDH. In CTIDH, the secret key is divided into multiple batches. The sign of each batch determines whether its isogeny operations proceed in a positive or negative direction. We explore the feasibility of adapting this attack to CTIDH to recover the secret keys. However, because of the unique isogeny operation in CTIDH, direct key recovery presents significant challenges. To address this, we propose an optimized Disorientation Attack for CTIDH, utilizing the differences in the number of points across batches. We also present the details and results of the implementation. Additionally, using the recovered secret keys from this optimized attack, we can enhance the existing Disorientation Attack to recover additional key values.