The update from Week 8

Security analysis based on FHE-DKSAP

In this part, we will conduct a security analysis of FHE-DKSAP. To begin, we will define the security requirements, and subsequently, we will demonstrate how FHE-DKSAP meets these security criteria.

1. Security Requirements:
   In FHE-DKSAP, confidential data encompasses the secret keys of both Alice and Bob. Furthermore, the generated stealth address must remain unlinkable to the recipient's original address. This scheme relies on the correct execution of encryption and decryption functions based on FHE, leveraging the inherent strengths of FHE to guard against quantum computing attacks. The details of the definitions can be found as follows.
   

2. Security Proof

   In this section, we will furnish formal proof for the security definitions associated with these requirements.

• Data confidentiality

  The security of a public key system lies in the fundamental principle that, given a public key, one cannot feasibly deduce the corresponding secret key. In the case of secp256k1, its security is anchored in the elliptic curve discrete logarithm problem (ECDLP). This problem is deemed computationally intractable for judiciously selected curves coupled with sufficiently large key sizes. With its 256-bit key size, secp256k1 is fortified against known threats, ensuring that secret keys remain confidential.

  Conversely, in the FHE-DKSAP scheme, public keys serve as the means to encrypt secret keys within the encryption function. Operating under the mechanics of FHE encryption, the retrieval of the original plaintext (in this instance, the secret keys) from its corresponding ciphertext is rendered unfeasible.

• Unlinkability
  Stealth addresses are typically one-time use addresses. Every time someone sends funds to a recipient, they compute and send it to a new stealth address. This ensures that there is no common address on blockchain to link multiple transactions to the same recipient. Since a new stealth address is created for every transaction, and it's not linked directly to the recipient's main public key on the blockchain, it becomes very difficult for an outsider to determine which transactions belong to a specific individual.

• Correctness
  Utilizing the additive homomorphic property, the decryption of

  $$C1+C2$$ will yield the result
  $$s{k}_1+s{k}_2$$. This is achieved without revealing or sharing the individual secret values. As a result, the stealth address can be computed accurately without compromising security.

• Quantum computing attack resistance
  Lattice-based cryptography stands out as a leading quantum-resistant contender. The robustness of these cryptographic algorithms hinges on the computational difficulty of lattice problems, notably the Shortest Vector Problem (SVP) and the Learning with Errors (LWE) problem. As previously touched upon in our introduction to FHE, its foundation lies in the LWE problem. Crucially, this problem presents formidable challenges to both classical and quantum computing platforms.