To prove the correctness of decryption for the LWE-based encryption scheme, we need to show that the decryption will retrieve the original message $m$ when the scheme's requirements are met. Specifically, given that the distribution $D_e$ satisfies $\sum_{i=1}^m |e_i| < q/4$, we want to demonstrate that decryption will succeed with high probability.

Outline of the LWE-based Encryption and Decryption Scheme

1. Encryption: The LWE encryption scheme encrypts a message $m \in \{0, 1\}$ as follows:

   • For a secret key $s$, choose $a$ uniformly from $\mathbb{Z}_q^n$ and noise $e$ from $D_e$.
   • Compute the ciphertext $(a, b)$ where $b = \langle a, s \rangle + e + \frac{q}{2} \cdot m$.

2. Decryption: The decryption process involves computing $b - \langle a, s \rangle$:

   • If $m = 0$, then $b - \langle a, s \rangle = e$.
   • If $m = 1$, then $b - \langle a, s \rangle = e + \frac{q}{2}$.
   • Since $e$ is sampled from $D_e$, the noise $e$ is small enough so that it satisfies $|e| < q/4$ with high probability.

Proof of Correctness for Decryption

1. Assumption: We assume that the sum of absolute values of the error terms $\sum_{i=1}^m |e_i| < q/4$ with high probability when $e_i$ is chosen from $D_e$.

2. Correctness for $m = 0$:

   • If $m = 0$, then $b = \langle a, s \rangle + e$.
   • When we compute $b - \langle a, s \rangle$ in decryption, we obtain $e$.
   • Since $|e| < q/4$, $e$ will not push the result over $q/4$, ensuring that decryption returns $m = 0$ correctly.

3. Correctness for $m = 1$:

   • If $m = 1$, then $b = \langle a, s \rangle + e + \frac{q}{2}$.
   • When we compute $b - \langle a, s \rangle$ in decryption, we obtain $e + \frac{q}{2}$.
   • Since $|e| < q/4$, the term $e + \frac{q}{2}$ will lie within the range $\left(\frac{q}{4}, \frac{3q}{4}\right)$, which is interpreted as $m = 1$ in decryption.

Conclusion

Since $|e| < q/4$ with high probability, the decryption function will interpret $b - \langle a, s \rangle$ correctly as either close to $0$ (for $m = 0$) or close to $\frac{q}{2}$ (for $m = 1$), thereby retrieving the original message $m$ correctly. This proves the correctness of the decryption in the LWE-based encryption scheme under the given assumption.

Would you like a further breakdown of any part of this proof?

Here are five additional questions that build on this explanation:

1. How does the choice of the modulus $q$ affect the security of the LWE-based encryption scheme?
2. Why is it necessary for the error $e$ to be chosen from a distribution with bounded values?
3. What would happen if $\sum_{i=1}^m |e_i|$ were not less than $q/4$?
4. How does the hardness of the LWE problem contribute to the security of this encryption scheme?
5. What are typical choices for $D_e$ in practice, and why?

Tip: When working with cryptographic proofs, always ensure to verify assumptions about noise bounds and their implications for security and correctness.