Binomial Theorem Given $a, b \in \mathbb{R}, n \in \mathbb{N}$, then $$ (a + b)^n = \sum_{k = 0}^{n} \binom{n}{k}a^k b^{n-k} $$ Proposition Let $p$ be prime. $\forall a, b \in \mathbb{Z}$, we have: $$ (a+b)^p \equiv a^p + b^p \mod p $$ Proof It suffices to show that $\binom{p}{k} \equiv 0 \mod p$ if $0 < k < p$. Observe: $p \mid p!, p \nmid k!, p \nmid (p-k)!$, hence $p \mid \binom{p}{k}$. $\blacksquare$ ...
The goal of this file is to help understand the GSW scheme and the implementation of GSWCiphertextin OnionPIR code. Let’s start with my understanding of GSW scheme. From TGSW to RGSW RGSW is a ring variation of GSW scheme. I do not see any formal paper defining RGSW. However, I do find this particular paper helpful: Faster Fully Homomorphic Encryption: Bootstrapping in less than 0.1 Seconds. This paper defines TLWE and TGSW. ...
The following is a note for my talk during Ling’s group meeting. What is FHE? Homomorphic encryption allows some computation (addition, scalar multiplication, ct-ct multiplication) directly on ciphertexts without first having to decrypt it. Partially Homomorphic Encryption support only one of those possible operation. RSA is an example: $$ \text{Enc}(m_1) \cdot \text{Enc}(m_2) = m_1^e \cdot m_2^e = (m_1 \cdot m_2)^e = \text{Enc}(m_1 \cdot m_2) $$ FHE supports Addition AND Scalar Multiplicaiton: $$ \begin{cases} \text{Enc}(m_1) + \text{Enc}(m_2) = \text{Enc}(m_1 + m_2)\\ \text{Enc}(m) \cdot c = \text{Enc}(m \cdot c) \end{cases} $$ Fancy! And it exsists! ...
The goal is to compare three methods for supporting keyword feature in PIR: Key-value filter in ChalametPIR, Sparse PIR, and the Cuckoo hashing method. In the beginning, we don’t want to start by comparing the detailed experimental performances, but we want to list their properties. What they are good / bad at. Metrics Client storage Client computation Online communication Download size Offline communication (if any) Server storage Server computation Ability to support multiple clients Notations: $m$: the number of key-value pairs. ...