Note

Finding a Close Point in a Lattice

Reference [Bab85] On Lovász’ lattice reduction and the nearest lattice point problem [Kle00] Finding the closest lattice vector when it’s unusually close [GPV08] Trapdoors for Hard Lattices and New Cryptographic Constructions [DP16] Fast Fourier Orthogonalization [GMRR22] Power analysis attacks on falcon.

Jul 10, 2023 6 min read Note

Principal Component Analysis (PCA)

Principal Component Analysis (PCA) can be viewed as a linear transform which maps a $n$-dimensional random vector to a smaller dimensional vector, but preserving most of its information.

Nov 15, 2022 3 min read Note

Singular Value Decomposition (SVD)

Singular Value Decomposition (SVD) is a way to decompose any real matrix $A \in \mathbb{R}^{m\times n}$ into a simple form. $$ A = U\Sigma V^T $$ where $U \in \mathbb{R}^{m \times m}$ and $V \in \mathbb{R}^{n \times n}$ and $\Sigma \in \mathbb{R}^{m \times n}$ is a matrix with only diagonal values.

Nov 15, 2022 3 min read Note

Barrett Reduction and Montgomery Reduction

Given a variable $a$ and a prime number $q$, we want to securely compute $a \text{ mod } q$.

The Barrett reduction is usually used in addition.

The Montgomery reduction is usually used in multiplication and reduction.

Oct 31, 2022 4 min read Note

Note on Number Theoretic Transform

NTT Github Repo: python-NTT FFT Recall for a sequence of signal $x=(x_0, x_1,\cdots,x_{N-1})$. The Fast-Fourier Transform does $$ \hat{x} =FFT(x)= \left ( \sum_{n=0}^{N-1} x_n e^{-2 \pi i\frac{ n m}{N}} \right)_{0\leq m\leq N-1} $$

Jun 18, 2022 3 min read Note

BIP32、BIP39、BIP44筆記

BIP(Bitcoin Improvement Protocals)是Bitcoin社群為了增進Bitcoin的使用效率，使其更於方便使用而建立，其中BIP32、BIP39、BIP44定義了HD Wallet的使用規範與細節。以下是個人閱讀白皮書的筆記，希望能幫助自己or他人理解這些協定與背後原理。

另外，有關BIP的相關文件可以至 bitcoin/bips查看。

Jul 4, 2021 5 min read Note