## We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. 我們建立了將任何減少的剩餘類別表示為素數和沒有小素數因子的整數的乘積的方法數量的估計。

Product of an integer free of small prime factors and prime in arithmetic progression

## Typically, the algorithms used have two parts – trial divisions aimed at eliminating numbers with small prime factors and primality tests based on an easy-to-compute statement that is valid for primes and invalid for composites. 通常，所使用的算法有兩個部分——旨在消除具有小素因數的數字的試驗部分和基於易於計算的對素數有效且對複合物無效的陳述的素數測試。

Fast Generation of RSA Keys using Smooth Integers

## In the third paper in a series of papers on autism savants, detection of giftedness and the use of mental arithmetic as an intervention in autism and a practice of metal wellness, we describe the use of python scripts towards primality detection exercises, of both small primes and arbitrary sized numbers and several other exercises including sequence prediction, inspired by branch prediction architectures. 在關於自閉症專家系列論文的第三篇論文中，天才的檢測和使用心算作為自閉症的干預和金屬健康的實踐，我們描述了使用 python 腳本進行素數檢測練習，兩個小素數和任意大小的數字和其他一些練習，包括受分支預測架構啟發的序列預測。

Autistic Savants and Mathematical sequence prediction.

## Real-world examples of periodical species range from cicadas, whose life cycles are large prime numbers, like 13 or 17, to bamboos, whose periods are large multiples of small primes, like 40 or even 120. 週期性物種的真實世界例子包括蟬，其生命週期是大素數，如 13 或 17，到竹子，其周期是小素數的大倍數，如 40 甚至 120。

Universal Evolutionary Model for Periodical Species

## We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. 我們考慮 {\it 噪聲多項式插值問題\/} 在殘差模 $p^k$ 的環 $\mathbb Z_{p^k}$ 上恢復未知 $s$-稀疏多項式 $f(X)$ ，其中 $p$ 是一個小素數，$k$ 是一個大整數參數，來自 $f(t) \in \mathbb Z_{p^k}$ 的殘差的近似值。

Noisy polynomial interpolation modulo prime powers

10.1088/1742-6596/1804/1/012027

## We say a non-zero submodule E of D is d-small prime if for each c ∈ R, d ∈ D, (d) «d D with cd ∈ E, then either d ∈ E or c ∈ [E: D] and an fi-module D is a d-small prime if annD = annE for each non-zero submodule E d-small in D. 我們說 D 的非零子模 E 是 d-small 素數，如果對於每個 c ∈ R, d ∈ D, (d) «d D with cd ∈ E, 那麼 d ∈ E 或 c ∈ [E: D]並且對於 D 中的每個非零子模 E d-small，如果 annD = annE，則 fi-模 D 是 d-small 素數。

On d-Small Prime Modules

10.1007/978-3-030-43229-4_15

## Small prime-sized discrete Fourier transforms appear in various applications from quantum mechanics, material sciences and machine learning. 小的素數離散傅里葉變換出現在量子力學、材料科學和機器學習的各種應用中。

Exploiting Symmetries of Small Prime-Sized DFTs

## The only work on construction of self-dual NMDS codes shows existence of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary self-dual NMDS codes of length <inline-formula> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> for odd prime power <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> or length up to 16 for some small primes <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$q\le 197$ </tex-math></inline-formula>. 構建自對偶 NMDS 代碼的唯一工作表明存在 <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary 自對偶 NMDS長度代碼 <inline-formula> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> 用於奇數次冪 <inline-formula> <tex-math notation= "LaTeX">$q$ </tex-math></inline-formula> 或一些小素數的長度可達 16 <inline-formula> <tex-math notation="LaTeX">$q$ </tex- math></inline-formula> 與 <inline-formula> <tex-math notation="LaTeX">$q\le 197$ </tex-math></inline-formula>。

Self-Dual Near MDS Codes from Elliptic Curves

## Let p be a small prime and n = n 1 n 2 > 1 be a composite integer. 設 p 為小素數， n = n 1 n 2 > 1 為合整數。

Faster Initial Splitting for Small Characteristic Composite Extension Degree Fields

10.1109/ACCESS.2019.2909113

## Hence, our attack does not require the incremental prime search assumption and is applicable when countermeasures against previous attacks are deployed since it also does not require the assumption of trial divisions with small primes on prime candidates. 因此，我們的攻擊不需要增量素數搜索假設，並且在部署針對先前攻擊的對策時適用，因為它也不需要假設在素數候選上具有小素數的試驗劃分。

A Practical Collision-Based Power Analysis on RSA Prime Generation and Its Countermeasure

10.1007/978-3-030-34621-8_13

## Such constructions are usually based on CLT13 multilinear maps, since CLT13 inherently provides a composite encoding space, with a plaintext ring $$\bigoplus _{i=1}^n \mathbb {Z}/g_i\mathbb {Z}$$ for small primes $$g_i$$’s. 這樣的構造通常基於 CLT13 多線性映射，因為 CLT13 固有地提供複合編碼空間，帶有明文環 $$\bigoplus _{i=1}^n \mathbb {Z}/g_i\mathbb {Z}$$ 用於小素數 $$g_i$$ 的。

Cryptanalysis of CLT13 Multilinear Maps with Independent Slots

10.1007/978-3-030-31578-8_20

## We first prove that the decision and search versions are equivalent provided q is a small prime. 我們首先證明如果 q 是一個小素數，決策和搜索版本是等價的。

Fractional LWE: a nonlinear variant of LWE