[數] 純量乘法；标量乘法
The ******st approach is to only permit multiplication and division by scalar numbers.
最簡單的辦法是我們隻允許對标量數字進行乘法和除法運算。
In order to improve the operation speed of the improved algorithm, we chose a better algorithm for scalar multiplication and a…
為了進一步提高改進算法的運算速度，文中選取了優異的快速标量乘法并對其效果進行了分析。
Random point scalar multiplication on binary field GF(2 m) is one of the costliest computations in elliptic curve cryptography.
隨機點點乘是橢圓曲線密碼體制中最耗時的運算。
The process of ECC includes the selection of base field and coordinates, scalar multiplication and field operation.
ECC算法中基域的選擇、坐标系的選擇、标量乘法和域算術運算的實現。
A field inversion is the most expensive operation on scalar multiplication, and the number of inversion determines the performance of scalar multiplication.
求逆是标量乘法中最耗時的運算，求逆運算次數的多少直接決定标量乘法的性能。
In these public key cryptographic algorithms, the kernel operations are modular exponentiation of multi-precision integer and elliptic curve scalar multiplication, which both are computing intensive.
這些公鑰密碼算法的關鍵操作為大整數模幂乘操作與橢圓曲線标量乘法操作，均屬于計算密集型運算。
By investigating the elliptic curve cryptosystems, the problems are reduced the fast computations of scalar multiplication of the elliptic curve.
通過對橢圓曲線密碼體制的研究，将快速實現橢圓曲線密碼的問題歸結為标量乘法的實現效率。
The fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication.
橢圓曲線密碼體制的實現速度依賴于曲線上标量乘法的運算速度。
We brought forward a new mixed coordinates system to denote the point over EC, thereby reduce the number of field multiplication in scalar multiplication.
提出了一種新型的混合坐标系統來表示橢圓曲線上的點，從而降低了标量乘中域乘法次數。
In order to accelerate operating speed, projective coordinates are used and point addition is improved to optimize the scalar multiplication algorithm.
為了提高運算速度，利用射影坐标思想，改進橢圓曲線上求兩點和運算公式，對标量乘算法進行優化。
Divisor scalar multiplication is the key operation in hyperelliptic curve cryptosystem.
除子标量乘是超橢圓曲線密碼體制中的關鍵運算。
Basic operation rules have been established, including addition, positive scalar multiplication, cancellation law for addition and so on.
基本的運算法則已經形成，包括加法運算、數乘運算、加法的消去律等。
The multi-scalar multiplication with two scalars is the most time consuming operations in elliptic curve cryptography(ECC).
在現代密碼系統中使用橢圓曲線密碼（ECC）最頻繁的一種方法是多點乘算法。
Secondly, the fast algorithms for scalar multiplication and multi-scalar multiplication on elliptic curves are designed based on integer splitting and precomputation technique.
其次，設計基于整數拆分與預計算相結合的橢圓曲線标量乘和多标量乘算法。
For example, compared with the Fixed-base Window method, it speed up 82.5% for the implementation speed of scalar multiplication of a 160-bit integer.
例如，對于160位的大整數标量乘，與固定基窗口方法相比，其實現速度提高了82.5%。
The definition of work suggests a third process of vector algebra, namely, scalar multiplication of two vectors.
功的定義用到矢量代數的第三種運算，即兩個矢量的标積。
The algorithm can greatly reduce the number of elliptic point addition, so the efficiency of scalar multiplication in elliptic curves is improved.
該算法比經典算法減少了點的加法的計算次數，從而加快了橢圓曲線上點的數乘的運算速度。
In this dissertation the elliptic curve cryptosystems and fast scalar multiplication algorithms are investigated.
本文主要研究了橢圓曲線公鑰密碼體制中标量乘法運算的快速算法。
The finite field arithmetic, elliptic curve scalar multiplication and the related algorithms are investigated in this dissertation.
本文主要研究有限域運算算法和橢圓曲線數乘運算算法。
The main operations of elliptic curve cryptosystem are scalar multiplication and multi-scalar multiplication for a pair of integers.
橢圓曲線密碼體制最主要的運算就是橢圓曲線上的标量乘和多标量乘，在各種密碼協議中起到了核心作用。
Firstly, based on the signed factorial expansions, . the fast algorithms for scalar multiplication and multi-scalar multiplication on elliptic curves are presented.
首先，設計基于帶符號階乘展開式的橢圓曲線标量乘和多标量乘算法。
Especially, the power attack is very severe for the security of scalar multiplication on Elliptic Curve.
本文主要研究了标量乘運算在能量攻擊下的安全性和效率，旨在給出快速安全的标量乘算法。
The center to the implementation of elliptic curve cryptosystems efficiently lies in the arithmetic of scalar multiplication and addition.
橢圓曲線密碼體制高速實現的關鍵是點的數乘與加法。
This algorithm greatly reduces times of addition operation which takes time for scalar multiplication algorithm by introducing signed and unsigned sliding window coding methods.
此算法通過引入有符號和無符號滑動窗口編碼方法，大大減少了标量乘算法中費時的加法運算次數。
Although there have been many SPA-resistant scalar multiplication algorithms, there are a few countermeasures for multiple scalar multiplication.
抗SPA攻擊的點乘算法較多，但對于多點乘算法相關措施較少。
The methods to speed up the scalar multiplication computation are mainly discussed in two ways: one is the number system, another is parallel algorithm.
主要從兩個方面來研究快速算法，一是研究數域系統以加快标量乘法運算，二是研究标量乘法運算的并行算法。
This paper presents a new fast scalar multiplication algorithm on elliptic curve cryptography.
基于橢圓曲線密碼，提出了一種快速标量乘算法。
Firstly, the field operations on OEF are analyzed, focusing on how to analyze and improve the multiplication which is most time-consuming in scalar multiplication.
首先分析最優擴域上域元素的運算，重點對标量乘中最費時的乘法進行分析和改進；
The scalar multiplication in elliptic curves is the basic to elliptic curve cryptosystem.
計算橢圓曲線上點的數乘是橢圓曲線密碼算法的基礎。
标量乘法（Scalar Multiplication） 是線性代數中的一種基本運算，指将一個标量（Scalar）（通常為實數或複數）與一個向量（Vector） 或矩陣（Matrix） 中的每個元素相乘的操作。其核心含義在于對向量或矩陣進行縮放（Scaling），即改變其大小（模長）而不改變其方向（對于非零向量而言）或結構。
對向量的标量乘法：
對矩陣的标量乘法：
标量乘法滿足以下基本性質（以向量為例，矩陣類似）：
标量乘法是線性代數的基礎操作之一，廣泛應用于：
Scalar multiplication（标量乘法）是線性代數中的基礎運算，指一個标量（實數或複數）與一個向量相乘的操作。其核心作用是通過标量改變向量的長度（模）和方向（若标量為負）。
假設标量為 ( c )，向量為 ( mathbf{v} = (v_1, v_2, dots, v_n) )，則标量乘法運算為： $$ c cdot mathbf{v} = (c cdot v_1, c cdot v_2, dots, c cdot v_n) $$ 即對向量的每個分量分别乘以标量。
标量乘法（( c cdot mathbf{v} )）與向量的點積（( mathbf{a} cdot mathbf{b} )）或叉積（( mathbf{a} times mathbf{b} )）不同，其結果仍為向量，而非标量或其他向量空間結構。
如果需要具體例子或進一步擴展，可以結合具體領域中的實際案例進行說明。
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