Exponentiation by squaring 第一次接触快速幂是解决一道n阶复数群的加密问题时,有个mul函数一直无法理解,学长说这就是曲线上的快速幂算法,作用呢就是加快幂运算的速度,在直接计算xa的情况下我们进行a-1次乘法操作,而利用快速幂算法只需log2a,复杂度由原来的O(n)降到了O(log(n)),可以说是质的飞跃了 算法流程 计算
And finally here’s the most optimized exponentiation by squaring algorithm I have seen around. It’s an iterative version where at each step you divide the exponent by two and square the base, and then for the iterations where the exponent is odd you multiply the result by the base. In ...
Binary exponentiation, also known as exponentiation by squaring and square-and-multiply algorithm, is used to calculate the values of large exponents, say 4103. It is a trick that uses base-2 numbers to compute the value of expressions involving large exponents. In exponentiation by squaring, we...
Comprehensive univariate polynomial class. All arithmetic performed symbolically. Some advanced features include: Arithmetic of polynomial rings over a finite field, the Tonelli-Shanks algorithm, GCD, exponentiation by squaring, irreducibility checking,
To compute log-table, one could modify the Pohlig–Hellman algorithm for the case when modulo is a power of $2$. Our main task here is to compute $x$ such that $g^x \equiv y \pmod{2^d}$, where $g=5$ and $y$ is a number of kind $2^n+1$. Squaring both parts $k$ ti...
The invariant squaring operation placed on Line 33 is performed first. After the squaring the result c is multiplied by the base g if and only if the most significant bit of the exponent is set. The shift on Line 47 moves all of the bits of the exponent upwards toward the most ...
The modular exponentiation can be performed by a sequence of modular squaring and multiplication based on the binary method. In this paper, we propose a combined algorithm to concurrently perform multiplication and squaring over GF(2~m) using the bipartite method and common operations. We expect ...
Provided that 2q≥ a, this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett reduction the value of a is bound by 0 ≤ a≤ (b−1)2, meaning that 2q≥ b2 is sufficient to ensure the reciprocal will have enough ...
Some advanced features include: Arithmetic of polynomial rings over a finite field, the Tonelli-Shanks algorithm, GCD, exponentiation by squaring, irreducibility checking, modular arithmetic (obviously) and polynomials from roots. csharp math polynomials symbolic arithmetic polynomial gcd exponentiation ...
The lines corresponding to a 1-bit represent a squaring and a multiplication. The lines corresponding to a 0-bit represent only a squaring. The algorithm uses 20 squarings and 10 additional multiplications, far fewer than the 1234566 multiplications needed by the naive technique. The square-and-...