## What is exponentiation algorithm?

Section 2.6 Exponentiation Algorithm. We present an algorithm for computing a power of an integer. We call this algorithm the Naive Exponentiation algorithm, since there is a more clever way of calculating powers which we will present with Algorithm 15.3.

## How do you calculate fast exponent?

How can we calculate A^B mod C quickly for any B?

1. Step 1: Divide B into powers of 2 by writing it in binary. Start at the rightmost digit, let k=0 and for each digit:
2. Step 2: Calculate mod C of the powers of two ≤ B. 5^1 mod 19 = 5.
3. Step 3: Use modular multiplication properties to combine the calculated mod C values.

Is squaring exponential?

Exponents and have special names. Raising a base to a power of is called “squaring” a number. Raising a base to a power of is called “cubing” a number. The inverse of squaring a number is finding the square root of a number.

What is the time complexity of binary exponentiation?

In reality, multiplication takes O(log N) time and hence, Binary exponentiation takes O(logN * logM) time and the normal approach takes O(M * logN) time.

### What is modular exponentiation method?

Modular exponentiation is exponentiation performed over a modulus. Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m.

### Is used for fast modular exponentiation calculation?

We can compute c using the “squares” method – this allows for fast computation of large positive integer powers of a number. For example, this allows a⁸, can be represented as ((a²)²)². … 7 multiplications are required (the exponent – 1).

What is the purpose of squaring a number?

ie. in the determination of a value; to square a number, perform some mathematical operations, then “unsquare” it. This must be a way to “take some aspect of something in to account” so as to be inclusive in determining a correct value and/or representation of what one is attempting to solve.

What does mod mean math?

Given two positive numbers a and n, a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus (or divisor) one is operating from.

#### How do you do modular arithmetic?

Modulus on a Standard Calculator

1. Divide a by n.
2. Subtract the whole part of the resulting quantity.
3. Multiply by n to obtain the modulus.

#### How do you calculate modular?

How to calculate the modulo – an example

1. Start by choosing the initial number (before performing the modulo operation).
2. Choose the divisor.
3. Divide one number by the other, rounding down: 250 / 24 = 10 .
4. Multiply the divisor by the quotient.
5. Subtract this number from your initial number (dividend).

What is optimal addition chain exponentiation?

In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. This corresponds to the sequence A003313 on the Online Encyclopedia of Integer Sequences.

What is an example of an addition chain?

As an example: (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since. Addition chains can be used for addition-chain exponentiation. This method allows exponentiation with integer exponents to be performed using a number of multiplications equal to the length of an addition chain for the exponent.

## Is the shortest addition chain NP-complete?

A number of other papers state that finding a shortest addition chain for a single number is NP-complete, citing this paper, but it does not claim or prove such a result. ^ a b c Otto, Martin (2001), Brauer addition-subtraction chains (PDF), Diplomarbeit, University of Paderborn, archived from the original (PDF) on 2013-10-19, retrieved 2013-10-19.