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.