From Diffie-Hellman Key Exchange to RSA private keys, the use of modular exponentiation in cryptography is endless.
The formula for modular arithmetic progression basically provides the key ingredients for the functioning of multiple public crypto-systems.
But, since this is an arithmetic progression, the performance of the program is highly affected by multiple layers of calculations and derivatives.
This is the challenge that we shall be talking about in detail.
From brute force to simple optimization of the power of exponent, we have covered nearly all the solutions that can be employed for performing a faster execution of mod exponentiation.
What is modular exponentiation in programming?
The arithmetic expression for modular exponentiation in programming depicts an integer value that is raised to the power of an exponent.
This integer is also referred to as the "base" which is represented by the character "b", while the exponent is represented by "e".
These characters are divided using a positive integer, also known as modulus (represented by "m").
Hence, the formula for modular exponentiation will look something as follows:
c = b2 mod m
In the formula mentioned above, we can clearly identify that the function of modular exponentiation is executed over a modulus in the program.
This algorithm is essentially used in computer science and more particularly, in the field of public-key cryptography.
In the context of public-key cryptography, we find the usage of modular exponentiation in both RSA public/private keys and Diffie-Hellman Key Exchange.
Now, other than being used in public-key cryptography, we can also find regular uses of the concept of modular exponentiation.
Let's find out about a few of them.
What are some real-time implications of modular exponentiation?
Simply put, the exponents are used for raising values of the integers by itself using a superscript to denote the value of the exponent.
You will find real-time uses of the exponents in the following situations and concepts:
Used in forming scientific scales
From pH to the Richter Scale, any time we find the depictions of scientific parameters on a scale, the exponents have definitely been used for determining the values.
Writing smaller or larger numbers
Sometimes in data science, it is required to represent a given data structure in highly smaller or larger numerical values.
Since the exponents can be represented in multiple ways, namely- subscripts and superscripts, they are essentially used for depicting data with smaller or larger scientific notations.
How to perform faster molecular exponentiation in programming?
The concept of modular exponentiation is essentially used in the concept of public key cryptography.
It basically computes the b i.e base value to the power of e i.e the exponent for mod m.
Hence, the formula would be written something as follows:
c ← b to the square e (mod m)
Now, we could employ the Brute Force Algorithm for solving the problem of performing the exponentiation problem faster just as we use it for merge two sorted arrays.
Method 1: Initiating the Brute Force Algorithm
- Start by multiplying the base value with itself for e-1 and taking the resultant output as mod m.
- Next up, we will be initiating floor division for the obtained output in order to generate modulo m as the result.
But, there are still a few problems associated with this approach. Such as follows:
- If the base and the exponents have larger values then the result of b to the power of e will be enormous. This essentially poses the problem of representing the data value as a result for the native language or system.
- Also, if the values of the base element and the exponent are larger than the decided amount then it would require infinite multiplication of the data to obtain the solution.
This level of computation would gradually require an extremely lengthy amount of time which ultimately makes the purpose of maintaining time efficiency Null and void.
Method 2: Using a slight optimised version of Naive Approach
Here's how we would implement the algorithm for this approach:
- We need to maintain the congruence of the mod m hence, we can apply the following formulas:
If, a = x (mod m)
a • k = x (mod m)
- Now, since we are multiplying these congruences a similar result as above would be obtained by multiplying all the factors and then considering the result for mod m
Hence, if a • b = x (mod m)
Then, a (mod m) • a (mod m) =
x (mod m)
Hence, in terms of modular exponentiation, by multiplying the results of mod m would gradually provide us smaller values of computation.
Method 3: Optimising the Exponent as power of two
Gradually, the exponent function performs better if its value remains a power of 2.
However, we need to optimize this solution for a general exponent that would work for any given number.
Here's how you can optimise the algorithm for this approach:
- The initial idea is to consider any number that is represented as a sum of the powers of two. This is luckily easier to attain since the modern computers represent the numbers in a similar format
- Next up, we will be creating a running total of all the required squares
- Gradually, we can arrive at the solution by working through the problem and considering the power of two to the maximum attainable size of the Exponent i.e e.
- Finally, by squaring the value of the base and adding these values to the output, the power of two will be initiated for the exponent.
For an arithmetic progression to be solved, the computer performs a profound number of calculations in order to reach the desired output.
The purpose of implementing faster mod exponentiation essentially solves this problem by providing an output in a format that is easier to comprehend for a programmer.
Although the use of mod exponentiation is more common in cryptography, we also find its implementation in the concept of arrays for instance, merging two sorted arrays.