Earlier this week I've discussed efficient algorithms for exponentiation. However, for real-life needs of number theoretic computations, just raising numbers to large exponents isn't very useful, because extremely huge numbers start appearing very quickly and these don't have much use.
What's much more useful is modular exponentiationraising integers to high powers . Luckily, we can reuse the efficient algorithms developed in the previous article, with very few modifications to perform modular exponentiation as well. This is possible because of some convenient properties of modular arithmetic.
Given two numbers, a and btheir product modulo n is. Such a number always exists, and we usually call it the remainder of dividing a by n. It follows from basic rules of modular arithmetic that .
Therefore, if we want to know the product of a and b modulo nwe just have to keep their remainders when divided by n. Note: a and b may be arbitrarily large, but x and y are always smaller than n. What is the most naive way you can think of for raising computing? Raise a to the power band then reduce modulo n.
Indeed, this is a very unsophisticated and slow method, because raising a to the power b can result in a really huge number that takes long to compute. As we've learned above, modular multiplication allows us to just keep the intermediate result at each step. Here's the implementation of a simple repeated multiplication algorithm for computing modular exponents this way:. It's much better than the naive algorithm, but as we saw in the previous article it's quite slow, requiring b multiplications and reductions modulo n.
We can apply the same modular reduction rule to the more efficient exponentiation algorithms we've studied before. We use exactly the same algorithm, but reduce every multiplication. So the numbers we deal with here are never very large. As I've noted in the previous articlethe RL method does a worse job of keeping its multiplicands low than the LR method.Even though Python natively supports big integers, taking the nth root of very large numbers can fail in Python.
When dealing with such large integers, you will need to use a custom function to compute the nth root of a number. The math module contains the expm1 -function that can compute the expression math. The improvement is significant in scientic computing. For example the Planck's law contains an exponential function minus However the result is different and using the exponential function directly is more reliable than builtin exponentiation with base math. For most all in Python 2. The math -module contains another math.
While the math -functions always convert it to a float and use the float-computation:. Supplying pow with 3 arguments pow a, b, c evaluates the modular exponentiation a b mod c :. For example one can use the 3-argument form of pow to define a modular inverse function:. While the math. The inverse of an exponentiation is exponentiation by the exponent's reciprocal. The math module contains the math. The math. It can also use. What's with the j? Since it has no imaginary part, b is 0.
Code Review Stack Exchange is a question and answer site for peer programmer code reviews. It only takes a minute to sign up. Note: this is not exactly the RSA problem.
For it to be the RSA problem, the modulus would have to be composite. Turns out, in this case, the modulus is prime. Note also: this is not the discrete log problem, as in the discrete log problem we are trying to find the exponent, not the base. You can check the answer by doing pow, and make sure it return If the modulus were not prime, you'd have to factor it to compute phi mod.
After that, everything else is the same. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Modular Exponentiation Ask Question. Asked 5 years, 6 months ago.
Active 5 years, 6 months ago. Viewed 1k times. What is the most efficient way to solve this problem programmatically? Juicy Juicy 4 4 silver badges 12 12 bronze badges. Factor the modulus ncompute phi n. Read the wikipedia article on RSA for details. It technically contains a working piece of code, but to efficiently compute this you will need to write completely different code. I think this would be a better fit of stackoverflow. This is a great fit for Code Review.
We often deal with performance issues. Suggesting alternative algorithms is commonplace here. Active Oldest Votes. That said, you can find the answer using the method that CodesInChaos recommended. Factor modeasy, mod is already prime, so it is factored.
Compute phi mod. This is the euler totient function. Use the extended euclidean algorithm to do this. There are python libraries that will do it for you pycrypto Crypto.
As the code runs perfectly but gives a TLE at the judge, I guess it is not fast enough. Is there is possible to optimize the below code to meet the judge requirement or it is impossible to beat the challenge using Python. Since there are zero accepted python solutions in a total of successful submissions, i think, no matter how much you optimise your solution, it is unlikely to manage to get accepted with python.
I don't know if this will help if you read the problem's comments you will see someone saying that it is impossible to solve in Python - that can happen on online judges with slower languages but you can optimize the code:.
Note that in 1you are computing pow n - 1, min k, n - 1 twice. You could compute it once and then use modular exponentiation only for what is left.
Same for 2. Learn more. Modular Exponentiation in Python Ask Question. Asked 4 years, 10 months ago. Active 4 years, 10 months ago. Viewed times. Akash Singh Akash Singh 61 1 1 silver badge 8 8 bronze badges. AndrewJaffe I had read the article before and used Python's inbuilt modular exponentiation in the above code but still it's not fast enough.
Active Oldest Votes. Rontogiannis Aristofanis Rontogiannis Aristofanis 7, 8 8 gold badges 34 34 silver badges 57 57 bronze badges. IVlad IVlad Thanks for the optimization but still not fast enough. Sign up or log in Sign up using Google.
Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.We have discussed recursive and iterative solutions for power. The problem with above solutions is, overflow may occur for large value of n or x. Therefore, power is generally evaluated under modulo of a large number.
Below is the fundamental modular property that is used for efficiently computing power under modular arithmetic. Below is the implementation based on above property. Time Complexity of above solution is O Log y. Modular exponentiation Recursive. This article is contributed by Shivam Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.Exponent Function - Python - Tutorial 23
Writing code in comment? Please use ide. Write an iterative O Log y function for pow x, y Modular Exponentiation Power in Modular Arithmetic Modular multiplicative inverse Euclidean algorithms Basic and Extended Program to find GCD or HCF of two numbers Minimum window size containing atleast P primes in every window of given range Median of an unsorted array using Quick Select Algorithm Sorting Algorithm Visualization : Merge Sort Count of smaller elements on right side of each element in an Array using Merge sort Expected number of moves to reach the end of a board Matrix Exponentiation Place the prisoners into cells to maximize the minimum difference between any two Smallest subarray with GCD as 1 Segment Tree Find the count of distinct numbers in a range Frequency of an integer in the given array using Divide and Conquer Minimum K such that sum of array elements after division by K does not exceed S Floyd-Rivest Algorithm Find N in the given matrix that follows a pattern.
Iterative Python3 program. Iterative Function to calculate. Update x if it is more. If y is odd, multiply. This code is contributed by Nikita Tiwari. Number of solutions to Modular Equations Find modular node in a linked list Using Chinese Remainder Theorem to Combine Modular equations Matrix Exponentiation Find Nth term A matrix exponentiation example Find the sum of power of bit count raised to the power B Expected number of moves to reach the end of a board Matrix Exponentiation Check if given number is a power of d where d is a power of 2.
Load Comments.Python is a fantastic tool for handling large amounts of data in an efficient way. This makes it a key tool for gleaning insights from a wide variety of sources.
Unsurprisingly, Python has a handy built-in repository of arithmetic and logical functions. It also supports more complex mathematics through modular libraries like NumPy and SciPy. While most of the built-in mathematical functions that Python supports are well-known to any high-school graduate, the MOD function stands out. Unlike trigonometric or exponential functions, MOD is something altogether more fundamental. The Python MOD function is easy to call.
You can do this with any expression that evaluates to a numeric type. Modular functions point to the value left out of a divided sum after division. The easiest way to understand the way modular functions is using the face of a clock. There are 24 hours in a day, but only 12 on a clock face. People tell the time of day by calculating the congruent number modulo 12 for the current hour of the day.
Immediately after noon, when the clock strikesit physically wraps around its perimeter to restart counting time from the start. This means that in the mathematical system the clock face asserts, 13 is congruent with 1. By the same system, 23 is congruent with In all cases, the remainder of the division equation is the answer of the modular function. In computer science, modular functions are incredibly important for a wide variety of reasons. They help programmers control bitwise operations involving cyclic data structures.
They allow for mathematical descriptions of logical behavior like the exclusive disjunction. The powerful thing about using Python MOD equations in computer programming is that high-level programming languages like Python allow you to process many all types of numeric operands — not just integers. Whenever you want to add or subtract a number and another number but have the result be within a certain range, modular functions are your key operator.
One of the fascinating things about the Python modulo operation is the fact that it can work with integers as well as floating-point numbers equally well. The only time when Python will generate an exception is when you attempt to divide by zero. Some immediate applications for modular functions obviously center around loop counters.
If you want to create a program that operates inside a loop but does not operate on every iteration of the loop, then you could design it to only operate on a specific modulo of that loop. That would look like this:.
This gives you a lot of power over timing certain program operations. Another option is to use modular functions to turn a list into a cyclic snippet of code.
Doing so would look like this:. Now the list in question will behave cyclically, returning to its beginning state at the end of every cycle. You could use this, for instance, to increase an index while wrapping back to the beginning after reaching the end of an array. Modular functions can become expensive in terms of operation time when it comes to very large numbers. The time it takes to calculate the modulo of a very large number gets longer as the number gets larger.
This led computer researchers and mathematicians to look for more efficient ways to perform these functions. In Python, the secret to performing modular functions on very large numbers is called modular exponentiation. This operation features logarithmic running time and is functionally similar to binary exponentiation. In Python, the built-in function pow performs modular exponentiation.
This makes modular exponentiation a key element of cryptography coding.
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Code Golf Stack Exchange is a question and answer site for programming puzzle enthusiasts and code golfers. It only takes a minute to sign up. The challenge is to write code in any open source, available on linux, language of your choice to perform modular exponentiation. The input will be two randomly chosen bit positive integers numbers x and y and a bit prime z. Here is a sample snippet of python to compare to. The code should accept a file with each of the three numbers on separate lines and output the result to standard out.
You may not use any libraries for modular exponentiation of course. For those who want really accurate timing, the best way may be for the timing code to actually be in the code you provide and for your code to simply repeat times without cheating :. This avoids any problems with start up overheads.
Runs in about 7ms on my laptop. Less than a factor of 2 away from GMPwhich is heavily optimized for this task. Uses Montgomery reduction to compute the modular reductions quickly.
Uses a python script to generate x86 assembly to do the bit ops multiply, add, etc. The code is about lines, a bit big to paste in an answer box. Download it here. Though it's simple, it should be pretty fast if built with GHC :. If the modulus is guaranteed to be a prime, but the exponent might be greater than or equal to the modulus, the last line should be changed to:. If you'd like it to handle multiple inputs per run, add import System. IO under the other import and the following two lines just under the print line:.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Fastest modular exponentiation Ask Question. Asked 6 years, 6 months ago. Active 6 years, 6 months ago. Viewed 3k times. The code that runs the fastest on average over runs on my computer wins. For example I don't know if this particular approach is good, but Active Oldest Votes. Keith Randall Keith Randall Olathe Olathe 4 4 bronze badges.