Discrete Logarithm Example Problems
More specifically say m 100 and t 17. For example let be the elliptic curve given by over the field.
How To Evaluate Logarithms
Time Complexity Exploration Eulers totient function.
Discrete logarithm example problems. This can be done in manywaysforexampleletusconsiderthefollowingproblem. The value of x being x x2 x3. 3 7 11 Output.
Dislog20Mod523 7 5 So the example above was far too simple. We recall that an integer is B-smooth if all its prime factors are at most B. The difficulty of this general discrete logarithm problem depends on the representation of the group.
Dislog18Mod523 6 12. Discrete logarithm problem is not always. The Elliptic Curve Discrete Logarithm Problem Problem 64 Elliptic Curve Discrete Log Problem Suppose is an elliptic curve over and.
Or is. Its easy to write a slow program to solve the discrete log problem. Log 2 19 9 mod 28 29 19 mod 29.
Given a a and b b what is the non-negative integer m m such that a bm a b m. For example the equation log 10 53 1724276 means that 10 1724276 53. But then computing logg t is really solving the congruence ng t mod m.
The discrete logarithm problem. We will see that such an answer does not always exists and describe an algorithm that gives an answer provided it exists. The discrete logarithm to the base b b of a a is the answer to the question.
Clearly the discrete logarithm problem for a general group G is exactly the problem of inverting the exponentiation function defined by where N is the order of. The discrete logarithm to the base g of h in the group G is defined to be x. A 2 b 3 m 5 The value which satisfies the above equation is 3 because 2 3 2 2 2 8 2 3 mod 5 8 mod 5 3 which is equal to b ie 3.
This is the currently selected item. In Example 92. It is an important open question whether determining B knowing just and is as hard as the discrete log problem in general.
Finding a discrete logarithm can be very easy. Let B N be such that N is B-smooth Then Algorithm 13 solves the DLP in G using OlogN 2BlogN group operations. The discrete logarithm problem is little more than an integer analog to a rotor-code cipher problem.
For example if the group is Z5 and the generator is 2 then the discrete logarithm of 1 is 4 because 2 4 1 mod 5. Take g as some other generator of ZmZ. Discrete Logarithm Problem Shanks Pollard Rho Pohlig-Hellman Index Calculus Discrete Logarithm Notation For example 215 27 mod 29 222 5 mod 29 This implies log 2 27 15 mod 28 log 2 5 22 mod 28 We can now write log 2 27 log 2 5 log 227 5 mod 29 mod 28 log 2 19 mod 28 15 22 9 mod 28 Therefore.
But if you have values for x a and n the value of b is very difficult to compute when the values of x a. There are better methods but we wont discuss them in this class. For example consider G to be the cyclic group of order N.
For example say G ZmZ and g 1. Dislogxg s sg. Lets make it harder.
The later is a problem in finding concurrent zeros for two periodic functions where the periodicity of one tracks the value of x and the other the value of y. Then logg t 17 or more precisely 17 mod 100. If taking a power is of Ot time then finding a logarithm is of O2t2 time.
And this can be made prohibitively large if t log 2 q is large. Given a group G a generator g of the group and an element h of G to find the discrete logarithm to the base g of h in the group G. Solving Discrete Logarithm Problem.
Given a multiple of the elliptic curve discrete log problem is to find such that. Discrete Logarithm Problem On the other hand given c and α finding m is a more difficult proposition and is called the discrete logarithm problem. 2 3 5 Output.
For example log 10 10000 4 and log 10 0001 3. For example if a 3 b 4 and n 17 then x 34 mod 17 81 mod 17 81 mod 17 13. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem because they involve non-integer exponents.
Lets try a slightly larger prime. Problem itself is not a decision problem the first step is to introduce a related decision problem whose hardness is essentially equivalent to computing discrete logarithms. Let p 31 g 3 and h 22.
See MaurerW for the latest references on this topic which will not be covered here. The value of y being y yp y2p ynp. The discrete logarithm problem is defined as.
The Discrete Logarithm Problem. If the discrete log problem for the group is easy an eavesdropper can compute either. The discrete logarithm problem is used in cryptography.
These are instances of the discrete logarithm problem. Solve the discrete logarithm problem of h to the base g using the Pohlig-Hellman method. Given values for a b and n where n is a prime number the function x ab mod n is easy to compute.
Let g G have order N.
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