Chapter 6. Discrete Logarithms (2023)

FAQs

How hard is the discrete logarithm problem? ›

So while the DLP is generally considered a “hard problem", its difficulty really depends not on the order of the group (or its structure), but on how the group is explicitly rep- resented. Every group of prime order p is isomorphic to Z/pZ; computing the discrete logarithm amounts to computing this isomorphism.

Finding a discrete logarithm can be very easy. For example, say G = Z/mZ and g = 1. More specifically, say m = 100 and t = 17. Then logg t = 17 (or more precisely 17 mod 100).

A discrete logarithm is just the inverse operation. For example, take the equation 3^k≡12 (mod 23) for k. As shown above k=4 is a solution, but it is not the only solution. Since 3²²≡1 (mod 23), it also follows that if n is an integer, then 3⁴⁺²²ⁿ≡12×1ⁿ≡12 (mod 23).

The discrete logarithm of x to the base g is the smallest non-negative integer a such that x = ga. We write logg x = a. The discrete logarithm problem in a cyclic group G is to find the discrete logarithm of x to the base g, when x has been chosen uniformly at random from the group.

Discrete math is fun.

Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring. Rarely is this the case with most discrete math topics.

Logarithms is one material that is difficult for students [1]. Another study on the difficulties in learning logarithms said that students are more focused on the procedural approaches and depended too much on rules rather than the concept of logarithm itself[2].

The discrete probability distribution variance gives the dispersion of the distribution about the mean. It can be defined as the average of the squared differences of the distribution from the mean, μ μ . The formula is given below: Var[X] = ∑(x - μ μ )² P(X = x)

There are several reasons why discrete math can be difficult, including its abstract nature, the use of unfamiliar mathematical notation, and the need to master multiple problem-solving strategies. Overall, it is safe to say that discrete math is indeed a challenging subject.

The most common types of logarithms are common logarithms, where the base is 10, binary logarithms, where the base is 2, and natural logarithms, where the base is e ≈ 2.71828.

What is the importance of discrete logarithms? ›

Aside from the intrinsic interest that the problem of computing discrete logarithms has, it is of considerable importance in cryptography. An efficient algorithm for discrete logarithms would make several authentication and key-exchange systems insecure.

Diffie- Hellman key exchange is based upon the discrete logarithm problem. g a n = determine x. There is no known efficient, non-quantum algorithm to solve the discrete logarithm problem.

Discrete logarithm is only the inverse operation. For instance, it can take the equation 3^k = 13 (mod 17) for k. In this k = 4 is a solution. Since 3¹⁶ ≡ 1(mod 17), it also follows that if n is an integer then 3⁴⁺¹⁶ⁿ ≡ 13 x 1ⁿ ≡ 13 (mod 17). Therefore, the equation has infinitely some solutions of the form 4 + 16n.

Many people will find discrete math more difficult than calculus because of the way they are exposed to both of the areas.

Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be easily be introduced at the middle school grade level.

First, Princi- ples and Standards recommends including discrete mathematics in the curriculum for all grades, from prekindergarten through grade 12.

Answer and Explanation: In general, logarithms are taught in high school. Normally, beginning high school students (grade 9) are introduced to logarithms.

When calculus is involved, natural logarithms are usually used. For special purposes, other logarithms are used, mainly logarithms with base 10 or with base 2.

In contrast, for single precision, both functions log and log2 are the same apart from division by ln2 in the log2 case, hence the same speed.

What are the 5 rules of logarithms? ›

Logarithm Rules and Properties

Product rule. Division rule. Power rule/Exponential Rule. Change of base rule.

Recall the general form of a logarithmic function is: f ( x ) = k + a log b ⁡ where a, b, k, and h are real numbers such that b is a positive number ≠ 1, and x - h > 0. A logarithmic function is transformed into the equation: f ( x ) = 4 + 3 log ⁡ .

Discrete mathematics is a branch of mathematics concerned with the study of objects that can be represented finitely (or countably).

A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.

"Discrete Math" is not the name of a branch of mathematics, like number theory, algebra, calculus, etc. Rather, it's a description of a set of branches of math that all have in common the feature that they are "discrete" rather than "continuous".

Can you learn discrete math on your own? Yes. The key to learning anything new is to have a desire to learn. But you also need to have the right resources.

Calculus is inherent in every other subject, even discrete structures. Discrete mathematics comes in mind. But calculus is already inherent in discrete mathematics. Combinatorics, set theory or graph theory are usually core elements in a discrete math course.

Re: What is the difference between log and log10 transformation in JMP? A common logarithm, Log10(), uses 10 as the base and a natural logarithm, Log(), uses the number e (approximately 2.71828) as the base.

What are the 7 properties of logarithms? ›

There are three laws of logarithms that are derived using the basic rules of exponents. The laws are the product rule law, quotient rule law, power rule law.

Currently, the best-known algorithm for computing discrete logarithms in Z∗p (for p prime) is the general number field sieve.

A logarithm (or log) is the mathematical expression used to answer the question: How many times must one “base” number be multiplied by itself to get some other particular number? For instance, how many times must a base of 10 be multiplied by itself to get 1,000? The answer is 3 (1,000 = 10 × 10 × 10).

Logarithms can be used to solve exponential equations and to explore the properties of exponential functions. They will also become extremely valuable in calculus, where they will be used to calculate the slope of certain functions and the area bounded by certain curves.

The discrete logarithm problem bears the same relation to these systems as factoring does to the RSA system: the security of these systems rests on the assumption that discrete logarithms are difficult to compute.

The discrete logarithm (logarithm) of an element β to the base α in G is an integer x such that α ^x = β. If x is restricted to the interval 0 ≤ x < #G then the discrete logarithm of β to the base α is unique.

Discrete logarithms are fundamental to a number of public-key algorithms, includ- ing Diffie-Hellman key exchange and the digital signature algorithm (DSA). Discrete logarithms are fundamental to a number of public-key algorithms, includ- ing Diffie-Hellman key exchange and the digital signature algorithm (DSA).

What is the difference between an index and a discrete logarithm? The two terms are synonymous. Which of the following states that if p is prime and a is a positive integer not divisible by p, then a^p−1≡1 (mod p)?

How is discrete logarithm evaluated for a number? ›

Finally, to compute the discrete logarithm d = loga b, assemble all of the individual values d modulo qc via the Chinese remainder theorem to obtain the value of d modulo p − 1. has small prime divisors, then we can evaluate all of the ingredients in the computation rapidly.

Many Diffie-Hellman implementations use numbers of a little over 300 digits long (1024 bits). These keys, the paper showed, can be cracked within a year for around 100 million US dollars. (Some people believe it can be done even more cheaply, but only the ballpark figure matters here.)

You can do this using GPO or Local security policy under Computer configuration -> Administrative Templates -> Network -> SSL Configuration Settings -> SSL Cipher Suite Order. Set this policy to enable. Each cipher suite should be separated with a comma. Remove as needed based on the list below.

So while the DLP is generally considered a “hard problem", its difficulty really depends not on the order of the group (or its structure), but on how the group is explicitly rep- resented. Every group of prime order p is isomorphic to Z/pZ; computing the discrete logarithm amounts to computing this isomorphism.

Many students find that discrete math is harder than calculus. Discrete math is a branch of mathematics that deals with objects that are discrete, meaning they can be counted or listed. This includes things like integers, graphs, and boolean values.

Reasons why discrete math can be a hard class

Discrete math often involves working with concepts that are not directly related to the physical world, such as graphs, sets, and logic. These abstract ideas can be difficult for students to grasp, especially if they are more comfortable with concrete, real-world concepts.

"NP-complete problems are difficult because there are so many different solutions." On the one hand, there are many problems that have a solution space just as large, but can be solved in polynomial time (for example minimum spanning tree).

Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general.

No. “NP-hard” applies to problems that receive an input of arbitrary length and produce a yes/no response as a decision for that input. Problems that require no input, such as “does P=NP”, can be solved in O(1) time (constant time).

What should I learn before discrete math? ›

What math do I need to learn before discrete mathematics? Students with a solid understanding of algebra, geometry, and precalculus will do very well in discrete math.

Discrete Mathematics provides an essential foundation for virtually every area of computer science, and its applications are correspondingly vast. At the most fundamental level, all of a computer's data is represented as bits (zeros and ones).

Calculus isn't really needed to understand discrete math, but if calculus is a prerequisite for the class, there are a number of good examples and homework problems that the professor might use that would indeed require calculus.

Typically, students in grade 11 take Algebra II (if they followed the traditional course sequence: Algebra I in 9th grade, and Geometry in 10th grade). However, some students may be able to take Algebra I while still in 8th grade. In those cases, both 11th and 12th grade become open for advanced math options.

1. Arrays. The first in our list of basic data structures is one of the simplest data structures. An array is a fixed-size structure that stores multiple items of the same kind of data sequentially.

I think discrete math is harder than calculus or linear algebra, but that may be a personal thing. Discrete math covers a diverse number of topics which have their own particular methods, whereas calculus relies on a handful of tricks which you can apply over and over again.

Statistics has gotten a reputation for being a very hard class, especially when taken in college, because it combines math concepts in order to form an analysis of a data set that can be used to understand an association in the data (whoo that was a mouthful).