Attacking Problems In Logarithms And Exponential Functions Pdf

attacking problems in logarithms and exponential functions pdf

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Add to Wishlist. By: David S. Product Description Product Details This original volume offers a concise, highly focused review of what high school and beginning college students need to know in order to solve problems in logarithms and exponential functions.

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Attacking Problems in Logarithms and Exponential Functions

Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Let G be any group. Denote its group operation by multiplication and its identity element by 1.

Let b be any element of G. For any positive integer k , the expression b k denotes the product of b with itself k times:.

Let a also be an element of G. This set G is a cyclic group under multiplication, and 10 is a generator. These are instances of the discrete logarithm problem. Other base logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. While integer exponents can be defined in any group using products and inverses, arbitrary real exponents in the real numbers require other concepts such as the exponential function. A similar example holds for any non-zero real number b.

This is the group of multiplication modulo the prime p. The k th power of one of the numbers in this group may be computed by finding its k th power as an integer and then finding the remainder after division by p. When the numbers involved are large, it is more efficient to reduce modulo p multiple times during the computation. Regardless of the specific algorithm used, this operation is called modular exponentiation. The discrete logarithm is just the inverse operation. In other words, the function.

The familiar base change formula for ordinary logarithms remains valid: If c is another generator of H , then. Can the discrete logarithm be computed in polynomial time on a classical computer? The discrete logarithm problem is considered to be computationally intractable.

That is, no efficient classical algorithm is known for computing discrete logarithms in general. This algorithm is sometimes called trial multiplication. It requires running time linear in the size of the group G and thus exponential in the number of digits in the size of the group. Therefore, it is an exponential-time algorithm, practical only for small groups G. More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization.

However none of them run in polynomial time in the number of digits in the size of the group. There is an efficient quantum algorithm due to Peter Shor. Efficient classical algorithms also exist in certain special cases. For example, in the group of the integers modulo p under addition, the power b k becomes a product bk , and equality means congruence modulo p in the integers.

The extended Euclidean algorithm finds k quickly. While computing discrete logarithms and factoring integers are distinct problems, they share some properties:. There exist groups for which computing discrete logarithms is apparently difficult. In some cases e. At the same time, the inverse problem of discrete exponentiation is not difficult it can be computed efficiently using exponentiation by squaring , for example.

This asymmetry is analogous to the one between integer factorization and integer multiplication. Both asymmetries and other possibly one-way functions have been exploited in the construction of cryptographic systems.

ElGamal encryption , Diffie—Hellman key exchange , and the Digital Signature Algorithm and cyclic subgroups of elliptic curves over finite fields see Elliptic curve cryptography. While there is no publicly known algorithm for solving the discrete logarithm problem in general, the first three steps of the number field sieve algorithm only depend on the group G , not on the specific elements of G whose finite log is desired.

By precomputing these three steps for a specific group, one need only carry out the last step, which is much less computationally expensive than the first three, to obtain a specific logarithm in that group. It turns out that much Internet traffic uses one of a handful of groups that are of order bits or less, e. The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a bit prime number, so called export grade.

The authors of the Logjam attack estimate that the much more difficult precomputation needed to solve the discrete log problem for a bit prime would be within the budget of a large national intelligence agency such as the U.

The Logjam authors speculate that precomputation against widely reused DH primes is behind claims in leaked NSA documents that NSA is able to break much of current cryptography. From Wikipedia, the free encyclopedia. This article includes a list of general references , but it remains largely unverified because it lacks sufficient corresponding inline citations.

Please help to improve this article by introducing more precise citations. October Learn how and when to remove this template message. See also: Discrete logarithm records. Unsolved problem in computer science :. Rosen, Kenneth H. Weisstein, Eric W. Wolfram Web. Retrieved 1 January Number-theoretic algorithms. Binary Euclidean Extended Euclidean Lehmer's. Cipolla Pocklington's Tonelli—Shanks Berlekamp. Public-key cryptography. Computational hardness assumptions. External Diffie—Hellman Sub-group hiding Decision linear.

Shortest vector problem gap Closest vector problem gap Learning with errors Ring learning with errors Short integer solution. Exponential time hypothesis Unique games conjecture Planted clique conjecture. Categories : Modular arithmetic Group theory Cryptography Logarithms Finite fields Computational hardness assumptions Unsolved problems in computer science. Hidden categories: Articles lacking in-text citations from October All articles lacking in-text citations Pages using RFC magic links.

Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Italics indicate that algorithm is for numbers of special forms.

Attacking problems in logarithms and exponential functions

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Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Let G be any group.

In the examples that follow, note that while the applications are drawn from many different disciplines, the mathematics remains essentially the same. Due to the applied nature of the problems we will examine in this section, the calculator is often used to express our answers as decimal approximations. Perhaps the most well-known application of exponential functions comes from the financial world.

Your answer seems reasonable. To practice mathematics, math workbooks are the good source. What is the annual interest rate? Problem 1.

6.5: Applications of Exponential and Logarithmic Functions

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When we multiply a number by itself, we call this process squaring.