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Chapter 4: Problem 57

Write out a table of discrete logarithms modulo 17 with respect to the primitive root \(3 .\)

Short Answer

Expert verified
Discrete logarithms modulo 17 with respect to the primitive root 3 are computed by finding x such that 3^x 鈮� n (mod 17) for n from 1 to 16.

Step by step solution

01

- Understand the Problem Statement

The problem requires finding the discrete logarithms modulo 17 with respect to the primitive root 3. This means determining all integers x such that 3^x 鈮� n (mod 17) for each n between 1 and 16.
02

- Identify the Primitive Root

Confirm that 3 is a primitive root of 17. A primitive root g of a prime p is an integer such that the powers of g generate all integers from 1 to p-1 modulo p.
03

- Calculate Powers of 3 Modulo 17

Compute 3^x (mod 17) for x ranging from 1 to 16 and record the results. This will help find the discrete logarithms.
04

- Create the Discrete Logarithm Table

Using the results from step 3, create a table where each integer n (from 1 to 16) is paired with the smallest exponent x such that 3^x 鈮� n (mod 17).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

primitive root
A primitive root is a foundation of many topics in number theory and especially in modular arithmetic. Given a prime number p, a number g is called a primitive root if g raised to different powers generates all numbers from 1 to p-1.
For example, we need to check if 3 is a primitive root modulo 17. If all powers of 3 generate each number from 1 to 16 (mod 17), then 3 is a primitive root.

Let's calculate:
3^1 鈮� 3 (mod 17)
3^2 鈮� 9 (mod 17)
3^3 鈮� 27 鈮� 10 (mod 17)
Continue till 3^16.
You will observe that the series covers all values from 1 to 16 modulo 17.
  • An essential fact: Not every integer has a primitive root.
  • Applications: Cryptographic systems like Diffie-Hellman key exchange use primitive roots.
modular arithmetic
Modular arithmetic is like clock arithmetic. It deals with the remainder when one number is divided by another.
For instance, a 鈮� b (mod n) means that when a is divided by n, it leaves the same remainder as b.

Consider 20 鈮� 3 (mod 17) because 20 - 3 = 17, which is a multiple of 17.
This concept is crucial in understanding discrete logs, as we're looking for an x such that 3^x 鈮� n (mod 17).
  • It's often used in cryptography, computer science, and number theory.
  • Provides the basis for understanding problems and solutions involving remainders.
discrete mathematics
Discrete mathematics deals with mathematical structures that are fundamentally distinct and not continuous. It includes topics such as combinatorics, graph theory, and, notably, number theory.
In our context, discrete logarithms fall under the purview of number theory.

Discrete Mathematics aids our comprehension of algorithms and computational systems. It applies to:
  • Encryption algorithms where securing data depends on discrete logarithms.
  • Advanced algorithms in computer science that process distinct, separate values.
Understanding discrete mathematics is a cornerstone for several theoretical and practical applications in today鈥檚 digital world.
logarithmic tables
Logarithm tables were used to simplify calculations before the advent of calculators. They convert multiplication and division into addition and subtraction.

When dealing with discrete logs, think of a logarithm as the power to which a base number (like the primitive root) must be raised to produce a given number鈥攕imilar to looking up values in logarithmic tables.
The table you create for discrete logs modulo 17 with respect to 3 will list:
  • Each integer n from 1 to 16.
  • The corresponding x such that 3^x 鈮� n (mod 17).
This concept is key in simplifying and solving multiplicative problems in modular systems.

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Most popular questions from this chapter

Show that \(\log _{2} 3\) is an irrational number. Recall that an irrational number is a real number \(x\) that cannot be written as the ratio of two integers.

The Vigen猫re cipher is a block cipher, with a key that is a string of letters with numerical equivalents \(k_{1} k_{2} \ldots k_{m},\) where \(k_{i} \in \mathbf{Z}_{26}\) for \(i=1,2, \ldots, m .\) Suppose that the numerical equivalents of the letters of a plaintext block are \(p_{1} p_{2} \ldots p_{m} .\) The corresponding numerical ciphertext block is \(\left(p_{1}+k_{1}\right)\) mod 26 \(\left(p_{2}+k_{2}\right) \bmod 26 \ldots\left(p_{m}+k_{m}\right)\) mod \(26 .\) Finally, we translate back to letters. For example, suppose that the key string is RED, with numerical equivalents \(1743 .\) Then, the plaintext ORANGE, with numerical equivalents \(141700130604,\) is encrypted by first splitting it into two blocks 141700 and 13 \(0604 .\) Then, in each block we shift the first letter by 17 , the second by \(4,\) and the third by \(3 .\) We obtain 52103 and 0410 \(07 .\) The cipherext is FVDEKH. Use the Vigen猫re cipher with key BLUE to encrypt the message SNOWFALL.

How many divisions are required to find gcd(34, 55) using the Euclidean algorithm?

Prove that the set of positive rational numbers is countable by setting up a function that assigns to a rational number \(p / q\) with \(\operatorname{gcd}(p, q)=1\) the base 11 number formed by the decimal representation of \(p\) followed by the base 11 digit \(\mathrm{A},\) which corresponds to the decimal number \(10,\) followed by the decimal representation of \(q\)

Describe a brute-force algorithm for solving the discrete logarithm problem and find the worst-case and averagecase time complexity of this algorithm.
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