83. | Learn Encryption Techniques with BASIC and C++

Learn Encryption Techniques with BASIC and C++
(Publisher: Wordware Publishing, Inc.)
Author(s): Gil Held
ISBN: 1556225989
Publication Date: 10/01/98

Table of Contents

Advantages and Disadvantages

The primary advantage associated with the use of public key cryptology is the increased level of security and convenience it provides, which reduces the effort involved in key management and key distribution associated with a private key system. Since the private key associated with the public key system does not have to be distributed, this both simplifies key management and reduces the possibility of key interception.

The ability to distribute a public key that can encrypt messages that can only be decrypted by the corresponding private key depends upon the selection of an appropriate algorithm. The algorithm must be a one-way function, which from a computational perspective is significantly easier to perform in one direction than from the reverse. For example, the one-way function might be able to be performed in seconds or minutes in the forward direction, but require months or years of trial and error to perform in the opposite direction.

As an example of a one-way function, consider the following equation:

      y = 5x⁴ + 27x³ + 14x² + 12

While it is relatively easy and fast to compute y given x, it is much more difficult to compute x given y. Although the time required to execute the one-way function in the forward direction is relatively fast in comparison to the process to perform the reverse, it requires significantly more than the amount of time required to perform private key encryption. This means that public key encryption may not be suitable for certain types of applications, such as high-speed encrypted communications. In such situations, public key cryptology can be used as a supplement to private key cryptology. For example, you could use a public key system to distribute private keys. In fact, a popular application for the use of a public key system by government, industry, and financial organizations is the distribution of private keys via a public key system.

There was a young fellow named Ben
Who could only count to modulo ten.
He said, ’When I go
past my last little toe
I shall have to start over again.’

-Anonymous

Understanding the Mathematics

The ability to obtain an appreciation for the manner by which public key cryptology operates requires an understanding of a few areas of mathematics that are used to create one-way functions. Thus, prior to turning your attention to the manner by which a popular public key system operates, let me turn your attention to a review of some mathematical concepts.

Most public key systems are based upon finite arithmetic in which numbers are manipulated according to rules that are different from ordinary arithmetic. Because one-way functions used in public key systems are based upon finite arithmetic, and modular arithmetic results in finite sets of numbers, let’s commence our review of mathematics with this topic. Since I previously discussed modular arithmetic when developing private key algorithms in this book, I will not turn to this topic in detail.

Modular Arithmetic

Modular arithmetic is a branch of mathematics in which you operate upon a finite set of numbers. For example, the finite set of numbers {0, 1, 2,. .. n-1} are referred to as modulo n arithmetic. Here, n can take any positive integer value except 1 and the result of any arithmetic operation belongs to the same set of n numbers.

Counting in modulo arithmetic goes from 0 to n-1, with the next numbers in the series being 0, 1, . . ., so that the sequence continues and repeats. In fact, when you perform modulo arithmetic, such operations as addition and multiplication are performed as with ordinary arithmetic, with the answer obtained by dividing by n and using the remainder as the result. The result is said to be modulo n, abbreviated as mod n. Thus writing x mod n references the remainder of x when divided by n. To obtain an appreciation for the operation of modulo arithmetic, let me turn your attention to modular addition and modular multiplication.

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