Note:  This document may also be found on-line
http://move.to/cryptography
or link here
abstract-algebra-project.html
because some of the links and software in the project are to on-line targets.
(The document is perfectly stand-alone in terms of readability though.)
 

• Introduction

 
     The need for digitally secured information has skyrocketed in recent years.  With the ever demanding requirements for better, more complicated encrypting techniques, cryptography has particularly experienced significant advances recently.  Although cryptographic methods can be traced back 4000 years to the time of the Egyptians, cryptography has more real world applications now than ever before.
    Traditionally, throughout history, cryptographic methods have mostly been utilized exclusively by governments and militaries.  In today's world of digital communication, the manners in which cryptographic manipulations can take place, really, are endless.
Although uses of cryptography directed for positive (that is, good) means can be sought after, unfortunately, it is usually evil intention that drives the demand for new cryptographic conventions.
    Beginning in the 1970's, particular mathematical methods involving the intractability of discrete logarithms took cryptographic methods to new heights.  Today, those same methods (along with many others) have been fully developed and standardized in hopes of establishing some sort of measure by which the fast growing industry can gauge itself.
    Although the cryptographic methods used by super computer systems today are quite involved, (not to mention conceptually impossible to follow), all cryptographic methods involve simple objectives, notions, and procedures.
    The following report serves as a basic summary outlining various terms, definitions, ideas, schemes, and applications involved with cryptography at a most fundamental level.  If the following terms (as relative to theory) and definitions (as relative to the mathematics involved) are generally understood, more complicated approaches to cryptographic systems (beyond this document) may be examined and understood.
 

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• Terms and definitions

 
Some of the following terms may seem a bit trivial.  However, the terms must be listed because the definitions of these terms may not exist as "dictionary explanations."  That is to say, the terms have specific meanings in relation to the study of cryptography.
 
• Access Control - A mechanism by which use of computer or network resources is limited depending on the security policy (Loshin).
• Active attack - An attempt to improperly modify data, gain authentication, or gain authorization by inserting false packets into the data stream or by modifying packets transiting the data stream (Loshin).
• Adversary - An entity in a two party communication which is neither the sender nor receiver, and which tries to defeat the information security service being provided between the sender and receiver (Menezes).
• Algorithm - The description of the mechanism by which plaintext is encrypted into ciphertext (Loshin).
• Anonymity - Concealing the identity of an entity involved in some process (Menezes).
• Asymmetric cryptography - An encryption system that uses different keys, for encryption and decryption (Loshin).
• *Authentication - The use of some mechanism by which one entity offers some data that proves its authorization to use a computer or network resource (Loshin).
• Authorization - Conveyance, to another entity, of official sanction to do or be something (Menezes).
• Certification - Endorsement of information by a trusted entity (Menezes).
• Channel - A means of conveying information from one entity to another (Menezes).
• Ciphertext - Data that has been encrypted (Loshin).
• *Confidentiality - Keeping information secret from all but those who are authorized to see it (Menezes).
• Confirmation - Acknowledgment that services have been provided (Menezes).
• Cryptanalysis - Refers to the pursuit of breaking ciphers (Loshin).
• Crypt analyst - Someone who engages in Cryptanalysis (Menezes).
• Cryptography - From Greek origin, literally means "hidden writing" and currently refers to the art and science of developing and using mechanisms by which to render unreadable to anyone but the intended recipient (Loshin).
• Cryptosystem - A general term referring to a set of cryptographic primitives used to provide information security services (Menezes).
• *Data Integrity - Ensuring information has not been altered by unauthorized or unknown means (Menezes).
• Decryption - Any procedure that takes as its input a stream of apparently random or meaningless information and produces as output a stream of recognizable information that is the same as the original plaintext (Loshin).
• Digital signature - This is a short string of data that can be associated with any other type of data, and that is used to certify that the communication has been sent by the signing entity and that it has not been tampered with in any way (Loshin).
• Encryption - Any procedure that takes as its input a stream of usable, recognizable, or understandable information and produces as output a stream of data that appears to be random and/or meaningless to anyone who does not have the proper key with which to decrypt it (Loshin).
• Entity or party - Someone or something which sends, receives, or manipulates information (Menezes).
• Entity authentication - Corroboration of the identity of an entity (person, computer terminal, credit card, etc.) (Menezes).
• Function - A mapping or a transformation (in this context) (Menezes).
• Integrity - The protection of information from unauthorized modification (Loshin).
• Key - A sequence of data used to encrypt a transmission or data file.  Knowledge of the correct key will allow an entity to decrypt an encrypted message or file (Loshin).
• Key certificate - A data structure consisting of a public key, the identity of the person, system, or role associated with that key, and information authenticating both the key and the association between that identity and that public key (Loshin).
• Man in the middle - Attack that occurs when a criminal attacker manages to insert them self between two end-points of a channel (Loshin).
• Message authentication - Corroborating the source of information; aka origin authentication (Menezes).
• Network sniffer - Can be used to actively scan all network data traffic (Loshin).
• *Non-repudiation - Concealing the identity of an entity involved in some process (Menezes).
• Ownership - A means to provide an entity with he legal right to use or transfer a resource to others (Menezes).
• Passive attack - An attack on an authentication system that inserts no data into the stream, but instead relies on being able to passively monitor information being sent between parties (Loshin).
• Plaintext - Data in recognizable format.  This refers to information that has not yet been encrypted or that has already been decrypted (Loshin).
• Receipt - Acknowledgment that information has been received (Menezes).
• Receiver - An entity in a two party communication which is the intended recipient of information (Menezes).
• Replay attack - An attack on an authentication system by recording and replaying previously sent valid messages (Loshin).
• Revocation - Retraction of certification or authorization (Menezes).
• Sender - An entity in a two party communication which is the legitimated transmitter of information (Menezes).
• Set - A set consists of distinct objects which are called elements of the set (Menezes).
• Signature - A means to bind information to an entity (Menezes).
• Steganography - The use of technologies or mechanisms to hide the existence of a message (Loshin).
• Symmetric encryption  (aka: secret key, single key, one key, private key, or conventional encryption) - The same key is used to encrypt and decrypt the ciphertext (Loshin).
• Time Stamping - Recording the time of creation or existence of information (Menezes).
• Trojan Horse - A program that may be used to actively eavesdrop on a computer (Loshin).
• Validation - A means to provide timeliness of authorization to use or manipulate information or resources (Menezes).
• Witnessing - Verifying the creation or existence of information by an entity other than the creator (Menezes).
 
*    The Menezes text lists these four elements as cryptographic goals of all information security objectives.  Specifically the authors write, "A fundamental goal of cryptography is to adequately address these four areas in both theory and practice.  Cryptography (then) is about the prevention and detection of cheating and other malicious activities" (4).
 


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• Mathematical Definitions

The following definitions come directly out of the Handbook of Applied Cryptography.
The math that the definitions involve is precisely taken from general Abstract Algebra theory; once again, terms may relate to specific cryptographic ideas (and thus not use general Abstract Algebra terms).  With an understanding of these basic definitions, a basic understanding of cryptography (the way it is developed and used) will be in place.
 
Definition:  A function is defined by two sets X and Y and a rule f which assigns to each element in X precisely one element in Y.  The set X is called the domain of the function and Y the codomain.  If x is an element of X the image of x is the element in Y which the rule f associates with x; the image y of x is denoted by y=f(x).  If y is an element in Y then a preimage of y is an element x of X for which f(x)=y.  The set of all elements in Y which have at least one preimage is called the image of f, denoted Im(f).
(Often, only the domain X and the rule f are given and the codomain is assumed to be the image of f.)
 
Definition:  A function (or transformation) is one-to-one if each element in the codomain Y is the image of at most one element in the domain X.
 
Definition:  A function (or transformation) is onto if each element in the codomain Y is the image of at least one element in the domain.
 
Definition:  If a function f for X to Y is one-to-one and Im(f)=Y, then f is called a bijection.
 
Definition:  If f is a bijection from X to Y then it is a simple matter to define a bijection g from Y to X as follows:  for each y element of Y define g(y)=x where x element of X and f(x)=y.  This function g obtained from f is called the inverse function of f.
 
Definition:  A function f from a set X to a set Y is called a one-way function if f(x) is "easy" to compute for all x elements of X but for "most" elements y of Im(f) it is "computationally infeasible" to find any x elements of X such that f(x)=y.
(This definition is intended to be simple.  The terms "easy," "most," and "computationally infeasible" should be taken loosely and in relation to theory only.)
 
Definition:  A trapdoor one-way function is a one-way function f where X to Y with the additional property that given some extra information (called trapdoor information) it becomes feasible to find for any given y element of Im(f) and x element of X such that f(x)=y.
 
Definition:  Let S be a finite set of elements.  A permutation p on S is a bijection from S to itself.
 
Definition:  Let S be a finite set and let f be a bijection from S to S.  The function f is called an involution if f=f inverse.  Ex. for a set (1,2,3) 1 goes to 2, 2 goes to 1, and 3 goes to 3.  Thus f=f inverse.
 

 

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• Applications

 
•  How large is large?

 
A key to making cryptography work depends upon how the designer of an encryption system tries to create a scheme for which the best approach to breaking it is through exhaustive search of the key space.  The key space must then be large enough to make an exhaustive search completely infeasible.  An important question then is "How large is large?".  In order to gain some perspective on the magnitude of numbers, the following table lists various items along with an associated magnitude (Menezes 44).
 
 

Reference	Magnitude (approximately)
seconds in a year	3 x 10^7
age of our solar system (years)	6 x 10^9
seconds since creation of solar system	2 x 10^17
clock cycles per year	1.6 x 10^15
binary strings of length 64	1.8 x 10^19 or 2^64
binary strings of length 128	3.4 x 10^38 or 2^128
binary strings of length 256	1.2 x 10^77 or 2^256
number of 75 digit prime numbers	5.2 x 10^72
electrons in the universe	8.37 x 10^77

 
    Table: How large is large? (Menezes 44)
 
 

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•     Example:
 
The following example provides a basic understanding of symmetric key encryption.  Although the example involves the basic ideas and procedures required for symmetric encryption, any basic encryption process can follow the outline sufficiently.
 
To begin, some common terminology will need to be applied in order to set up the example.   Although these terms and ideas are quite general indeed, the Handbook of Applied Cryptography  can be used as a reference to this specific example (Menezes sections 1.4-1.6).
 
Definitions:
 
A denotes a finite set called the alphabet of definition.
 
C denotes a set called the ciphertext space. C consists of strings of symbols from an alphabet of definition, which may differ from that of M.  An element of C is called ciphertext.
 
D denotes a decryption function or decryption transformation.  Each element d of K determines a bijection from C to M.
 
E denotes an encryption function or encryption transformation.  Each element e of K determines a bijection from M to C.
 
K denotes a set called the key space.  An element of K is called a key.
 
M denotes a set called the message space.  M consists of strings of symbols from an alphabet of definition.  An element of M is called a plaintext message or simply a plaintext.
 
Now that the appropriate definitions are in place (in the context of this example) the example itself may be given.
 
Let A = {a,b,c,...,x,y,z} be the English alphabet.
 
Let M and C be the set of all strings of length five over A.
 
The key e is chosen to be a permutation on A.
 
To encrypt, an English message is broken up into groups each having five letters (with appropriate padding if the length of the message is not a multiple of five) and a permutation e is applied to each letter one at a time.
 
To decrypt, the inverse permutation d = e^-1 is applied to each letter of the ciphertext.
 
Note:  In a symmetric key encryption, d = e and e = d, because the encryption and decryption processes are similar.
 
In this example, key e is chosen to be the permutation which maps each letter to the one which is three positions to its right, like this. . .
 
e =

 then, some message
 
m = goodl uckon final sthis isnot secur eakqe
 
can be encrypted to
 
c = E(m) = jrrgo xfnrq ilqdo vxklv lvqrw vhfxu hdnth
 
 
Once again, although the nature of this example is quite easy, very difficult encryption techniques follow the same basic ideas.  By obtaining a clear conception of how easy encryption schemes work, more difficult cryptography methods can be more easily understood.

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• Conclusion
 
     In short, the aim of this report is simply to give a general introduction to the field of cryptography.  With the terms, mathematical definitions, applications, and short example given in this document, the path for further exploration of cryptography has been laid.  In the links and software sections directly following this report are links to on-line sources related to the notions discussed above.  The links and software links not only concern topics in cryptography and Abstract Algebra, but in many math-related areas.

• Mathematical Markup Language: A hypertext document explaining the growing need for a WWW based language for mathematical notation.
 
• Cryptography.com:  A link to an internet destination concerning cryptography in the world today (concerns academia as well as industry).
 
• ABSTRACT ALGEBRA ON LINE:  A good source of general definitions, notations, etc.
 
• Symbolic Mathematical Computation Information Center:  A source describing the aim to automate mathematical computations of all sorts.
 
• Crypto-Log:  The internet guide to cryptography.
 
• A general link to a page provided by the University of Wisconsin that lists thousands of mathematics resources.

• Macsyma: Easy-to-use, powerful, mathematical and finite element software

• Maple: Waterloo Maple innovative technical software solutions.

• Mathcad: A good web source serving a multitude of math purposes.

• Mathematica: A well developed source for industry as well as academia.

• Matlab: MathWorks is the leading developer and supplier of technical computing software in the world.

and John L. Rhodes.  New York:  Academic Press.  1968.

Company.  1998.

Macmillan Company.  1960.

Menezes, Alfred J.  Paul C. van Oorschot.  Scott A. Vanstone.  Handbook of Applied

Cryptography.  New York:  CRC Press.  1997.