Oliver Pell
Cryptology
This is an essay I wrote for a mathematics essay prize. We had a number of possible topics to choose from and I choose cryptology, because I already had a passing interest in thanks to my attempts to code encryption algorithms for my computer programs. It won, by the way ;). This is the original essay with a few corrected spelling mistakes (though there are probably some left). I am currently studying at Imperial College, my home page is here.
Oliver Pell
Introduction
Useful Terminology
A Brief
History of Cryptography & Cryptanalysis
Early Cryptographic Systems
Cryptography During The Two World Wars
Cryptography In The Modern Age
Cryptography
Single Key Cryptography
Substitution Ciphers
Transposition Ciphers
Product Ciphers
Block Ciphers
Stream Ciphers
Data Encryption Standard (DES)
Public Key Cryptography
The Key Distribution Problem
A Solution To The Key Distribution Problem
Properties For A TwoKey Cryptosystem
Knapsack Problem
RivestShamirAdleman (RSA) Algorithm
"Real World" Usage Of The RSA Algorithm
Cryptography in the "Real World"
Applications Of Cryptography
Politics Of Cryptography
Cryptanalysis
Types Of Cryptanalysis
Types Of Cryptanalytic Attacks
Frequency Tables
Cryptanalysis Of Public Key Ciphers
A Triumph of Cryptanalysis  Enigma
What Was Enigma?
Initial Work On Cryptanalysing Enigma
What Made It Possible?
The Turing Bombe
Endnotes
Sources
Authors Notes
The goal of this essay is to introduce the reader, with as little jargon as possible, to the basic theories of cryptography and (to a lesser extent) cryptanalysis.
The science of cryptology is the science of secure communications, formed from the Greek words kryptós, "hidden", and lógos, "word".
Much of the terminology of cryptography can be linked back to the time when only written messages were being encrypted and the same terminology is still used regardless of whether it is being applied to a written message or a stream of binary code between two computers.
CIPHERTEXT 
The encrypted form of the PLAINTEXT. 
CODE 
An unvarying rule for replacing a piece of information with another object, not necessarily of the same sort e.g. ASCII. 
CRYPTANALYSIS 
The science (and art) of recovering information from ciphers without knowledge of the key. 
CRYPTOGRAPHY 
The science of the enciphering and deciphering of messages in secret code or cipher. 
CRYPTOSYSTEM 
A system for encrypting information. 
DECRYPTION 
The process of converting the CIPHER back into PLAINTEXT. 
ENCRYPTION 
The process of converting the PLAINTEXT into a CIPHER. 
KEY 
The secret information known only to the transmitter and the receiver which is used to secure the PLAINTEXT. 
MONOALPHABETIC SUBSTITUTION 
A method of encryption where a letter in the plaintext is always replaced by the same letter in the ciphertext. 
PLAINTEXT 
The source information to be secured. 
POLYALPHABETIC SUBSTITUTION 
A method of encryption where a letter in the plaintext is not always replaced by the same letter in the ciphertext. 
N.B. Ciphers, as in the case of codes, also replace a piece of information (an element of the plaintext that may consist of a letter or word or string of symbols) with another object. The difference is that the replacement is made according to a rule defined by a secret key known only to the transmitter and legitimate receiver(s) in the expectation that an outsider, ignorant of the key, will not be able to undo the replacement and retrieve the original plaintext.
A Brief History of
Cryptography & Cryptanalysis
It seems reasonable to assume that people have tried to conceal information in written form since writing was developed and examples survive in stone inscriptions and papyruses showing that many ancient civilisations including the Egyptians, Hebrews and Assyrians all developed cryptographic systems. The first recorded use of cryptography for correspondence was by the Spartans who (as early as 400 BC) employed a cipher device called a "scytale" to send secret communications between military commanders. The scytale consisted of a tapered baton around which was wrapped a piece of parchment inscribed with the message. Once unwrapped the parchment appeared to contain an incomprehensible set of letters, however when wrapped around another baton of identical size the original text appears.
The Greeks were therefore the inventors of the first transposition cipher and in the fourth century BC the earliest treatise on the subject was written by a Greek, Aeneas Tacticus, as part of a work entitled On the Defence of Fortifications. Another Greek, Polybius later devised a means of encoding letters into pairs of symbols using a device known as the Polybius checkerboard which contains many elements common to later encryption systems. In addition to the Greeks there are similar examples of primitive substitution or transposition ciphers in use by other civilisations including the Romans.
The Polybius checkerboard consists of a five by five grid containing all the letters of the alphabet. Each letter is converted into two numbers, the first is the row in which the letter can be found and the second is the column. Hence the letter A becomes 11, the letter B 12 and so forth.
The Arabs were the first people to clearly understand the principles of cryptography and to elucidate the beginning of cryptanalysis. They devised and used both substitution and transposition ciphers and discovered the use of letter frequency distributions in cryptanalysis. As a result of this by approximately 1412 alKalkashandi could include in his encyclopaedia Subh ala’sha a respectable if elementary treatment of several cryptographic systems. He also gave explicit instructions on how to cryptanalyze ciphertext using letter frequency counts including examples illustrating the technique.
European cryptography dates from the Middle Ages during which it was developed by the Papal and Italian city states. The earliest ciphers involved only vowel substitution (leaving the consonants unchanged). Circa 1379 the first European manual on cryptography, consisting of a compilation of ciphers, was produced by Gabriele de Lavinde of Parma, who served Pope Clement VII. This manual contains a set of keys for correspondents and uses symbols for letters and nulls with several two character code equivalents for words and names. The first brief code vocabularies, called nomenclators, were expanded gradually and for several centuries were the mainstay of diplomatic communications for nearly all European governments. In 1470 Leon Battista Alberti described the first cipher disk in Trattati in cifra and the Traicté de chiffres, published in 1586 by Blaise de Vigernère contained a square table commonly attributed to him as well as descriptions of the first plaintext and ciphertext autokey systems.
By 1860 large codes were in common use for diplomatic communications and cipher systems had become a rarity for this application however cipher systems prevailed for military communications (except for highcommand communications because of the difficulty of protecting codebooks from capture or compromise). During the US Civil War the Federal Army extensively used transposition ciphers. The Confederate Army primarily used the Vigenère cipher and on occasional monoalphabetic substitution. While the Union cryptanalysts solved most of the intercepted Confederate ciphers, the Confederacy in desperation, sometimes published Union ciphers in newspapers, appealing for help from readers in cryptanalysing them.
Cryptography During The Two World Wars
During the first world war both sides employed cipher systems almost exclusively for tactical communications while code systems were still used mainly for highcommand and diplomatic communications. Although field cipher systems such as the U.S. Signal Corps cipher disk lacked sophistication some complicated cipher systems were used for highlevel communications by the end of the war. The most famous of these was the German ADFGVX fractionation cipher.
In the 1920s the maturing of mechanical and electromechanical technology came together with the needs of telegraphy and radio to bring about a revolution in cryptodevices  the development of rotor cipher machines. The concept of the rotor had been anticipated in the older mechanical cipher disks however it was an American, Edward Hebern, who recognised that by hardwiring a monoalphabetic substitution in the connections from the contacts on one side of an electrical rotor to those on the other side and cascading a collection of such rotors, polyalphabetic substitutions of almost any complexity could be produced. From 1921 and continuing through the next decade, Hebern constructed a series of steadily improving rotor machines that were evaluated by the U.S. Navy. It was undoubtedly this work which led to the United States’ superior position in cryptology during the second world war. At almost the same time as Hebern was inventing the rotor cipher machine in the United States, European engineers such as Hugo Koch (Netherlands) and Arthur Scherbius (Germany) independently discovered the rotor concept and designed the precursors to the most famous cipher machine in history  the German Enigma machine which was used during World War 2. These machines were also the stimulus for the TYPEX, the cipher machine employed by the British during World War 2.
The United States introduced the M134C (SIGABA) cipher machine during World War 2. The Japanese cipher machines of World War 2 have an interesting history linking them to both the Hebern and the Enigma machines. After Herbert Yardley, an American cryptographer who organised and directed the U.S. government’s first formal codebreaking efforts during and after the first world war, published The American Black Chamber in which he outlined details of the American successes in cryptanalysing the Japanese ciphers, the Japanese government set out to develop the best cryptomachines possible. With this in mind, it purchased the rotor machines of Hebern and the commercial Enigmas, as well as several other contemporary machines, for study. In 1930 the Japanese’s first rotor machine, code named RED by U.S. cryptanalysts, was put into service by the Japanese Foreign Office. However, drawing on experience gained from cryptanalysing the ciphers produced by the Hebern rotor machines the U.S. Army Signal Intelligence Service team of cryptanalysts succeeded in cryptanalysing the RED ciphers. In 1939 the Japanese introduced a new cipher machine, codenamed PURPLE by U.S. cryptanalysts, in which the rotors were replaced by telephone stepping switches.
The greatest triumphs of cryptanalysis occurred during the second world war  the Polish and British cracking of the Enigma ciphers and the American cryptanalysis of the Japanese RED, ORANGE and PURPLE ciphers. These developments played a major role in the Allies’ conduct of World War 2.
Cryptography In The Modern Age
After World War 2 the electronics that had been developed in support of radar were adapted to cryptomachines. The first electrical cryptomachines were little more than rotor machines where the rotors had been replaced by electronic substitutions. The only advantage of these electronic rotor machines was their speed of operation and they inherited the inherent weaknesses of the mechanical rotor machines.
There is little information available regarding the secret cipher machines of the 1960s and it is likely that this subject will remain the shrouded in rumour until the relevant information is declassified.
The era of computers and electronics has meant an unprecedented freedom for cipher designers to use elaborate designs which would be far too prone to error if handled by pencil and paper, or far to expensive to implement in the form of an electromechanical cipher machine. The main thrust of development has been in the development of block ciphers, beginning with the LUCIFER project at IBM, a direct ancestor of DES (Data Encryption Standard).
Cryptographic systems are generally grouped according to three facts about them:
A substitution cipher is one in which the units of the plaintext (usually letters or numbers) are replaced with other symbols or groups of symbols. The actual order of the units of the plaintext is not changed.
The simplest substitution cipher is one where the alphabet of the cipher is merely a shift of the plaintext alphabet, for example, A might be encrypted as B, C as D and so forth. Of this type of cipher, the most well known is the Caesar cipher, used by Julius Caesar in which A becomes D etc. It is easy to guess that cyclicalshift substitution ciphers of this sort are not at all secure because individual letter frequencies are left completely intact.
There are primarily two approaches that have been used with substitution ciphers to reduce the extent to which the structure of the plaintext, including the letter frequencies, survives into the ciphertext. One of these methods is to treat more than a single letter as one element i.e. two or three successive letters are treated as one unit. The other method is to use several different cipher alphabets.
Historically, for encrypting elements of a plaintext made up of more than a single letter only digraphs (two successive letters) have ever been used. By treating two successive letters as a single unit the extent to which the original letter frequency distribution survives is reduced, thus making the job of the cryptanalysts harder, but not impossible since it can be shown that digraphs themselves have a high degree of correlation. The most well known digraph substitution cipher is the Playfair cipher, invented by Sir Charles Wheatstone.
Charles Wheatstone was a 19^{th} century English physicist, born on February 6^{th}, 1802. As well as devising the Playfair cipher he also invented the Wheatstone bridge, a device for accurately measuring electrical resistance which became widely used in laboratories. He also initiated the usage of electromagnets in electric generators and devised the stereoscope, a device for viewing pictures in three dimensions still used today. The Playfair cipher was named for Lyon Playfair, the first Baron Playfair of St. Andrews, who championed its usage at the British Foreign Office (although he was unsuccessful). 
Here is an example of a Playfair cipher. The aid used to carry out the encryption is a 5 x 5 square matrix similar to a Polybius checkerboard in that it contains all the letters of the alphabet (I and J are treated as the same letter); however a keyword is placed first and then the remaining letters are placed in alphabetical order. 
If the plaintext contains an odd number of letters then an X is appended to the last word to make it an even number. Also, if any of the digraphs consist of identical letters e.g. SUMMER, then an extra letter is placed between them.
The first step in performing the encryption is to locate the two letters from the plaintext in the matrix. There are then several different substitution rules depending on their positioning:
Thus, we can now encrypt the phrase "Merchant Taylors’ School":
Plaintext: 
ME 
RC 
HA 
NT 
TA 
YL 
OR 
SZ 
SC 
HO 
OL 
Ciphertext: 
SC 
OF 
LM 
BI 
AB 
AR 
PU 
BX 
ME 
OV 
RH 
(The last S of "TAYLORS" is paired with a Z to separate it from the first S of "SCHOOL").
The other approach to concealing plaintext structure in the ciphertext involves using several different substitution ciphers. The resulting ciphers, which are generically known as polyalphabetics, have a long history of usage. The bestknown polyalphabetic ciphers are the simple Vigenère ciphers which are named after the 16^{th} century French cryptographer Blaise de Vigenère (see History). Blaise de Vigenère actually produced a more sophisticated autokey cipher, but through an accident of history his name has become attached to this weaker cipher.
For many years this cipher was thought to be impregnable and it is rumoured that a well known scientific magazine pronounced it "uncrackable" as late as 1917, despite the fact that it had been broken by then.
In the simplest system of the Vigenère type the key is a word or a phrase which is repeated over and over again. The plaintext is encrypted using the table shown as Figure 4. The ciphertext letter is found at the intersection of the column headed by the plaintext letter and the row indexed by the key letter. To decrypt the plaintext letter is found at the head of the column determined by the intersection of the diagonal containing the cipher letter and the row containing the key letter. 
It is the periodicity of the repeating key which leads to the weaknesses in this method and its vulnerabilities to cryptanalysis. This periodicity of a repeating key can be eliminated by the use of a runningkey Vigenère cipher, produced when a nonrepeating key is used. However, even though runningkey ciphers eliminate periodicity it is still possible to cryptanalyse them by means of several methods, however the job of the cryptanalyst is made much harder and a cryptanalyst would require a much larger segment of ciphertext to solve a runningkey cipher than one with a repeating key. In fact, the work of Major Joseph Mauborgne of the U.S. Army eventually led to the realisation that the only cryptographic system that is totally secure is that with a onetime completely random key.
Here is how the words "Merchant Taylors School" can be encrypted using this cipher:
Plaintext: 
M 
E 
R 
C 
H 
A 
N 
T 

T 
A 
Y 
L 
O 
R 
S 

S 
C 
H 
O 
O 
L 
Key: 
D 
O 
N 
T 
S 
T 
A 
N 

D 
A 
L 
O 
N 
E 
D 

O 
N 
T 
S 
T 
A 
Ciphertext: 
P 
S 
E 
V 
Z 
T 
N 
G 

W 
A 
J 
Z 
B 
V 
V 

G 
P 
A 
G 
H 
L 
Transposition ciphers rearrange the letters of the plaintext without changing the letters themselves. For example, a very simple transposition cipher is the rail fence, in which the plaintext is staggered between two rows and then read off to give the ciphertext. In a two row rail fence the message MERCHANT TAYLORS’ SCHOOL becomes:
M 
R 
H 
N 
T 
Y 
O 
S 
C 
O 
L 
E 
C 
A 
T 
A 
L 
R 
S 
H 
O 

Which is read out as: MRHNTYOSCOLECATALRSHO.
The rail fence is the simplest example of a class of transposition ciphers called route ciphers. These were quite popular in the early history of cryptography. Generally, in route ciphers the elements of the plaintext (usually in this case single letters) are written on a prearranged route into a matrix agreed upon by the transmitter and receiver. The example above has a two row by ncolumn matrix in which the plaintext is entered sequentially by columns, the encryption route is therefore to read the top row and then the lower.
Obviously, to even approach an acceptable level of security, the route would have to be much more complicated than the one in this example. One form of transposition that has enjoyed widespread use relies on identifying the route by means of an easily remembered keyword. This can be done in several ways. One way, as in this example, is to define the order in which each column is written depending on the alphabetical position of each letter of the keyword relative to the other letters.
Using the keyword CIPHER, a matrix can be written out like the one below:
C 
I 
P 
H 
E 
R 
1 
4 
5 
3 
2 
6 
M 
E 
R 
C 
H 
A 
N 
T 
T 
A 
Y 
L 
O 
R 
S 
S 
C 
H 
O 
O 
L 
Z 
Z 
Z 
Unlike the previous example the plaintext has been written into the columns from left to right as normal, and the ciphertext will be formed by reading down the columns. The order in which the columns are written to form the ciphertext is determined by the key.
This matrix therefore yields the ciphertext: MNOOHYCZCASZETRORTSLALHZ.
The first column is first because C is the earliest in the alphabet, followed by the second to last column because E is the next in the alphabet.
The security of this method of encryption can be significantly improved by reencrypting the resulting cipher using another transposition. Because the product of the two transpositions is also a transposition, the effect of multiple transpositions is to define a complex route through the matrix which would not by itself by easy to define with a simply remembered mnemonic.
When decrypting a route cipher, the receiver simply enters the ciphertext into the agreedupon matrix
according to the encryption route and then simply reads out the plaintext.
In modern cryptography transposition cipher systems serve mainly as one of several methods used as a step in forming a product cipher.
In the days of manual cryptography i.e. without the aid of a computer product ciphers were a useful device for the cryptographer and double transposition ciphers on keywordbased matrices were, in fact, widely used. There was also some use of a particular class of product ciphers called fractionation systems. In a fractionation system a substitution is first made from symbols in the plaintext to multiple symbols (usually pairs, in which case the cipher is called a biliteral cipher) in the ciphertext, which is then superencrypted by a transposition.
One of the most famous field ciphers ever was a fractionation system  the ADFGVX cipher which was employed by the German Army during the first world war. This system was so named because it used a 6 ´ 6 matrix to substitutionencrypt the 26 letters of the alphabet and 10 digits into pairs of the symbols A, D, F, G, V and X. The resulting biliteral cipher is only an intermediate cipher, it is then written into a rectangular matrix and transposed to produce the final cipher which is the one which would be transmitted. 
Here is an example of enciphering the phrase "Merchant Taylors" with this cipher using the key word "Subject".

A 
D 
F 
G 
V 
X 
A 
S 
U 
B 
J 
E 
C 
D 
T 
A 
D 
F 
G 
H 
F 
I 
K 
L 
M 
N 
O 
G 
P 
Q 
R 
V 
W 
X 
V 
Y 
Z 
0 
1 
2 
3 
X 
4 
5 
6 
7 
8 
9 
Plaintext: 
M 
E 
R 
C 
H 
A 
N 
T 

T 
A 
Y 
L 
O 
R 
S 
Ciphertext: 
FG 
AV 
GF 
AX 
DX 
DD 
FV 
DA 

DA 
DD 
VA 
FF 
FX 
GF 
AA 
This intermediate ciphertext can then be put in a transposition matrix based on a different key.
C 
I 
P 
H 
E 
R 
1 
4 
5 
3 
2 
6 
F 
G 
A 
V 
G 
F 
A 
X 
D 
X 
D 
D 
F 
V 
D 
A 
D 
A 
D 
D 
V 
A 
F 
F 
F 
X 
G 
F 
A 
A 
The final cipher is therefore: FAFDFGDDFAVXAAFGXVDXADDVGFDAFA.
Generally, ciphers transform pieces of plaintext of a fixed size into ciphertext. In older, manual systems, these pieces were usually single letters or characters (or sometimes, as in the Playfair cipher, digraphs), since these were the largest units that could be easily encrypted or decrypted by hand. Although systems which operated on sets of three characters and other, larger groups of numbers, were proposed and understood to potentially be more secure they were never implemented because of the extra difficulty in the manual encryption or decryption process. In modern, single key cryptography however, the units of information can be much larger.
A block cipher is a type of symmetrickey encryption algorithm that changes a fixedlength block of the plaintext into the same length of ciphertext. The encryption works by means of a key. Decryption is simply the reverse of the encryption process using the same secret key. The fixed length is called the block size and for modern block ciphers is usually 64 bits. As processors become more sophisticated, however, it is likely that this block size will increase to 128 bits.
Since different plaintext blocks are mapped to different ciphertext blocks, a block cipher effectively provides a permutation of the set of all possible messages. The actual permutation produced during any particular operation is of course secret, and determined by the key.
An iterated block cipher encrypts a plaintext block using a process with several stages (rounds). At each stage the same process (known as a round function) is applied to the data using a subkey (the set of subkeys usually being derived from a user provided key). The number of rounds in an iterated block cipher depends on the desired security level of the encrypted ciphertext and the tradeoff that must be made with performance; fairly obviously a iterated block cipher with a large number of rounds will require more processing time. It is worth noting that in some cases the number of rounds required to provide an accurate level of security will be too large for the cipher to be practical.
An example of an iterated block cipher is a Feistel cipher. Feistel ciphers are a special class of iterated block ciphers. In this type of cipher the ciphertext is calculated from the repeated application of the same round function.
A stream cipher also breaks the plaintext into units, this time it is normally a single character. It then encrypts the n^{th} unit of the plaintext with the n^{th} unit of the key stream. Stream ciphers can be designed to be exceptionally fast, much faster than any block cipher. While the encryption of any particular plaintext with a block cipher will result in the same ciphertext when the same key is used; with a stream cipher, the transformation of the smaller plaintext units will vary, depending on when they are encountered during the encryption process.
A stream cipher generates what is known as a keystream  a sequence of bits, which is used as a key. The encryption process involves combining the keystream with the plaintext.
The keystream can be generated in two ways:
The majority of stream cipher designs are for synchronous stream ciphers.
Interest in stream ciphers is currently attributed to the appealing properties of the onetime pad. A onetime pad, which is sometimes called the Vernam cipher, uses a keystream which is the same length as the plaintext message and consists of a series of bits generated completely at random. Theoretically this should produce ciphertext which is the most secure possible, because since the keystream is random even a cryptanalyst with infinite computational resources can still only guess at the underlying plaintext. While the onetime pad has occasionally seen use in wartime for ultra secret transmissions the fact that the key is as long as the message introduces severe practical problems and so, while theoretically perfectly secure, the onetime pad is generally impractical. Stream ciphers were developed as an approximation to the onetime pad.
At this time there is no de facto standard for stream ciphers although the most widely used stream cipher is RC4, a stream cipher designed by Rivest for RSA Data Security Inc. It is a variable keysize stream cipher with an algorithm based on the use of a random permutation.
Strangely, certain modes of operation of a block cipher transform it into a keystream generator and so, in this way, any block cipher can be used as a stream cipher. Stream ciphers with a dedicated design and typically much faster, however.
One method for generating a keystream is a Linear Feedback Shift Register (LFSR). This is a mechanism for generating a sequence of binary bits. LFSRs are easy to implement and fast operating in both hardware and software however a single LFSR is not secure because over the years a mathematical framework has been developed which allows for the analysis of their output. This problem can be solved by using a shift register cascade, a set of LFSRs connected together so that the behaviour of one of them depends on another. The detailed operation of LFSRs and shift register cascades is beyond the scope of this essay.
Data Encryption Standard (DES)
Originally developed by IBM under the name of LUCIFER, the American NSA (National Security Agency  the US equivalent of GCHQ) and the National Institute of Standards and Technology played a substantial role in the final stages of developing DES. DES is the most well known and widely used symmetric algorithm in the world. The NIST has recertified DES every five years and it was last certified in 1993 but NIST have indicated that they would not recertify DES again, AES (Advanced Encryption Standard) is expected to replace DES.
DES has a 64bit block size and uses a 56bit key during encryption. DES is a 16round Feistel cipher and was originally designed for implementation in hardware. Because it is a singlekey cryptosystem, when used for communication both sender and receiver must know the same secret key which can be used to encrypt or decrypt the message. DES can also be used by a singleuser, for example to store files on a hard disk securely.
No easy attack on DES has yet been discovered, despite research efforts over many years. There is no feasible way to "break" DES other than an exhaustive search  a process which takes 2^{55} steps on average. However, cryptanalysis methods which rely on knowledge of some of the plaintext have had some success. The consensus of the cryptography community is that, if it is not currently so, DES will soon be insecure. As a result of this as of November of 1998 DES was no longer used by the U.S. Government.
Up to this point all the examples have assumed that the encryption process is undertaken with the same key as the decryption process, and that only the sender and the receiver possess the secret key. However, this can cause problems.
A major problem in the practical use of singlekey cryptography is the key distribution problem. This problem basically occurs because both the sender and receiver must hold a copy of the key, but they must also prevent others from gaining a copy of the key.
This does not, on the face of it, appear to be a problem, but it can be one as is probably best illustrated by an example.
Suppose that two individuals, O and L, wish to exchange information securely but can not guarantee the security of the transmission itself. They would probably use some sort of encryption to ensure that even if the message was intercepted its contents would remain secret. In order for this to operate they would both have to know a secret key which could be used to encrypt the data. However, since the transmission medium is not secure they would have to meet in person to decide upon the key. Once this was done, however, they could exchange information happily, encrypted with this secret key, in the knowledge that to anyone without the key it would simply look like garbage. O and L must protect the security of the key in their possession since if a cryptanalyst, C, obtained it, he could read, alter or fake a message from either O or L.
This appears to be working perfectly, except a problem could soon arrive. Suppose that O or L then wanted to correspond with another individual, T. If they were to give T the key they would further compromise it because C would now have another source from which he might obtain it. In order to ensure that each message is not intercepted by another party both O and L would have to create different keys to exchange with T so that they would all each hold two keys (because otherwise O could alter T’s message to L and so forth).
Now consider a system with 1,000 members, all of whom wish to communicate in secret with each other. In this case, each individual would need to hold a key for every individual besides himself, in other words 999 keys for other people. Each individual would also have to protect those 999 keys from being compromised.
It is possible to calculate the number of keys present in a system with any number of members using these facts as I have done below, calculating the number of keys required by multiplying the number of members minus one by the number of members and divided by two.
Keys = [n ´ (n1)]/2
Members 
#/Members^{2} 
# of keys required 
2 
4 
1 
3 
9 
3 
4 
16 
6 
5 
25 
10 
6 
36 
15 
7 
49 
21 
8 
64 
28 
9 
81 
36 
10 
100 
45 
100 
10000 
4950 
1000 
1000000 
499500 
The number of keys is divided by two because some of the keys are duplicated (the key A uses to send to B is the same which B uses to send to A). The reason why I have included a column containing the number of members squared is made obvious below.
Here is a graph of the number of members squared against the number of keys required (I have excluded the final two rows to ensure that there is a practical scale). 
As you can see the number of different keys required is nearly proportional to the square of the number of members in the system. This is the key distribution problem.
A Solution To
The Key Distribution Problem
A solution to the key distribution problem can be found in public key, or twokey, cryptography. Public Key cryptography is based on the idea that a user can possess two keys  one public and one private key. The public key can only be used to encrypt the data to be sent and the private key can only be used to decrypt it. Although singlekey cryptography has been in use for centuries, public key cryptography is a relatively new invention with the first discussion about the subject in open literature being in 1976.
The fact that anyone can use a single "locking" key to encrypt a message which they are confident can still only be read by a single authorised user means that the number of keys required can be greatly reduced. For example, in the 1000 member system above the number of keys necessary could be reduced from 499,500 to 2000 (1000 public and 1000 private keys).
For example I will use a far simpler three user system. An individual, say P, could distribute his public key to two other individuals, A and C. A could then send a message to P by encrypting it with the public key. When P receives the message he could then decrypt and read the message using his private key. Supposing C was a nosy individual then he might attempt to spy on the communications between A and P, however, because he only possesses the encrypting key, he can not decrypt the message.
There is a problem with this solution however. In this example, although C can not read or alter a message sent to P by A, he or she could easily fake a message because C has the same public key as A. Therefore, with a public key system the ability to authenticate messages has been given up in return for privacy. In many cases this will not be a problem however there are times when it will be so.
There is an alternative to this method. A could send P and C his or her private (unlocking) key and keep the public key secret. A could then encrypt messages with his public key and send them to either P or C. This system has the advantage that the person receiving the message knows that it must have come from A; however, both P and C can now decrypt the message. In this example secrecy has been sacrificed in order to maintain an ability to authenticate the message.
This is a problem with the public key system which can only be solved by increasing the number of keys in the system. It can be solved however, by combining the two methods outlined above. If each user had two sets of public and private keys and distributed one key from each set then the capability to authenticate messages and to keep them secret would be maintained.
For example, another three individuals P, D and W are using this system. Each user holds, in addition to their normal private (unlocking) key and everyone else’s public keys, a public key which is paired with a private key which they then distribute. P could then encrypt a message with his own public key, thus authenticating it, and then encrypt it with the public key of whoever the message was intended for (thus ensuring secrecy).
Under this system each message is encrypted twice, once in a way which only the intended receiver can decrypt it, and once in which only the authentic sender could have encrypted it. Even though the number of keys required has been increased it still does not approach the number of keys which would be required for a singlekey system of the same size.
It is this idea of nested encryption processes with public and private keys which makes publickey cryptography so attractive.
Properties
For A TwoKey Cryptosystem
For twokey cryptography to be possible a cryptosystem must have the following properties:
At present no cryptosystem ever devised has satisfied all of these conditions in a provable way. However, cryptographers have devised cryptosystems of this sort by starting with a difficult mathematical problem such as factoring a product of large prime numbers and attempting to make the cryptanalysis of the scheme be equivalent to solving the problem. If this can be done then the security of the scheme is at least as good as the underlying problem is hard to solve.
There are a number of problems that are commonly used as the basis for public key cryptosystems.
Ralph Merkle, XEROX and Martin Hellman, Stanford University, put forward one of the best known proposals for a publickey cryptosystem in 1976. In 1997 the six cofounders of public key cryptography were awarded the Paris Kanellakis Theory and Practice Award, named after a researcher who died in 1995, for their work. Hellman and Merkle were among those who received the award. 
They suggested using the knapsack, or subsetsum, problem as the basis for a public key cryptosystem. This problem entails determining whether a number can be expressed as a sum of some subset of a given sequence of numbers and, more importantly, which subset has the desired sum.
Given a sequence of numbers A, where A = (a_{1} ... a_{n}), and a number C, the knapsack problem is to find a subset of a_{1} ... a_{n} which sums to C.
For a specific example:
n = 5, C = 14, A = (1, 10, 5, 22, 3)
Solution = 14 = 1 + 10 + 3
In general, all the possible sums of all subsets can be expressed by:
m_{1}a_{1} + m_{2}a_{2} + m_{3}a_{3} + ... + m_{n}a_{n} where each m_{i} is either 0 or 1.
The solution is therefore a binary vector M = (1, 1, 0, 0, 1).
There is a total number 2^{n} of such vectors (in this example 2^{5} = 32)
Obviously not all values of C can be formed from the sum of a subset and some can be formed in more than one way. For example, when A = (14, 28, 56, 82, 90, 132, 197, 284, 341, 455, 515) the figure 515 can be formed in three different ways but the number 516 can not be formed in any ways.
The only known solution to the general knapsack problem (as above) is a brute force test  testing all the possible subsets until either the sum is found or you run out of subsets.
However, there is a special case if A is sorted and when each progressive term is larger than the one which follows it (called the simple knapsack problem) and there is an easy solution (this type of sequence is described as superincreasing). The value of C is compared to each value of A, starting with the largest. If A is larger than C then that number is not in the subset, if it is smaller then that value is in the subset. The next value is then compared to C less the sum of all the values found already.
For example:
A = (1, 3, 5, 10, 22), C = 14
14 £ 22 m_{5} = 0
14 ³ 10 m_{4} = 1
4 £ 5 m_{3} = 0
4 ³ 3 m_{2} = 1
1 ³ 1 m_{1} = 1
M = (1, 1, 0, 1, 0)
The cryptographer uses a secret transformation to turn a simple knapsack into a hard one. Legitimate users, knowing the secret transformation, could easily invert the hard knapsack back to the simple knapsack. This can be done using a private multiplier and a private modulus, the original sequence is multiplied by the multiplier and then dividing by the modulus (finds the remainder). This has the effect of scrambling the values. The plaintext is then converted into a set of binary bits which are then multiplied by the values of the public sequence and the result is recorded. To decrypt them is the equivalent of the hard knapsack problem for anyone who doesn’t know the secret information.
For example:
Taking the following private information:
A = (1, 3, 5, 10), b = 20, k = 7 (b is the private modulus, k is the private multiplier)
\ A = (7 ´ 1 mod 20, 7 ´ 3 mod 20, 7 ´ 5 mod 20, 7 ´ 10 mod 20)
Public key = (7, 1, 15, 10)
To encrypt the plaintext we must convert in into a binary number. For this simple example I will convert the letter to a value describing its position in the alphabet.
Let plaintext P = m (= 13) = 1101
Assigning each of these values a "size" from the public key and multiplying produces the following code, C.
C = 1 ´ 7 + 1 ´ 1 + 0 ´ 15 + 1 ´ 10 = 18 (= r)
The code letter is therefore r.
To decrypt this code letter without the secret transformation would be computationally infeasible, so we must perform the operation on the letter r (18).
To decrypt, use the inverse key k^{1}.
Let k^{1} = 3 (any value for which k ´ k^{1} mod 20 = 1)
Let C = 18
C’ = 3 ´ 18 mod 20 = 14
Now we can use the value of 14 to solve the simple knapsack problem.
(1, 3, 5, 10)
14 ³ 10 m_{4} = 1
4 £ 5 m_{3} = 0
4 ³ 3 m_{2} = 1
1 ³ 1 m_{1} = 1
\ M = (1, 1, 0, 1) corresponds to binary 1101 = 13 = m
Unfortunately, it has been shown that it is possible, in a reasonable length of time, to derive the private key from the public key thus knapsackbased cryptosystems have been shown not to be secure. However, although of little practical use, they are an excellent introduction to the idea of public key encryption.
RivestShamirAdleman (RSA)
Algorithm
The main contenders for twokey cryptosystems in the mid1980s were those based on factoring large integers. Of these, the best known is the RSA algorithm, developed by Ron Rivest, Adi Shamir, and Leonard Adleman in 1977.
In this system a user chooses a pair of prime numbers so large that factoring the product is beyond all computing capabilities. This is possible because testing for primes is easy whereas factoring the product is very difficult.
In practise, RSA works as follows. The user takes two large primes, p and q, and computes their product n = pq. n is called the modulus. The user then chooses a number, e, less than n and with no common factors (except 1) with (p1)(q1). Another number, d, is then found, subject to the condition that (ed1) is divisible by (p1)(q1). e is called the public exponent, d is called the private exponent. The public key is therefore the pair n and e, the private key is the pair n and d. The factors of n, p and q, can either be kept secret with the private key or destroyed.
It is currently virtually impossible to obtain the private key, d, from the public key (n and e). However, if n could easily be factored into p and q then a cryptanalyst could obtain the private key, d. The security of RSA is therefore based upon the assumption that factoring is difficult.
Here are the stages in sending a message using this method:
c = m^{e} mod n (where e and n are Ms public key pair).
m = c^{d} mod n
For example:
Let p = 5, q = 11, n = pq = 55
The least common multiple of (p1)(q1) is 20 = 2^{2} ´ 5.
Therefore, in this case, any key, e, not divisible by 2 or 5 will have a matching key, d (e must be relatively prime to (p1)(q1) for it to be the key).
Let e = 7
(ed 1) mod (p1)(q1) = 0 \ d = 3
Let the plaintext message, m = b = 2
\ The ciphertext, c = m^{e} mod n = 2^{7} mod 55 = 18
To decrypt this information it is necessary to know d.
m’ = c^{d} mod n = 18^{3} mod 55 = 2 \ The message has been successfully decrypted.
This example used small primes so it can be seen that the product, n, is not at all difficult to factor to retrieve the original primes. In using RSA it has always been suggested to use "strong" primes which have certain properties making their product especially difficult to factor using certain factoring methods. However, advances in factoring techniques over the last decade have near completely negated the advantage of strong primes, the elliptic curve factoring algorithm is one such advance. Therefore, it is not choosing traditionally "strong" primes which matters but rather choosing large primes.
In 1997, a specific assessment of the security of 512bit RSA keys shows that one may be factored for less than $1,000,000 in cost and eight months of effort. It is therefore believed that 512bit keys provide insufficient security for anything other than shortterm needs. RSA Laboratories currently recommends key sizes of 768 bits for personal use, 1024 bits for corporate use, and 2048 bits for extremely valuable keys like the rootkey pair used by a certifying authority. Security can be increased by changing a users keys regularly and it is typical for a user’s key to expire after two years (the opportunity to change keys also allows for a longer length key to be chosen).
It should be noted that the key sizes for RSA (and other publickey techniques) are much larger than those for block ciphers like DES, but the security of an RSA key cannot be compared to the security of a key in another system purely in terms of length. It should also be noted that although increasing the length of the modulus may increase the security of the system it also significantly increases the amount of time required to perform encryption, decryption and authentication  doubling the modulus will, on average, quadruple the amount of time taken for encryption and increase the time taken for decryption by a factor of eight.
"Real
World" Usage Of The RSA
Algorithm
DES, and other block ciphers, are much faster in operation than RSA. Typically, in software, DES is approximately 100 times faster than RSA and in hardware can be between 1000 and 10000 times faster depending on the implementation. Because of the length of time taken for public key encryption a public key system such as RSA is often used in conjunction with a secretkey system such as DES. In this case the message is encrypted with a randomly chosen DES key and the key itself is encrypted and sent with RSA. This method combines the low number of keys required for RSA with the highspeed operation of DES.
RSA is the most widely used publickey cryptosystem available currently and has often been referred to as a defacto standard regardless of official recognition. In fact, RSA is part of many official standards worldwide. These include ISO 9796 which lists RSA as a compatible cryptographic algorithm and many internet standards and proposals including S/MIME.
Cryptography in the "Real World"
In the information dependent world in which we now live cryptography can be found all around us, often in places where you would not expect it. When people think about encryption they tend to think about vast computer banks processing military and diplomatic communications, or a world war two rotor cipher machine slowly deciphering an order. In reality, cryptography  although obviously essential for the military and diplomatic services  has many commercial uses and applications. From protecting confidential company information, to protecting a telephone call, to allowing someone to order a product on the Internet without the fear of their credit card number being intercepted and used against them, cryptography is all about increasing the level of privacy of individuals and groups. For example, cryptography is often used to prevent forgers from counterfeiting winning lottery tickets. Each lottery ticket can have two numbers printed onto it, one plaintext and one the corresponding cipher. Unless the counterfeiter has cryptanalysed the lottery’s cryptosystem he or she will not be able to print an acceptable forgery.
In a world where virtually all data of any importance is held on a computer system the necessity of cryptography cannot be disputed.
Widespread use of cryptosystems is something most governments are not particularly happy about  precisely because it threatens to give more privacy to the individual, including criminals. For many years, police forces have been able to tap phone lines and intercept mail, however, in an encrypted future that may become impossible.
This has lead to some pretty strange decisions on the part of governments, particularly the United States government. In the United States, cryptography is classed as a munition and the export of programs containing cryptosystems is tightly controlled. In 1992, the Software Publishers Association reached agreement with the State Department to allow the export of software that contained RSA's RC2 and RC4 encryption algorithms, but only if the key size was limited to 40 bits as opposed to the 128 bit keys available for use within the US. This significantly reduced the level of privacy produced. In 1997 this was increased to 56 bits. The US government has proposed several methods whereby it would allow the export of stronger encryption, all based on a system where the US government could gain access to the keys if necessary, for example the clipper chip.
The resolution of this issue is regarded to be one of the most important for the future of ecommerce.
It is beyond the scope of this essay to deal with all types of cryptanalysis so I will give a brief overview and an example of where cryptanalysis has been successful. Unlike cryptography which is a clearly defined science, cryptanalysis is as much an art as it is a science. Success in cryptanalysising a cipher is a flash of inspiration almost as often as it the result of using crytanalysis techniques alone.
There are several distinct types of cryptanalytic attack. The type used depends on the type of cipher and how much information the cryptanalyst has.
Types Of
Cryptanalytic Attacks
A standard cryptanalytic attack is to determine the key which maps a known plaintext to a known ciphertext. This plaintext can be known because it is standard or because it is guessed. If the plaintext segment is guessed it is unlikely that its exact position is known however a message is generally short enough for a cryptanalyst to try all possible positions in parallel. In some systems a known ciphertextplaintext pair will compromise the entire system however a strong encryption algorithm will be unbreakable under this type of attack.
A brute force attack requires a large amount of computing power and a large amount of time to run. It consists of trying all possibilities in a logical manner until the correct one is found. For the majority of encryption algorithms a brute force attack is impractical due to the large number of possibilities.
Another type of brute force attack is a dictionary attack. This essentially involves running through a dictionary of words in the hope that the key (or the plaintext) is one of them. This type of attack is often used to determine passwords since people usually use easy to remember words.
In a ciphertext only attack the cryptanalyst has only the encoded message from which to determine the plaintext, with no knowledge whatsoever of the actual message. A ciphertext only attack is presumed to be possible, if not easy. In fact, an encryption techniques resistance to a ciphertext only attack is considered the basis for its cryptographic security.
In a chosen plaintext attack the cryptanalyst has the capability to find the ciphertext corresponding to an arbitrary plaintext message of his or her own choosing. The likelihood of this type of attack being possible is not much. Codes which can survive this attack are considered to be very secure.
In a chosen ciphertext attack the cryptanalyst can choose an arbitrary ciphertext and find the corresponding decrypted plaintext. This attack can be used in public key systems, where it may reveal the private key.
In an adaptive chosen plaintext attack the cryptanalyst can determine the ciphertext of chosen plaintexts in an iterative process based on previous results. This is the general name for a method of attacking product ciphers called "differential cryptanalysis".
The cryptanalysis of singlekey cryptosystems depends on one simple fact  that some traces of the original structure of the plaintext may be visible in the ciphertext. For example, in a monoalphabetic substitution cipher where each letter in the plaintext is replaced by a letter in the ciphertext which is the same each time, a simple analysis of a sizeable portion of ciphertext can be used to retrieve most of the plaintext.
Here is a monoalphabetic substitution cipher of a random paragraph of English:
UFMDHQAQTMGRG BX GRAZTW PWM
UFMDHBGMGHWOG VWDWAVG BA BAW
GRODTW XQUH AQOWTM HCQH HFQUWG
BX GHFIUHIFW BF DQHHWFA RA HCW
DTQRAHWLH OQM GIFJRJW WAUFMDHRBA
QAV SW VRGUWFARSTW RA HCW
URDCWFHWLH.
W occurs 20 times in the cipher, H occurs 16 times. From this information and using the frequency table to the left it would be possible for a cryptanalyst to recover the majority of the plaintext. The usefulness of analysing the structure of the ciphertext can be reduced by any encryption procedure that attempts to disguise the structure. However, eliminating the underlying structure is harder than it would first seem. Digraphs, for example, show a strong frequency distribution  TH occurs very often, about 20 times more often than HT and so on. With tables of digraph frequencies it is possible to recover the underlying plaintext, however, the amount of ciphertext required would be much greater. In the heyday of manual cryptanalysis huge volumes of word patterns were compiled. Although these only contained the most obvious and easily recognised word patterns they were still of importance if they could provide that vital clue which could break the entire cipher. 
Cryptanalysis
Of Public Key Ciphers
Public key cryptography requires a fundamentally different type of cryptanalysis than is used for single key cryptanalysis. Because public key cryptography relies on "hard" mathematical problems, their cryptanalysis is essentially research into solving the underlying mathematical problems. Cryptanalysis of public key ciphers is therefore virtually indistinguishable from research into any other area of mathematics.
For example, to "crack" the RSA algorithm and obtain the private key from the public key would essentially involve research into factoring algorithms. Factoring is a very active field of research among mathematicians and computer scientists. The best generalpurpose factoring algorithm today is the Number Field Sieve but even so, the running time to search through all the possibilities is very long.
In 1980 it was possible to factor a 50digit number with 1,000,000,000,000 computer operations. By 1984 factoring algorithms had advanced to the stage where a 75digit number could be factored in the same number of operations. If a mathematical advance was made which enabled a 150 (or more) digit number to be factored in the same number of operations it would allow cryptanalysts to break many RSA implementations.
A Triumph of Cryptanalysis  Enigma
No look at cryptanalysis, however brief, with even the slightest element of historical content, could finish without at least briefly covering the tale of how the Enigma was broken.
The efficiency of the German armed forces during World War 2 would not have been possible without radio communication. Messages sent over the radio had to be encrypted and the encryption system they used was adapted from one which was commercially available before the war. Despite the fact that it was modified so as to make it harder to crack it was broken by a group working at the GCHQ.
The Enigma machine consisted of a 26 letter keyboard for input. The output was read off 26 lamps which each corresponded to a letter. The encipherment was performed by a device called a "scrambler" that was made of three rotating wheels on a common spindle and a plugboard known as a "Steckerboard" that added an additional level of security. 
The keyboard and lampboard were both connected to a common cabling bus. When no keys were being held down all the lamps were connected to the bus and the current present on any particular wire would cause its lamp to glow. When a key was pressed a current was placed on its wire but the corresponding lamp was disconnected from the bus.
Since no letter could ever be enciphered to itself this was a good setup. When a key was pressed one wire on the bus would have current applied to it and the other 25 could respond with the enciphered letter. The one that actually did respond was the result of the other components.
The original, commercial, Enigma did not have the plugboard. The 26 wire bus was connected to a circular set of contacts that sat to the right hand side of the three rotor scrambler. These three rotors were identical except in their internal wiring. The rotors had two sets of 26 connectors, one on the left and one on the right. All three rotors had to be rotated by hand and by doing so the "key" could be changed. Because each rotor could be set independently there were 26 ´ 26 ´ 26 (17576) possible rotor settings. This would not have provided a secure ciphering system so every time a key was pressed one of the rotors (known as the "fast" rotor) was advanced one position so that even if the same key was pressed three times the result would be three different characters. When the fast rotor got to a certain position it would cause the middle rotor to be rotated to one position, which in turn could cause the slow rotor to move one position.
Initial Work
On Cryptanalysing Enigma
The initial work on cryptanalysing Enigma was done by the Polish. The earliest Polish work on the intercepting German machine ciphers had begun in 1928, right after the systems introduction by the German Army. However, no progress was made during the first four years. Then the Polish Cipher Bureau decided to recruit three young mathematicians. Among them was Marian Rejewski. On September 1, 1932 Rejewski and his two somewhat younger colleagues began work at the Cipher Bureau in Warsaw. In early October Rejewski began work on the Enigma and was supplied with an obsolete commercial Enigma machine which had been brought in Germany.
The whole, complicated process of mastering the German Enigma was ultimately concluded in the early days of 1933. This included combinations of mathematics, statistics, computational ability and guesswork. The breaking of Enigma involved two distinct processes. The first was the theoretical reconstruction of the Enigma cipher device itself including the internal wiring and the interdependence between different components of the machine. This knowledge of the machine enabled the Poles to build doubles of the Enigma, The second was the creation of methods for recovering the Enigma keys (starting positions) based exclusively on the basis of intercepts.
At the beginning of the second world war, realising that a German invasion was imminent, the Polish handed over all the information including the duplicate Enigma models over to the British.
With the level of sophistication of the Enigma machines it should have been unbreakable. However, the Germans had a number of procedural flaws which allowed the British and Polish to break the cipher.
Some of these were just stupid. For example, the Germans reused the monthly code book settings. Others were less obvious. It was discovered that the German military messages were very similar in nature and, once they were decrypted, very careful records were kept regarding how the messages tended to be formatted. When the British thought they had a likely idea of what the start of the message was, they would use this to reduce the many trillions of possible Enigma settings to a small enough number to be tested individually.
The design of the Turing bombe is normally credited to two British mathematicians  Alan Turing and Gordon Welchman, working in secret at Bletchley Park. The information handed over to the British by the Polish was much more advanced than anything the British (or anyone else) had accomplished . One of the things the Polish had developed was a machine they called a Bomba (the Polish word for bomb  probably named because of the ticking sound it made). The Polish machine and the British one worked on different algorithms but were mechanically very similar. This provided Turing with a starting point. 
The Bombe was effectively a collection of Enigma machines that were all working together. According to Welchman there were twelve sets of Enigma rotors in the Bombe, but there is also other evidence that larger Bombes existed. None of the Bombes survived the war so it is difficult to be sure.
The theorem the Bombe was based on the loop that exists between the crib (a guessed sequence of letters) and the equivalent cipher text. The method Turing devised to test this was based upon the mathematical technique reductio ad adsurdum. For every one of the possible rotor settings Turing would start by assuming that this was the correct setting and then try to disprove this. If the procedure did disprove the setting the machine would automatically advance to the next setting to be tested. If it did not the setting was recorded, and then the procedure continued to try and find other possible settings.
Although Moore’s Law may cease to hold at some point in the near future it is still an inevitability that the power of computer processors will increase with each passing year.
Within a matter of a few decades computers many more times powerful than those available today will be in use. These more powerful processors will allow more complicated encryption algorithms to be run within a reasonable time span. This will inevitably lead to the steadily increasing power of encryption systems. The same computing power which makes these computers useful to cryptographers will make them useful to cryptanalysts. The computer shown in Figure 10 is one in use at the Government Communications Headquarters in Cheltenham, with computers hundreds of times more powerful it will become plausible to decrypt many ciphers by a simple brute force attack where it had not been so before. This will probably spell the end of existing cryptosystems, however, other, more complicated ones will be developed in their place. We have already seen this happen. In November 1998 the US Government discontinued its usage of the DES system because what was considered secure upon the algorithms conception is no longer so. A variant of DES called tripleDES will be used until AES (Advanced Encryption System) is ready. 
It also seems likely that better factoring algorithms will be developed in the future. Coupled with increasing computing power this may mean the end of RSA, or simply that people will be using larger and larger primes to stay ahead of the cryptanalysts.
There is an inherent danger in attempting to make any definite statements about the future, more often than not they are hilariously inaccurate. However, in my opinion, I believe that the balance between cryptographer and cryptanalyst will be roughly maintained although I think it is likely that the actual volume of encrypted traffic will increase due to individuals using cryptography to protect their privacy. This is of course the great fear of law enforcement agencies who have, for many years, enjoyed the ability to tap phone lines. Attempts to setup some sort of escrow key system such as what the United States government attempted with the clipper clip may be successful, or they may not. Whatever happens, however, cryptology promises to remain an area of interesting developments for a long time yet.
CDs
Encyclopedia Britannica CD 99
Web Sites
Forensic Files
A Cryptography Compendium
Beginners Guide to Cryptography
RSA Laboratories
Ralph Merkle’s Homepage
German Enigma Cipher Machine
Enigma and the Turing Bombe
Sci.Crypt FAQ
This list represents all the sources which I actually used information from in producing this essay. This is not a complete list of all the information sources I consulted.