1. The University of Adelaide, School of Computer Science
14 April 2017
Security in Computing, 4th Ed, Pfleeger
Chapter 2
Elementary Cryptography
Chapter 2 — Instructions: Language of the Computer

2. In this chapter Concepts of encryption
Cryptanalysis: how encryption systems are "broken"
Symmetric (secret key) encryption and the DES and AES algorithms
Asymmetric (public key) encryption and the RSA algorithm
Key exchange protocols and certificates
Digital signatures
Cryptographic hash functions

3. Cryptography
Cryptography (secret writing) is the strongest tool for controlling against many kinds of security threats.
Well-disguised data cannot be read, modified, or fabricated easily.
Cryptography is rooted in higher mathematics
Group and field theory, computational complexity, and even real analysis, not to mention probability and statistics.
Fortunately, it is not necessary to understand the underlying mathematics to be able to use cryptography.

4. Terminology and Background
Consider the steps involved in sending messages
from a sender, S
to a recipient, R
If S entrusts the message to T, who then delivers it to R, T then becomes the transmission medium.
If an outsider, O, wants to access the message (to read, change, or even destroy it), we call O an interceptor or intruder.
Encryption is a means of maintaining secure data in an insecure environment.

5. Terminology
Encryption is the process of encoding a message so that its meaning is not obvious
Decryption is the reverse process, transforming an encrypted message back into its normal, original form.
Alternatively, the terms encode and decode or encipher and decipher are used instead of encrypt and decrypt
A system for encryption and decryption is called a cryptosystem.

6. Terminology
The original form of a message is known as plaintext, and the encrypted form is called ciphertext.

7. Terminology
The original form of a message is known as plaintext, and the encrypted form is called ciphertext.

8. Terminology
For convenience, we denote a plaintext message P as a sequence of individual characters
P = <p1, p2, …, pn>.
Similarly, ciphertext is written as
C = <c1, c2, …, cm>.
We write C = E(P) and P = D(C), where C represents the ciphertext, E is the encryption rule, P is the plaintext, and D is the decryption rule.
What we seek is a cryptosystem for which P = D(E(P)). In other words, we want to be able to convert the message to protect it from an intruder, but we also want to be able to get the original message back so that the receiver can read it properly.

9. Encryption Algorithms
The cryptosystem involves a set of rules for how to encrypt the plaintext and how to decrypt the ciphertext.
The encryption and decryption rules, called algorithms, often use a device called a key, denoted by K, so that the resulting ciphertext depends on the original plaintext message, the algorithm, and the key value.
C = E(K, P)

10. Encryption Algorithms
It would be very expensive for you to contract with someone to invent and make a lock just for your house.
Also, you would not know whether a particular inventor's lock was really solid or how it compared with those of other inventors.
A better solution is to have a few well-known, well-respected companies producing standard locks that differ according to the (physical) key
Then, you and your neighbor might have the same model of lock, but your key will open only your lock.
In the same way, it is useful to have a few well-examined encryption algorithms that everyone could use, but the differing keys would prevent someone from breaking into what you are trying to protect.

11. Encryption Algorithms
Sometimes the encryption and decryption keys are the same, so P = D(K, E(K,P)). This form is called symmetric encryption because D and E are mirror-image processes.
At other times, encryption and decryption keys come in pairs. Then, a decryption key, KD, inverts the encryption of key KE so that
P = D(KD, E(KE,P)). Encryption algorithms of this form are called asymmetric
An encryption scheme that does not require the use of a key is called a keyless cipher.

12. Encryption Algorithms

13. cryptology
Cryptography means hidden writing, and it refers to the practice of using encryption to conceal text.
Cryptanalyst studies encryption and encrypted messages, hoping to find the hidden meanings.

14. cryptology
Both a cryptographer and a cryptanalyst attempt to translate coded material back to its original form. Normally, a cryptographer works on behalf of a legitimate sender or receiver, whereas a cryptanalyst works on behalf of an unauthorized interceptor.
Cryptology is the research into and study of encryption and decryption; it includes both cryptography and cryptanalysis.

15. Cryptanalysis A cryptanalyst's chore is to break an encryption.
cryptanalyst attempts to deduce the original meaning of a ciphertext message.
Better yet, he or she hopes to determine which decrypting algorithm matches the encrypting algorithm so that other messages encoded in the same way can be broken.

16. Cryptanalyst can attempt to do …
Break a single message
Recognize patterns in encrypted messages, to be able to break subsequent ones by applying a straightforward decryption algorithm
Infer some meaning without even breaking the encryption, such as noticing an unusual frequency of communication or determining something by whether the communication was short or long

17. Cryptanalyst can attempt to do …(cont.)
Deduce the key, to break subsequent messages easily
Find weaknesses in the implementation or environment of use of encryption
Find general weaknesses in an encryption algorithm, without necessarily having intercepted any messages

18. Information needed by a cryptanalyst
A cryptanalyst works with a variety of pieces of information: encrypted messages, known encryption algorithms, intercepted plaintext, data items known or suspected to be in a ciphertext message, mathematical or statistical tools and techniques, properties of languages, computers, and plenty of ingenuity and luck.
Each piece of evidence can provide a clue, and the analyst puts the clues together to try to form a larger picture of a message's meaning in the context of how the encryption is done.

19. Attack models Attack models for the cryptanalysis Ciphertext-only:
is an attack model for cryptanalysis where the attacker is assumed to have access only to a set of ciphertexts.
The attack is completely successful if the corresponding plaintexts can be deduced, or even better, the key.
Known-plaintext
is an attack model for cryptanalysis where the attacker has samples of both the plaintext and its encrypted version (ciphertext). These can be used to reveal further secret information such as secret keys.
Chosen-plaintext
is an attack model for cryptanalysis which presumes that the attacker has the capability to choose arbitrary plaintexts to be encrypted and obtain the corresponding ciphertexts.[1] The goal of the attack is to gain some further information which reduces the security of the encryption scheme.

20. Breakable Encryption
An encryption algorithm is called breakable when, given enough time and data, an analyst can determine the algorithm.
However, an algorithm that is theoretically breakable may in fact be impractical to try to break.
Ex., consider a 25-character message that is expressed in just uppercase letters. A given cipher scheme may have 2625 (approximately 1035) possible decipherments
If your computer could perform on the order of 1010 operations per second, finding this decipherment would require on the order of 1025 seconds.
Infeasible to compute

21. Breakable Encryption
Two other important issues must be addressed when considering the breakability of encryption algorithms.
First, the cryptanalyst cannot be expected to try only the hard, long way.
ingenious approach might require only 1015 operations. => 1015 operations take slightly more than one day
Second, estimates of breakability are based on current technology.
Things that were infeasible in 1940 became possible by the 1950s
A conjecture known as "Moore's Law" asserts that the speed of processors doubles every 1.5 years, and this conjecture has been true for over two decades.
It is risky to pronounce an algorithm secure just because it cannot be broken with current technology, or worse, that it has not been broken yet.

22. Representing Characters
We begin with the encryption of messages written in the standard 26-letter English-alphabet, A through Z.
Convention: plaintext is written in UPPERCASE letters, and ciphertext is in lowercase letters
Because most encryption algorithms are based on mathematical transformations, they can be explained or studied more easily in mathematical form.

23. Representing Characters
Consider performing arithmetic on the "letters" of a message
Expressions such as A + 3 = D or K - 1 = J have their natural interpretation.
Arithmetic is performed as if the alphabetic table were circular (modular arithmatic)
every result of an arithmetic operation is between 0 and 25
Ex. Y + 3 = B (and B – 3 = Y)
Two simple forms of encryption:
substitutions, in which one letter is exchanged for another
transpositions, in which the order of the letters is rearranged

24. Substitution Ciphers The Caesar Cipher ci = E(pi) = pi + 3
A full translation chart of the Caesar cipher is shown here.
Using this encryption, the message
TREATY IMPOSSIBLE
would be encoded as
T R E A T Y I M P O S S I B L E
w u h d w b l p s r v v l e o h
The pattern pi + 3 was easy to memorize and implement, however, it is easy break

25. Cryptanalysis of the Caesar Cipher
Many clues on the "TREATY IMPOSSIBLE” ciphertext
the space between the two words is preserved in the ciphertext
double letters are preserved: The SS is translated to vv
when a letter is repeated, it maps again to the same ciphertext as it did previously. So the letters T, I, and E always translate to w, l, and h.
These clues make this cipher easy to break.

26. Cryptanalysis of the Caesar Cipher
Suppose you are given the following ciphertext message, and you want to try to determine the original plaintext.
Ciphertext: “wklv phvvdjh lv qrw wrr kdug wr euhdn”
27-symbol alphabet: A through Z plus the "blank" character
Start with small words: English has relatively few small words, such as am, is, to, be, he, we, and, are, you, she, and so on.
substitute known short words at appropriate places in the ciphertext until you have something that seems to be meaningful.
Once the small words fall into place, you can try substituting for matching characters at other places in the ciphertext.
There is a strong clue in the repeated r of the word wrr.
two very common three-letter words having the pattern xyy are see and too. other less common possibilities are add, odd, and off

27. Cryptanalysis of the Caesar Cipher
Note that the combination wr appears in the ciphertext
if wrr is SEE, wr would have to be SE, which is unlikely
However, if wrr is TOO, wr would be TO, which is quite reasonable.
Substituting T for w and O for r, the message becomes
The OT could be cot, dot, got, hot, lot, not, pot, rot, or tot; a likely choice is not. Unfortunately, q = N does not give any more clues because q appears only once in this sample.
The word lv is also the end of the word wklv, which probably starts with T.
Likely two-letter words that can also end a longer word include so, is, in, etc.
However, so is unlikely because the form T-SO is not recognizable;
IN is ruled out because of the previous assumption that q is N
A more promising alternative is to substitute IS for lv throughout, and continue to analyze the message in that way.
By now, you might notice that the ciphertext letters uncovered are just three positions away from their plaintext counterparts.

28. Cryptanalysis of the Caesar Cipher
The cryptanalysis described here is ad hoc, using deduction based on guesses instead of solid principles.
But you can take a more methodical approach, considering
which letters commonly start words
which letters commonly end words
which prefixes and suffixes are common
Cryptanalysts have compiled lists of common prefixes, common suffixes, and words having particular patterns.
(For example, sleeps is a word that follows the pattern abccda.)

29. Other Substitutions
In substitutions, the alphabet is scrambled, and each plaintext letter maps to a unique ciphertext letter.
mathematical way description:
permutation is a reordering of the elements of a sequence
For instance, we can permute the numbers l to 10 in many ways, including:
π1 = 1, 3, 5, 7, 9, 10, 8, 6, 4, 2; and π2 = 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
A permutation is a function, so we can write expressions such as π1(3) = 5
meaning that the letter in position 3 is to be replaced by the fifth letter
If the set is the first ten letters of the alphabet, π1(3) = 5 means that C is transformed into e

30. Other Substitutions Alternative to using the permutation (π)
One way to scramble an alphabet is to use a key
a word that controls the permutation
For instance, if the key is word, the sender or receiver first writes the alphabet and then writes the key under the first few letters of the alphabet.
The sender or receiver then fills in the remaining letters of the alphabet, in some easy-to-remember order, after the keyword.

31. Complexity of Substitution Encryption and Decryption
An important issue in using any cryptosystem is the time it takes to turn plaintext into ciphertext, and vice versa.
it is essential that the scrambling and unscrambling not deter the authorized parties from completing their missions
The timing is directly related to the complexity of the encryption algorithm
encryption and decryption with substitution ciphers can be performed by direct lookup in a table illustrating the correspondence
Transforming a single character can be done in a constant amount of time, so we express the complexity of the algorithm by saying that the time to encrypt a message of n characters is proportional to n ( O(n) )

32. Cryptanalysis of Substitution Ciphers
The techniques described for breaking the Caesar cipher can also be used on other substitution ciphers
Short words, words with repeated patterns, and common initial and final letters all give clues for guessing the permutation.
breaking the code is a lot like working a crossword puzzle. You try a guess and continue to work to substantiate that guess until you have all the words in place or until you reach a contradiction
Using brute force attack, the cryptanalyst could try all 26! permutations of a particular ciphertext message
We can use our knowledge of language to simplify this problem. For example, in English, some letters are used more often than others. The letters E, T, O, and A occur far more often than J, Q, X, and Z, for example.
Encryption, even in a simple form, will deter the casual observer.

33. The Cryptographer's Dilemma
An encryption algorithm must be regular for it to be algorithmic and for cryptographers to be able to remember it. Unfortunately, the regularity gives clues to the cryptanalyst
There is no solution to this dilemma

34. One-Time Pads
A one-time pad is sometimes considered the perfect cipher
large, nonrepeating set of keys is written on sheets of paper, glued together into a pad.
if the keys are 20 characters long and a sender must transmit a message 300 characters in length
the sender would tear off the next 15 pages of keys
The sender would write the keys one at a time above the letters of the plaintext and
encipher the plaintext with a prearranged chart (called a Vigenère tableau) that has all 26 letters in each column, in some scrambled order

35. One-Time Pads The one-time pad method has two problems:
the need for absolute synchronization between sender and receiver, and
the need for an unlimited number of keys.
key
Plaintext
ciphertext
because row M column i is u, row A column a is a, and so on.

36. Transpositions (Permutations)
The goal of substitution is confusion
the encryption method is an attempt to make it difficult for a cryptanalyst or intruder to determine how a message and key were transformed into ciphertext.
A transposition (permutation) is an encryption in which the letters of the message are rearranged.
the cryptography aims for diffusion

37. Columnar Transpositions
rearrangement of the characters of the plaintext into columns
The following set of characters is a five-column transposition.

38. Columnar Transpositions
For instance, suppose you want to write the plaintext message THIS IS A MESSAGE TO SHOW HOW A COLUMNAR TRANSPOSITION WORKS. We arrange the letters in five columns
The resulting ciphertext would then be read down the columns as

39. Encipherment/Decipherment Complexity
This cipher involves no additional work beyond arranging the letters and reading them off again.
Therefore, the algorithm requires a constant amount of work per character, and the time needed to apply the algorithm is proportional to the length of the message.
we cannot produce output characters until all the message's characters have been read. This restriction occurs because all characters must be entered in the first column before output of the second column can begin, but the first column is not complete until all characters have been read.
Thus, the delay associated with this algorithm also depends on the length of the message, as opposed to the constant delay we have seen in previous algorithms

40. Digrams, Trigrams, and Other Patterns
Just as there are characteristic letter frequencies, there are also characteristic patterns of pairs of adjacent letters, called digrams.
Letter pairs such as -re-, -th-, -en-, and -ed- appear very frequently.

41. Cryptanalysis by Digram Analysis
The first step in analyzing the transposition is computing the letter frequencies.
If we find that in fact all letters appear with their normal frequencies, we can infer that a transposition has been performed.
The problem is to find where in the ciphertext a pair of adjacent columns lies and where the ends of the columns are

42. Cryptanalysis by Digram Analysis
Assume the block being compared is seven characters
The first comparison is c1 to c8, c2 to c9, …, c7 to c14. Then, we try a distance of eight characters, and so the window of comparison shifts and c1 is compared to c9, c2 to c10, and continuing..
For each window position, we ask two questions. First, do common digrams appear, and second, do most of the digrams look reasonable?
Figure 2-5  Moving Comparisons.

43. Combinations of Approaches
Substitution and transposition can be considered as building blocks for encryption.
A combination of two ciphers is called a product cipher.
Product ciphers are typically performed one after another, as in E2(E1(P,k1), k2)

44. Making "Good" Encryption Algorithms
What Makes a "Secure" Encryption Algorithm?
What does it mean for a cipher to be "good"?
The meaning of good depends on the intended use of the cipher
A cipher to be used by military personnel in the field has different requirements from one to be used in a secure installation with substantial computer support
In this section, we look more closely at the different characteristics of ciphers

45. Shannon's Characteristics of "Good" Ciphers
The amount of secrecy needed should determine the amount of labor appropriate for the encryption and decryption.
reiteration of the principle of timeliness from Chapter 1
The set of keys and the enciphering algorithm should be free from complexity
If the process is too complex, it will not be used
we should restrict neither the choice of keys nor the types of plaintext on which the algorithm can work
For instance, an algorithm that works only on plaintext having an equal number of A's and E's is useless.
Similarly, it would be difficult to select keys such that the sum of the values of the letters of the key is a prime number.
Furthermore, the key must be transmitted, stored, and remembered

46. Shannon's Characteristics of "Good" Ciphers
The implementation of the process should be as simple as possible
formulated with hand implementation in mind
A complicated algorithm is prone to error or likely to be forgotten
With the development and popularity of digital computers, algorithms far too complex for hand implementation became feasible
Still, the issue of complexity is important. People will avoid an encryption algorithm whose implementation process severely hinders message transmission
And a complex algorithm is more likely to be programmed incorrectly.

47. Shannon's Characteristics of "Good" Ciphers
Errors in ciphering should not propagate and cause corruption of further information in the message
One error early in the process should not throw off the entire remaining ciphertext
For example, dropping one letter in a columnar transposition throws off the entire remaining encipherment
The size of the enciphered text should be no larger than the text of the original message
ciphertext that expands dramatically in size cannot possibly carry more information than the plaintext
it gives the cryptanalyst more data from which to infer a pattern
longer ciphertext implies more space for storage and more time to communicate

48. Properties of "Trustworthy" Encryption Systems
When we say that encryption is "commercial grade," or "trustworthy," we mean that it meets these constraints:
It is based on sound mathematics
It has been analyzed by competent experts and found to be sound
It has stood the "test of time.“
Three algorithms are popular in the commercial world and meet the above criteria: DES (data encryption standard), RSA (Rivest Shamir Adelman, named after the inventors), and AES (advanced encryption standard).

49. Symmetric and Asymmetric Encryption Systems
Two basic kinds of encryptions: symmetric (also called "secret key") and asymmetric (also called "public key")
Symmetric
One key for enrcyption and decryption
Usually, the decryption algorithm is closely related to the encryption one
Ex., Caesar cipher: encryption: Pi + 3; decryption Ci - 3
provide a two-way channel to their users
A and B share a secret key, and they can both encrypt information to send to the other as well as decrypt information from the other
the system also provides authentication proof that a message received was not fabricated by someone other than the declared sender

50. Symmetric Encryption Systems
The symmetry of this situation is a major advantage of this type of encryption
But, has key distribution problem
How do A and B obtain their shared secret key?
In general, n users who want to communicate in pairs need
n * (n - 1)/2 keys
By the nature of the public key approach, you can send a public key in an message or post it in a public directory
Only the corresponding private key, which presumably is kept private
So, for all encryption algorithms, key management is a major issue
involves storing, safeguarding, and activating keys

51. Stream and Block Ciphers
Most of the ciphers we have presented so far are stream ciphers
convert one symbol of plaintext immediately into a symbol of ciphertext
The exception is the columnar transposition cipher
The transformation depends only on the symbol, the key, and the control information of the encipherment algorithm
skipping a character in
the key during encryption,
affect the encryption of all
future characters

52. Stream and Block Ciphers
A block cipher encrypts a group of plaintext symbols as one block
Ex., The columnar transposition
entire message is translated as one block
Block ciphers work on blocks of plaintext and produce blocks of ciphertext

53. Confusion and Diffusion
Two additional important concepts are related to the amount of work required to perform an encryption
An algorithm providing good confusion has a complex functional relationship between the plaintext/key pair and the ciphertext
it will take an interceptor a long time to determine the relationship between plaintext, key, and ciphertext
Ex1: Caesar cipher is not good for an analyst who deduces the transformation of a few letters can also predict the transformation of the remaining letters, with no additional information
Ex2: one-time pad is good because one plaintext letter can be transformed to any ciphertext letter at different places in the output

54. Confusion and Diffusion
Two additional important concepts are related to the amount of work required to perform an encryption
Confusion
Diffusion: distributing the information from single plaintext letters over the entire output

55. Cryptanalysis Breaking Encryption Schemes
Assume that the attacker knows the encryption/decryption algorithm then she may use three types of attacks
Ciphertext only attack
The attacker has only the ciphertext and wants to know the plaintext (and the key if possible)
Known plaintext attack
The attacker has both the plaintext and its corresponding ciphertext. The goal is to find the key
Chosen plaintext attack
The attacker may ask that specific plaintexts be enciphered and she is given the corresponding ciphertexts. Her goal is to find the key that was used

56. The Data Encryption Standard (DES)
developed for the U.S. government
1976
intended for use by the general public
accepted as a cryptographic standard both in the United States and abroad
many hardware and software systems have been designed with the DES
However, recently its adequacy has been questioned

57. The Data Encryption Standard (DES)
Overview
combination of two fundamental building blocks of encryption: substitution and transposition
derives its strength from repeated application of these two techniques
one on top of the other, for a total of 16 cycles
Hard to trace a single bit through 16 iterations
The algorithm begins by encrypting the plaintext as blocks of 64 bits
The key is 64 bits long
in fact it is only 56-bit (the other bits are used to check digits)

58. The Data Encryption Standard (DES)
Overview
Leverages the two techniques Shannon identified to conceal information: confusion and diffusion
ensuring that the output bits have no obvious relationship to the input bits and spreading the effect of one plaintext bit to other bits in the ciphertext
Substitution provides the confusion, and transposition provides the diffusion

59. The Data Encryption Standard (DES)
 A Cycle in the DES.

60. DES (Cont.)
Types of Permutations.

61. DES (Cont.)
Details of a Cycle.

62. DES (Cont. )

63. The Data Encryption Standard (DES)
Double and Triple DES
the DES 56-bit key length is not long enough for some people to feel comfortable
Computing power has increased dramatically
some researchers suggest using a double encryption for greater secrecy
Take two keys, k1 and k2
perform two encryptions, one on top of the other: E(k2, E(k1,m)).
Does this make the key as powerful as 112-bit key? NO
the cryptanalyst works plaintext and ciphertext toward each other
It only becomes 57-bit key
However, a simple trick does indeed enhance the security of DES
The so-called triple DES procedure is C = E(k3, E(k2, E(k1,m))).
This process gives a strength equivalent to a 112-bit key
differential and linear cryptanalysis (self study)

64. The Advanced Encryption Standards(AES)
To solve the DES security problems
Contest to develop a new algorithm
Rijndael (pronounced RINE dahl) algorithm
Won the contest and became the AES
Key lengths are: 128, 192, 256 (and possibly more) bits
1999
the U.S. government has approved AES for protecting Secret and Top Secret classified documents

65. Public Key Encryption
In 1976, Diffie and Hellman proposed a new kind of encryption system
Each user has two keys:
One is public and the other is private
Also, use one to encrypt and the other to decrypt
In symmetric key system, each pair of users needs a separate key

66. Public Key Encryption
In a public key or asymmetric encryption system, each user has two keys:
a public key and a private key.
The user may publish the public key freely
The keys operate as inverses
one key undoes the encryption provided by the other key
Ex. let kPRIV be a user's private key, and let kPUB be the corresponding public key
we can write the relationship as
P = D(kPRIV, E(kPUB, P)) = D(kPUB, E(kPRIV, P))

67. Rivest Shamir Adelman (RSA) Encryption
Public key system
RSA has been the subject of extensive cryptanalysis: no serious flaws have yet been found
Based on an underlying hard problem
Determining the prime factors of a large number
An area of mathematics known as number theory
mathematicians study properties of numbers such as their prime factors
Operates with arithmetic mod n

68. Rivest Shamir Adelman (RSA) Encryption
The two keys used in RSA, d and e, are used for decryption and encryption
They are actually interchangeable
Either can be chosen as the public key (the other must be kept private)
P = E(D(P)) = D(E(P))
Any plaintext block P is encrypted as Pe mod n
factoring Pe to uncover the encrypted plaintext is difficult
legitimate receiver who knows d simply computes (Pe)d mod n = P and recovers P without having to factor Pe.

69. Rivest Shamir Adelman (RSA) Encryption
Here is how it works
take two large primes, p and q, and compute their product n = pq
n is called the modulus
Choose a number, e, less than n and relatively prime to (p-1)(q-1)
i.e., e and (p-1)(q-1) have no common factors except 1
Find another number d such that (ed - 1) is divisible by (p-1)(q-1)
The values e and d are called the public and private exponents, respectively.
The public key is the pair (n, e); the private key is (n, d).
The factors p and q may be destroyed or kept with the private key.
It is currently difficult to obtain the private key d from the public key (n, e).
If one could factor n into p and q, then one could obtain the private key d.
Example: Suppose Alice wants to send a message m to Bob. Alice creates the ciphertext c by exponentiating: c = me mod n, where e and n are Bob's public key. She sends c to Bob. To decrypt, Bob also exponentiates: m = cd mod n; the relationship between e and d ensures that Bob correctly recovers m. Since only Bob knows d, only Bob can decrypt this message.

70. RSA Example Choose p = 3 and q = 11 Compute n = p * q = 3 * 11 = 33
the number of positive integers less than n that are co-prime to n
i.e., no common factors with n except 1
1 is included
Choose e such that 1 < e < φ(n) and e and n are co-prime. Let e = 7
Compute a value for d such that (d * e) % φ(n) = 1. One solution is d = 3 [(3 * 7) % 20 = 1]
Public key is (e, n) => (7, 33)
Private key is (d, n) => (3, 33)
The encryption of m = 2 is c = 27 % 33 = 29
The decryption of c = 29 is m = 293 % 33 = 2

71. Rivest Shamir Adelman (RSA) Encryption

72. The Uses of Encryption Cryptographic Hash Functions Key Exchange
Digital Signatures
Certificates

73. The Uses of Encryption
Encryption implements protected communications channels
it can also be used for other duties/applications
Cryptographic Hash Functions, Key Exchange, Digital Signatures, Certificates
Public key algorithms are useful only for specialized tasks
very slow; take 10,000 times as long to perform as a symmetric encryption
underlying modular exponentiation depends on multiplication and division
slower than the bit operations (addition, exclusive OR, substitution, and shifting) on which symmetric algorithms are based
symmetric encryption is the cryptographers' "workhorse," and public key encryption is reserved for specialized, infrequent uses, where slow operation is not a continuing problem

74. Cryptographic Hash Functions

75. Cryptographic Hash Functions
The most widely used cryptographic hash functions are
MD4, MD5 (where MD stands for Message Digest)
MD5 is an improved version of MD4
Any message will have 128-bit digest
SHA/SHS (Secure Hash Algorithm or Standard).
it produces a 160-bit digest
cryptanalysis attacks on SHA, MD4, and MD5
For SHA, the attack is to find two plaintexts that produce the same hash digest (collision)
263 steps, far short of the 280 steps that would be expected of a 160-bit hash function

76. Birthday Attack
In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday.
By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February 29).
However, 99% probability is reached with just 57 people, and 50% probability with 23 people.
These conclusions are based on the assumption that each day of the year (except February 29) is equally probable for a birthday.
The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of cracking a hash function.

77. Birthday Attack
A list of 23 people, comparing the birthday of the first person on the list to the others allows 22 chances for a matching birthday, the second person on the list to the others allows 21 chances for a matching birthday, third person has 20 chances, and so on. Hence total chances are: = 253, so comparing every person to all of the others allows 253 distinct chances (combinations): in a group of 23 people there are (23 * 22) / 2 = 253 pairs.

78. Key Exchange We talk about symmetric keys here
The problem is almost circular: To establish an encrypted session, you need an encrypted means to exchange keys.

79. Key Exchange Public key cryptography can help
To see how, suppose S and R want to derive a shared symmetric key
kPRIV-S, kPUB-S, kPRIV-R, and kPUB-R, are the private and public keys for S and R, respectively
S chooses any symmetric key K
S sends E(kPRIV-S,K) to R
R takes S's public key, removes the encryption, and obtains K
Ooops, any eavesdropper who can get S's public key can also obtain K
let S send E(kPUB-R, K) to R. Then, only R can decrypt K
Ooops, R has no assurance that K came from S
The solution is for S to send to R:
E(kPUB-R, E(kPRIV-S, K))

80. Key Exchange

81. Key Exchange Another key exchange approach
Diffie-Hellman key exchange protocol
S and R use some simple arithmetic to exchange a secret
They agree on a field size n and a starting number g
they can communicate these numbers in the clear
Each thinks up a secret number, say, s and r.
S sends to R gs and R sends to S gr
S computes (gr)s and R computes (gs)r ,which are the same, so grs = gsr becomes their shared secret
computations are done over a field of integers mod n (omitted for simplicity)
Diffie-Hellman, however, does NOT provide authentication
You can not be sure if you are talking to the right person

82. Digital Signatures
A digital signature is a protocol that produces the same effect as a real signature:
It is a mark that only the sender can make
but other people can easily recognize as belonging to the sender

83. Digital Signatures Two conditions It must be unforgeable:
If person P signs message M with signature S(P,M), it is impossible for anyone else to produce the pair [M, S(P,M)]
It must be authentic: If a person R receives the pair [M, S(P,M)] purportedly from P, R can check that the signature is really from P
Only P could have created this signature, and the signature is firmly attached to M

84. Digital Signatures

85. Digital Signatures Two more properties
It is not alterable. After being transmitted, M cannot be changed by S, R, or an interceptor.
It is not reusable. A previous message presented again will be instantly detected by R.

86. Digital Signatures Public Key Protocol
ideally suited to digital signatures.
E: use the public key in transformation
D: use the private key in transformation

87. Certificates
A public key and user's identity are bound together in a certificate, which is then signed by someone called a certificate authority, certifying the accuracy of the binding.

88. The University of Adelaide, School of Computer Science
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Certificates
The algorithms to generate a matched pair of public and private keys are publicly known, and software that does it is widely available.
So if Alice wanted to use a public key cipher, she could generate her own pair of public and private keys, keep the private key hidden, and publicize the public key.
But how can she publicize her public key— assert that it belongs to her—in such a way that other participants can be sure it really belongs to her?
Chapter 2 — Instructions: Language of the Computer

89. The University of Adelaide, School of Computer Science
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Certificates
A complete scheme for certifying bindings between public keys and identities— what key belongs to who—is called a Public Key Infrastructure (PKI).
A PKI starts with the ability to verify identities and bind them to keys out of band. By “out of band,” we mean something outside the network and the computers that comprise it, such as in the following scenarios.
If Alice and Bob are individuals who know each other, then they could get together in the same room and Alice could give her public key to Bob directly, perhaps on a business card.
Chapter 2 — Instructions: Language of the Computer

90. The University of Adelaide, School of Computer Science
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Certificates
If Bob is an organization, Alice the individual could present conventional identification, perhaps involving a photograph or fingerprints.
If Alice and Bob are computers owned by the same company, then a system administrator could configure Bob with Alice’s public key.
A digitally signed statement of a public key binding is called a public key certificate, or simply a certificate
Chapter 2 — Instructions: Language of the Computer

91. The University of Adelaide, School of Computer Science
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Certificates
One of the major standards for certificates is known as X.509. This standard leaves a lot of details open, but specifies a basic structure. A certificate clearly must include
the identity of the entity being certified
the public key of the entity being certified
the identity of the signer
the digital signature
a digital signature algorithm identifier (which cryptographic hash and which cipher)
Chapter 2 — Instructions: Language of the Computer

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Certificates
Certification Authorities
A certification authority or certificate authority (CA) is an entity claimed (by someone) to be trustworthy for verifying identities and issuing public key certificates.
There are commercial CAs, governmental CAs, and even free CAs.
To use a CA, you must know its own key. You can learn that CA’s key, however, if you can obtain a chain of CA-signed certificates that starts with a CA whose key you already know.
Then you can believe any certificate signed by that new CA
Chapter 2 — Instructions: Language of the Computer