1. Public Key vs. Private Key Cryptosystems
Cryptography
Public Key vs. Private Key
Cryptosystems

2. Department of Mathematics State University of West Georgia
by William M. Faucette
Department of Mathematics
State University of West Georgia

3. What is Cryptography?
Cryptography is a scientific mix of mathematical theory and computational application which allows the confidential transfer of information.

4. What is Cryptography?
Please allow me to introduce the main characters in our drama:
Alice and Bob wish to perform some form of communication while Eve is an eavesdropper who wishes to spy on or tamper with the communications between Alice and Bob.

5. What is Cryptography?
Cryptography is concerned with four facets of data transfer:
Confidentiality
Authenticity
Integrity
Non-repudiation

6. Confidentiality
A message sent from Alice to Bob cannot be read by anyone else.

7. Authenticity
Bob knows that only Alice could have sent the message he has just received.

8. Integrity
Bob knows that the message from Alice has not been tampered with in transit.

9. Non-Repudiation
It is impossible for Alice to turn around later and say she did not send the message.

10. Cryptography in Ancient Times

11. Cryptography in Ancient Times
Perhaps one of the most ancient methods of cryptography, attributed to Julius Caesar, involves fixing an alphabet and choosing a “shift index”.

12. Cryptography in Ancient Times
The “shift index” tells you how many letters down the alphabet to shift a letter in order to encode it.

13. Cryptography in Ancient Times
For example, if we use the standard 26-letter English alphabet and choose a shift index of 4, then
A is encoded to E,
B is encoded to F,
C is encoded to G,
and so forth.

14. Cryptography in Ancient Times
For letters at the end of the alphabet, we simply wrap around to the beginning of the alphabet:
V is encoded to Z,
W is encoded to A,
X is encoded to B,
and so forth.

15. A Modern Description of this Cryptosystem

16. A Modern Description of this Cryptosystem
Take each letter, A through Z, and assign it a number in the ring Z/26Z by taking A to 1, B to 2, C to 3, , Y to 25, and Z to 0.
This allows us to convert any string of text, called plaintext, into a string of numbers between 0 and 25.

17. A Modern Description of this Cryptosystem
Once we have the message as a string of digits, to encode the message, we simply apply the function
where n is the shift index.

18. A Modern Description of this Cryptosystem
The encoding is completed by turning the resulting string of digits back into characters using the original correspondence.

19. Oops!

20. Oops!
The only problem with this cryptosystem is that it is easily broken. That is, it is possible for an unauthorized person to convert the ciphertext back to plaintext.

21. Oops!
In order to break this code, you need only perform a frequency analysis, counting the number of times each letter occurs in the ciphertext.

22. Oops!
Knowing that the letter E is the most commonly occurring letter in English text, we can (probably) assume that the letter E maps to the most commonly occurring letter in the ciphertext.

23. Oops!
Knowing the correspondence of one plaintext letter to one ciphertext letter gives you enough information to decode the intercepted ciphertext.

24. A Better Cryptosystem
A Digraph Cipher

25. A Better Cryptosystem
One problem with the preceding cryptosystem is that it takes one letter and encodes it to the same letter every time. This enables us to conduct a frequency analysis and break the cipher.

26. A Better Cryptosystem
Rather than encode one letter at a time, we can encode blocks of letters at a time. For example, we can encode pairs of letters. Such a cryptosystem is known as a digraph cipher.

27. Digraph Cipher
Use the same function taking the English alphabet into the ring Z/26Z. For a pair of plaintext letters, this gives us a pair of integers modulo 26. We can consider this ordered pair as a vector in (Z/26Z)2.

28. Digraph Cipher
To encipher this vector, v, we need an enciphering matrix, M. That is, a 2x2 matrix with entries in Z/26Z which is invertible in Z/26Z.
Such a matrix is invertible if and only if its determinant is relatively prime to 26.

29. Digraph Cipher
The enciphering is then accomplished by multiplying the vector v by the enciphering matrix M, and then converting the resulting vector back into letters.

30. Example

31. Example Start with the plaintext West Georgia
This message has an odd number of letters, so we add a random letter ‘x’ and break the message into digraphs:
WE ST GE OR GI AX

32. Example Next, we convert the digraphs to vectors in Z/26Z: WE (23, 5)
GE (7, 5)
OR (15, 18)
GI (7, 9)
AX (1, 24)

33. Example
For our enciphering matrix, we’ll use the matrix

34. Example
We encipher all the vectors at once using matrix multiplication:

35. Example The product of these two matrices is
remembering that we are working in Z/26Z.

36. Example
Converting these vectors back into digraphs, we get the ciphertext
IKTGCOFMOEVU

37. Example IKTGCOFMOEVU WESTGEORGIAX Comparing the ciphertext
with the plaintext
WESTGEORGIAX
we see that the two Es go to two different letters, K and O, making breaking this cipher more difficult.

38. Variations on a Theme

39. Other Variations
Of course, there’s nothing special about digraphs: We can divide the plaintext into blocks of k letters and use a kxk enciphering matrix.

40. Other Variations
We can also add a fixed vector b after multiplying by the enciphering matrix M.
If P is the plaintext message, the ciphertext message is given by
MP+b mod 26

41. Private Key Cryptography

42. Private Key Cryptography
The cryptosystems we have described so far are all private key cryptosystems.

43. Private Key Cryptography
The enciphering keys in the last variation are the matrices M and the vector b.
These keys must be kept private because knowing the enciphering keys allows one to compute the deciphering keys.

44. Private Key Cryptography
For example, if the cryptosystem uses the enciphering function
C=MP+b
Then we can solve this matrix equation for P to get
P=M-1(C-b)=M-1C-M-1b

45. Private Key Cryptography
So, we see that if the data (M, b) are the enciphering keys, the deciphering keys are (M-1M-1b).
From this we see that anyone who knows the enciphering keys can compute the deciphering keys.

46. Public Key Cryptography

47. Public Key Cryptography
In contrast, with public key cryptography, knowledge of the enciphering key does not allow one to compute the deciphering key.

48. Public Key Cryptography
Similarly, knowledge of the deciphering key does not allow one to compute the enciphering key.

49. Why Would Someone Use Public Key Cryptography?

50. Why Would Someone Use Public Key Cryptography?
If knowledge of an enciphering key allows one to compute the corresponding deciphering key, it is possible for this party to intercept and read a ciphertext message intended for another party. This defeats confidentiality.

51. Why Would Someone Use Public Key Cryptography?
If knowledge of a deciphering key allows one to compute the corresponding enciphering key, it is possible for this party to code and send a ciphertext message to a third party. This defeats authenticity.

52. When Would Someone Use Public Key Cryptography?

53. When Would Someone Use Public Key Cryptography?
Public key cryptography tends to be slower than private key cryptography, so why would anyone use it?

54. When Would Someone Use Public Key Cryptography?
Public key cryptography is used in an auxiliary capacity, say to agree upon keys for a traditional private key cryptosystem.

55. When Would Someone Use Public Key Cryptography?
It is possible for two parties to initiate secret communications without ever having had any prior contact, without having established any prior trust, without exchanging any preliminary information.

56. How Does Public Key Cryptography Work?

57. How does Public Key Cryptography Work?
In order to implement public key cryptography, each person, Alice and Bob, has a public enciphering key, KE, and a private deciphering key, KD.

58. How does Public Key Cryptography Work?
The public keys are published and made available to the public, while the private keys are kept confidential.

59. How does Public Key Cryptography Work?
Since the enciphering keys are made public, in order to ensure the security of the cryptosystem, it must be computationally infeasible to find the private keys from the public keys.

60. How does Public Key Cryptography Work?
Computationally infeasible does not mean that the computation is impossible. Rather, it means that the amount of computer time necessary to perform the computation is prohibitively long.

61. How does Public Key Cryptography Work?
So, in order to implement public key cryptography, we must have some function that is easy to compute, but whose inverse function cannot be computed in any reasonable sense.

62. How does Public Key Cryptography Work?
That is, in order to implement public key cryptography, we must have a trapdoor function.

63. Trapdoor Functions

64. Trapdoor Functions
A trapdoor function is a function f which is easy to compute, but whose inverse function f-1 is impossible to compute without performing a prohibitively lengthy computation.

65. Trapdoor Functions
Two types of trapdoor functions that are used in the RSA cryptosystem and Elliptic Curve cryptosystems are these:
The prime factorization problem
The discrete logarithm problem

66. The Prime Factorization Problem

67. The Prime Factorization Problem
The Fundamental Theorem of Arithmetic states that every natural number can be factored (essentially) uniquely into a product of prime numbers.

68. The Prime Factorization Problem
However, given a very large number n, say on the order of 10100, it is computationally infeasible to factor n.

69. A Little Computation
In order to factor n, one systematic way which is easily implemented on a computer is to divide n by
2, 3, 4, , n1/2
to test for a divisor.

70. A Little Computation
If we try this approach with a natural number of the order of 10100, this technique would take 1050 operations to complete.

71. A Little Computation
In 1997, the Department of Energy announced the world’s fastest computer performed one trillion floating point operations per second, a teraflop.

72. A Little Computation
This computer would take more than 3x1033 years to factor a 100 digit number by this systematic method.

73. The Discrete Logarithm Problem

74. The Discrete Logarithm Problem
To describe the discrete logarithm problem, we start with a finite abelian group G of very large order.

75. The Discrete Logarithm Problem
Typically, G is a group such as (Z/nZ)*, the group of invertible elements in the ring Z/nZ, or Fq*, the group of nonzero elements in the finite field with q elements.

76. The Discrete Logarithm Problem
For a fixed element b in G consider the map from the natural numbers into G given by n maps to bn.

77. The Discrete Logarithm Problem
For any element y in G, the discrete logarithm of y base b is the smallest natural number n so that bn=y.

78. The Discrete Logarithm Problem
Like the prime factorization problem, the discrete logarithm problem is believed to be difficult and also to be the hard direction of a trapdoor function.

79. The Discrete Logarithm Problem
The discrete logarithm problem has received much attention in recent years. The best discrete logarithm problems have expected running times similar to those of the best factoring algorithms.

80. The Discrete Logarithm Problem
Rivest has analyzed the expected time to solve the discrete logarithm problem both in terms of computing power and cost.
See R.L. Rivest. Response to NIST's proposal. Communications of the ACM, 35: 41-47, July 1992.

81. The Discrete Logarithm Problem
The discrete logarithm problem appears to be much harder over arbitrary groups than over finite fields; this is the motivation for cryptosystems based on elliptic curves.

82. Next Time
In the next two lectures, we will systematically look at two public key cryptosystems: The RSA cryptosystem and elliptic curve cryptosystems.

83. Thanks for Attending