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Cryptography

Cryptography. Public Key vs. Private Key Cryptosystems. by William M. Faucette. Department of Mathematics State University of West Georgia. What is Cryptography?.

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Cryptography

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  1. Cryptography Public Key vs. Private Key Cryptosystems

  2. 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 Descriptionof 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) ST (19, 20) 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 Comparing the ciphertext IKTGCOFMOEVU 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.

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