Cryptology

1
Cryptology

1. Definitions
2. Substitution Ciphers
3. Transpositions Ciphers
4. The DES Algorithm
5. Public-Key Cryptology

2
Definitions

• code thousands of words, phrases or symbols
  that form codewords that replace plaintext
  elements.
• cipher a method of secret writing
• cryptography art of devising ciphers
• cryptoanalysis art of breaking ciphers
• cryptology art of devising breaking ciphers

3
Substitution Ciphers

• Monoalphabetic (26! possible ciphers)
• Caesar cipher
• Newspapers Daily Cryptoquote
• Polyalphabetic
• Vigenere cipher
• Playfair cipher

4
Caesar Cipher

• ABCDEFGHIJKLMNOPQRSTUVWXYZ
• DEFGHIJKLMNOPQRSTUVWXYZABC
• ATTACK AT DAWN
• would be encoded as
• DWWDFN DW GDZQ

5
Newspapers Daily Cryptoquote

• OGR MWRZNVMD YXMP GMC URRD M CQWUBX BY XNURZOQ
  MDK YZRRKBW.

6
Newspapers Daily Cryptoquote

• OGR MWRZNVMD YXMP GMC URRD M CQWUBX BY XNURZOQ
  MDK YZRRKBW.
• THE AMERICAN FLAG HAS BEEN A
• SYMBOL OF LIBERTY AND FREEDOM.

7
Percentages of the English Language

• Letters__ Diagrams__ Trigrams__
  Words___
• E 13.05 TH 3.16 THE 4.72 THE 6.42
• T 9.02 IN 1.54 ING 1.42 OF 4.02
• O 8.21 ER 1.33 AND 1.13 AND 3.15
• A 7.81 RE 1.30 ION 1.00 TO 2.36
• N 7.28 AN 1.08 ENT 0.98 A 2.09
• I 6.77 HE 1.08 FOR 0.76 IN 1.77
• R 6.64 AR 1.02 TIO 0.75 THAT 1.25
• S 6.46 EN 1.02 ERE 0.69 IS 1.03
• H 5.85 TI 1.02 HER 0.68 I 0.94
• D 4.11 TE 0.98 ATE 0.66 IT 0.93
• L 3.60 AT 0.88 VER 0.63 FOR 0.77
• C 2.93 ON 0.84 TER 0.62 AS 0.76
• F 2.88 HA 0.84 THA 0.62 WITH 0.76
• U 2.77 OU 0.72 ATI 0.59 WAS 0.72
• M 2.62 IT 0.71 HAT 0.55 HIS 0.71
• P 2.15 ES 0.69 ERS 0.54 HE 0.71
• Y 1.51 ST 0.68 HIS 0.52 BE 0.63
• W 1.49 OR 0.68 RES 0.50 NOT 0.61

8
Polyalphabetic Substitutions

• Use a different alphabet for each letter in the
  plaintext.
• Defeats attacks based upon common English
  frequency charts.

9
Vigenere Cipher

• ABCDEFGHIJKLMNOPQRSTUVWXYZ
• BCDEFGHIJKLMNOPQRSTUVWXYZA
• CDEFGHIJKLMNOPQRSTUVWXYZAB Key COOKIEMONSTERCOOKI
  EMONSTER
• DEFGHIJKLMNOPQRSTUVWXYZABC Plaintext ATTACKATDAWN
  PLEASE
• EFGHIJKLMNOPQRSTUVWXYZABCD
• FGHIJKLMNOPQRSTUVWXYZABCDE
• GHIJKLMNOPQRSTUVWXYZABCDEF 1. Use key letter to
  select row
• HIJKLMNOPQRSTUVWXYZABCDEFG 2. Use plaintext
  letter to select column
• IJKLMNOPQRSTUVWXYZABCDEFGH 3. Ciphertext letter
  is found at selected row column
• JKLMNOPQRSTUVWXYZABCDEFGHI
• KLMNOPQRSTUVWXYZABCDEFGHIJ Ciphertext CHHKKOMHQSP
  RGNSOCM
• LMNOPQRSTUVWXYZABCDEFGHIJK
• MNOPQRSTUVWXYZABCDEFGHIJKL
• NOPQRSTUVWXYZABCDEFGHIJKLM
• OPQRSTUVWXYZABCDEFGHIJKLMN
• PQRSTUVWXYZABCDEFGHIJKLMNO
• QRSTUVWXYZABCDEFGHIJKLMNOP
• RSTUVWXYZABCDEFGHIJKLMNOPQ
• STUVWXYZABCDEFGHIJKLMNOPQR

10
Playfair Cipher

• 1) Group plaintext into pairs of letters. The
  letters I and J are
• considered to be the same letter. If any pair
  contains identical M B Q Z A
• letters insert a Q. If odd number of
  letters, add an X. D R G F S
• 2) If the 2 letters are in same row, take the
  pair of letters N H U E K
• to the right of the plaintext letters V T L W I
• 3) If the 2 letters are in the same column, take
  the pair of O X C P Y
• of letters below the plaintext letters.
• 4) If the 2 letters form the corners of a
  rectangle, Take the 2
• letters at the opposite corners of the
  rectangle. The letter
• in the same row as the first plaintext letter
  is taken as the
• first cipher letter.
• Plaintext Now is the time for all good men
• Grouped NO WI ST HE TI ME FO RA LQ LG OQ OD ME
  NX
• Cipher VM IV RI UK LV ZN DP SB CG CU CM MN ZN HO

11
Transposition Ciphers

• Railfence Transpositions
• Columnar Transpositions
• Double Transpositions

12
Railfence Transpositions

• Plaintext IS THIS A GOOD CIPHER
• I I O I R
• Railfence S H S G O C P E
• T A D H
• Ciphertext IIOIRSHSGOCPETADH

13
Columnar Transpositions

• MEGABUCK
• PLEASETR Key determines number of columns.
• ANSFERON
• EMILLION Ciphertext is written using columns in
• DOLLARST alphabetical order of letters in key.
• OMYSWISS
• BANKACCO Ciphertext AFLL SKSO SELA WAIA
• UNTSIXTW TOOS SCTC LNMO MANT ESIL YNTW
• OTWOABCD RNNT SOWD PAED OBUO ERIR ICXB

14
Double Transpositions

• POLITICS MONEY
• COMETOTH TTEDR Columns of first matrix are
• EAIDOFTH OFYMI entered into the second matrix.
• EPARTY AOAPC Columns of second matrix yield
• EEHHT the ciphertext.
• OT
• Plaintext COME TO THE AID OF THE PARTY
• Ciphertext DMPH TOAEO EYAH TFOET RICT

15
The DES Algorithm

• Data Encryption Standard was adopted by National
  Bureau of Standards in 1977
• Plaintext is 1st grouped into blocks of 64 bits
• 56-bit key
• 19 distinct stages
• Initial key independent transposition
• 16 substitution steps using 56-bit key
• Final 2 stages involve more transpositions
• Decryption uses same key with stages in reverse
  order

16
The DES Algorithm
64 bit plaintext
Li-1
Ri-1
Initial Transposition
Iteration 1
Iteration 2
Li-1 f(Ri-1,Ki)
56 bit key
Iteration 16
32 bit swap
32 bits
32 bits
Final Transposition
Detail of one iteration
64 bit ciphertext
17
The DES Algorithm

• IBMs original design used 128 bits
• U.S. National Security Agency requested reduction
  to 56 bits
• Reason for change has not been made public
• Reasons for particular choices for iteration
  functions has remained secret as well
• Requires key distribution

18
The DES Algorithm

• DES has been replaced with Triple DES
• This newer version uses a 112-bit key.
• AES (Advanced Encryption Standard)
• According to the U.S. Commerce Department all
  federal departments must use AES by May 29, 2002.
  This should influence commercial use as well.
  AES was developed by Belgian researchers and is
  based upon a 128-bit key.

19
Public-Key Cryptography(The RSA algorithm is
most famous example.)

• Relationship between the plaintext and the
  ciphertext

Named for developers Rivest, Shamir, and
Adleman.
20
Public-Key Cryptography

• Selecting a public key
• 1) Select 2 distinct primes, p q (preferably
  extremely large).
• 2) Form the product, n p q.
• 3) Compute f (p-1) (q-1).
• 4) Select any integer e, with the property that
  GCD(e,f) 1.
• The pair of integers, e and n, comprise the
  public key.
• Example If p 3 and q 11, then n 33 and f
  20. We could choose
• e 7, since GCD(7,20) 1. Thus, our public
  key
• would be the pair e 7 and n33.
• 10,967,535,067

21
Public-Key Cryptography

• Selecting a private key
• Using the value for e and f found earlier,
• find d such that (ed) mod f 1.
• The pair of integers, d and n, comprise the
  private key.
• Continuing previous example
• Since e 7 and f 20, d must be 3 (73 mod 20
  1).
• Thus, our private key would be the pair d 3
  and n 33.

22
Public-Key Cryptography

• Ciphertext is generated using c pe mod n.
• Text Numeric p7 c p7 mod 33
• S 19 893,871,739 13
• U 21 1,801,088,541 21
• N 14 105,413,504 20
• D 4 16,384 16
• A 1 1 1
• Y 25 6,103,515,625 31
• Ciphertext 13 21 20 16 01 31

23
Public-Key Cryptography

• Plaintext is recovered using p cd mod n.
• Numeric c3 c p3 mod 33 Text
• 13 2,197 19 S
• 21 9,261 21 U
• 20 8,000 14 N
• 16 4,096 4 D
• 1 1 1 A
• 31 29,791 25 Y
• Plaintext SUNDAY

24
Public-Key Cryptography

• The security is dependent upon the difficulty of
  finding the prime factors of a very large
  integer. No efficient algorithm has yet been
  found.
• Factoring 200 digit integers requires 4.3106
  years.
• Factoring 300 digit integers requires 5.51012
  years.
• Factoring 500 digit integers requires 4.71022
  years.
• (Assumes a computer that uses 1 nanosecond per
  instruction.)

25
Public-Key Cryptography

• E-mail signatures
• A encodes his personal ID using his private key
• If B can decode the personal ID using As public
  key, then B knows that A sent message.
• C E(ID,private_keyA) P D(C, public_keyA)

26
Public-Key Cryptography

• Encrypted signatures.
• A encodes personal ID using As private key.
• A encodes result using Bs public key.
• Upon receipt, B decodes by first using Bs
  private key.
• B then verifies signature by decoding using As
  public key.
• As steps C E( E(ID,privateA), publicB)
  Transmit C
• Bs steps Receive C P D( D(C,privateB),
  publicA)
• If P equals As ID then B is confident that
  message came from A
• furthermore A is protected because only B can
  decode the message.

27
Public-Key Cryptography

• PGP (Pretty Good Privacy)
• Uses Public-Key cryptography
• Used by many to encrypt their e-mail and
  implement signatures
• Inexpensive (free version available for personal
  use)
• http//www.pgpi.org
• http//web.mit.edu/network/pgp.html
• http//www.pgp.com

28
References

• The Codebreakers, by David Kahn, 1973.
• Excellent account of the history of cryptology
  with special emphasis during World War II
• Cryptanalysis for Microcomputers, by Caxton C.
  Foster, 1982.
• Codes, Ciphers, and Computers, by Bruce Bosworth,
  1982.
• Computer Networks, Andrew Tanenbaum, 1988.
• Cryptology, NSF Chautauqua Program taught by
  Robert E. Lewand at Christian Brothers University
  in Memphis, TN on June 28-30, 1998.