Cryptography : From Demaratus to RSA

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From Demaratus in ancient Sparta using wax covered tablets to the German Enigma to Diffie-Helman and RSA cryptography has always been at the cutting edge.

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Cryptography : From Demaratus to RSA

  1. 1. Cryptography 101
  2. 2. What is Cryptography? Encryption is the process of: 1.Transforming data (x) 2.Using an algorithm (e) 3.To make it unreadable to anyone (y) 4.except those possessing the key. (k) K={k1,…,kn} The Quick Brown Fox Me (mod N) where M=The Quick Brown Fox Uif Rvjdl Cspxo Gpy Kerchhoffs’ principle: A cryptosystem should be secure even if the Attacker knows all the details about the system, with the exception of The key.
  3. 3. We shall extend the empire of Persia such that its boundaries will be God's own sky, so the sun will not look down upon any land beyond the boundaries of what is our own -Xerxes (Ahasuerus) ~450 B.C.
  4. 4. (Spartan) Scytale Rail Fence Cipher Route Cipher Transposition Ciphers
  5. 5. The Quick Brown Fox GSV JFRXP YILDM ULC Substitution Ciphers The Quick Brown Fox ZIT JXOEA WKGVF YGB Shift Cipher (Caesar) The Quick Brown Fox SGD PTHBJ AQNVM ENW At-Bash
  6. 6. Modular ciphers a = r mod m 42 = 9*4 + 6 r = a – m*q 42 = 6 mod 9 42 = q*9 + 6 6 = 42 – q*9 q = 0, r = 42 q = 1, r = 33 q = 2, r = 24 q = 3, r = 15 q = 4, r = 6 (0<q<m-1) q = 5, r = -3 q = 6, r = -12 12 + 7 = 19 => 1 mod 9 14 – 2 = 12 => 3 mod 9 11 * 8 = 88 => 7 mod 9 15/5 = 3 !=> 3 mod 9 If the multiplicative inverse exists for a number then we can divide by that number 5*2=10 => 1 mod 9 2 is the multiplicative inverse of 5 (and vice versa) 15*2 = 30 => 3 mod 9 If x is coprime with modulus then it has an inverse.
  7. 7. Caesar Cipher Encryption: ek (x) = x + k mod 26 Decryption: ek (y) = y – k mod 26 The quick brown fox k=3 t=20, 20 + 3 = 23 mod 26 h=8, 8 + 3 = 11 mod 26 e=5, 5 + 3 = 8 mod 26 Affine Cipher k=(a,b) Encryption: ek (x) = a*x + b mod 26 Decryption: ek (y) = a-1 * (y – b) mod 26 The quick brown fox k=(5, 3) t=20, 5*20 + 3 = 103 = 25 mod 26 h=8, 5*8 + 3 = 43 = 17 mod 26 e=5, 5*5 + 3 = 28 = 2 mod 26 21 * 5 = 105 = 1 mod 26 21 * 25-3 = 462 = 20 mod 26 21 * 17-3 = 294 = 8 mod 26 21 * 2-3 = -21 = 5 mod 26
  8. 8. Brute-Force Attacks Given: y = SGD PTHBJ AQNVM ENW Keyspace = {1,…,25} Decryption : ki(y) =? x
  9. 9. Abu Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī 850 C.E. Cryptanalysis Letter Frequency Short word and letter combinations the be to of and that have for not http://www.richkni.co.uk/php/crypta/
  10. 10. Normal English letter frequency Ciphertext letter frequency
  11. 11. Vigenère cipher A polyalphabetic cipher Key = KING The sun and the man in the moon Dpr yev ntn buk wia ox buk wwbt 4 possible ways to spell the word “the” K – DPR I - BUK N – GNO G - ZRM http://www.simonsingh.net/The_Black_Chamber/vigenere_cracking_tool.html http://sharkysoft.com/vigenere/
  12. 12. Enigma http://cryptoclub.math.uic.edu/shiftcipher/shiftcipher.php http://enigma.louisedade.co.uk/enigma.html
  13. 13. AXP AVC .. IOV NKZ .. HSA PYT .. PPZ LEX FZD YQO .. IZL NQL .. NNQ CMA .. GUH BIS FGT YHD .. KDY GNV .. NBJ COQ .. GOI BKK MIW MRI .. VWG EZG .. SYX SJB .. TVB KFM DJG UDG .. OJN QDE .. SNH SMS .. TLI KPK LNK TMF .. ZAO RXJ .. SXV SVZ .. TYO KJJ XKN JAE .. CTL OUL .. ERS XWU .. WHJ WBQ BHG DBG .. CMM OTY .. EAA XXT .. JQR ISH RZU ZQN .. UKM HAY .. YCE FGR .. JEY ICV RTC ZUW .. QFF VLP .. PII LRK .. JCE IGP Loops(1,4) (LTKGBDUHP) (XJINCOQVE) (FY) (RZ) (A) (M) (S) (W) Loops(2,5) (XVFLPECGHBOKA) (ZQSYJDNMTUIRW) Loops(3,6) (PCWIKF) (DOJQAT) (NERHSU) (VZXBMY) (L) (G) Loops(1,4) 8, 9, 9, 2, 2, 1, 1, 1, 1 Loops(2,5) 2, 13, 13 Loops(3,6) 6, 6, 6, 6, 6, 1, 1 By the end of WWII enigma had a key space of 159 sextillion (159*1021)
  14. 14. Confusion and Diffusion Claude Shannon Confusion The relationship between the key and the ciphertext as complex and as involved as possible. e.g. Enigma & complex substitution (S-boxes) 011011 Diffusion Statistics of the plaintext is "dissipated" in the statistics of the ciphertext. If we change a character of the plaintext, then several characters of the ciphertext should change. http://en.wikipedia.org/wiki/Permutation_box P-Box
  15. 15. Left Right ABCDEF GHIJKL ABCDEF F() = HJLGIK Xor = JIHGKL JIHGKL ABCDEF DES Data Encryption Standard (1973) 56 bit (Lucifer cipher) Key Length Security Estimation 56-64 bits A few hours or days 112-128 bits Several decades (w/o QC) 256 bits Several decades (w QC)
  16. 16. AES Advance Encryption Standard (2001) Currently accepted industry standard Supports 128, 192 and 256 bit keys. In 1997 National Institute of Standards and Technology (NIST) Called for proposals for AES • Rijandel • Mars • RC6 • Serpent • Twofish In 2001 Rijandel was adopted and renamed AES.
  17. 17. Diffie-Hellman Key Exchange (DHKE) Discrete Logarithm Problem Used in: SSH TLS IPSec
  18. 18. Diffie-Hellman Key Exchange (DHKE) Discrete Logarithm Problem 1. Choose a prime modulus P. 17 2. Choose an integer A that will be known as the generator. 3 3. Alice and Bob both choose a private number Ax mod P Alice a – 15 Bob b – 13 315 mod 17 = 6 313 mod 17 = 12 6 <- 12 12 15 mod 17 = 10 6 13 mod 17 = 10 Hacker knows: Alice - Ax mod P = 6 Bob - Ax mod P = 12 A is specially chosen to induce the discrete logarithm problem and ensure a one way function. Exponentiation is commutative: k = (Ax)y = (Ay)x
  19. 19. RSA Rivest, Shamir, Adleman Discrete logarithm and integer factorization Set up 1. Choose two large primes: p=3 and q=11 2. n = p*q = 33 3. Θ(n) = (p-1)(q-1)=(3-1)(11-1)=20 4. Find a number e where gcd(e, Θ(n)) = 1 e=3 5. Find the number d where e*d = 1 mod Θ(n) d=7 Public key (n, e) = (33, 3) Private key(d) c = me mod n m = cd mod n Alice Bob m=4 43 mod 33 = 31 31-> 317 mod 20 = 4

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