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Applied Cryptography ( Public key ) (I)

Applied Cryptography ( Public key ) (I). ECE 693 BD Security.

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Applied Cryptography ( Public key ) (I)

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  1. Applied Cryptography(Public key) (I) ECE 693 BD Security

  2. John wrote the letters of the alphabet under the letters in its first lines and tried it against the message. Immediately he knew that once more he had broken the code. It was extraordinary the feeling of triumph he had. He felt on top of the world. For not only had he done it, had he broken the July code, but he now had the key to every future coded message, since instructions as to the source of the next one must of necessity appear in the current one at the end of each month. —Talking to Strange Men, Ruth Rendell

  3. Confidentiality using Symmetric Encryption • traditionally symmetric encryption is used to provide message confidentiality

  4. Placement of Encryption • have two major placement alternatives • link encryption • encryption occurs independently on every link • implies must decrypt traffic between links • requires many devices, but paired keys • end-to-end encryption • encryption occurs between original source and final destination • need devices at each end with shared keys

  5. Placement of Encryption

  6. Placement of Encryption • when using end-to-end encryption must leave headers in clear • so network can correctly route information • hence although contents protected, traffic pattern flows are not • ideally want both at once • end-to-end protects data contents over entire path and provides authentication • link protects traffic flows from monitoring

  7. Placement of Encryption • can place encryption function at various layers in OSI Reference Model • link encryption occurs at layers 1 or 2 • end-to-end can occur at layers 3, 4, 6, 7 • as move higher less information is encrypted but it is more secure though more complex with more entities and keys

  8. Encryption vs Protocol Level

  9. Random Numbers • many uses of random numbers in cryptography • nonces in authentication protocols to prevent replay • session keys • public key generation • keystream for a one-time pad • in all cases its critical that these values be • statistically random, uniform distribution, independent • unpredictability of future values from previous values

  10. Pseudorandom Number Generators (PRNGs) • often use deterministic algorithmic techniques to create “random numbers” • although are not truly random • can pass many tests of “randomness” • known as “pseudorandom numbers” • created by “Pseudorandom Number Generators (PRNGs)”

  11. Linear CongruentialGenerator • common iterative technique using: Xn+1 = (aXn + c) mod m • given suitable values of parameters can produce a long random-like sequence • suitable criteria to have are: • function generates a full-period • generated sequence should appear random • efficient implementation with 32-bit arithmetic • note that an attacker can reconstruct sequence given a small number of values • have possibilities for making this harder

  12. Using Block Ciphers as PRNGs • for cryptographic applications, can use a block cipher to generate random numbers • often for creating session keys from master key • Counter Mode Xi = EKm[i] • Output Feedback Mode Xi = EKm[Xi-1]

  13. ANSI X9.17 PRG See a good website for software: https://www.cryptosys.net/rng_algorithms_old.html It uses date/time & seed inputs and 3 triple-DES encryptions to generate a new seed & random value; DTi - Date/time value at the beginning of ith generation stage Vi - Seed value at the beginning of ith generation stage Ri - Pseudorandom number produced by the ith generation stage K1, K2 - DES keys used for each stage Then compute successive values as: Ri= EDE([K1, K2], [Vi XOREDE([K1, K2], DTi)]) Vi+1 = EDE([K1, K2], [Ri XOR EDE([K1, K2], DTi)]) EDE: triple DES (Encryption – Decryption – Encryption)

  14. Blum Shub Generator • based on public key algorithms • use least significant bit from iterative equation: • xi = xi-12 mod n • where n=p.q, and primes p,q =3 mod 4 • unpredictable, passes next-bit test • security rests on difficulty of factoring N • is unpredictable given any run of bits • slow, since very large numbers must be used • too slow for cipher use, good for key generation

  15. Natural Random Noise • best source is natural randomness in real world • find a regular but random event and monitor • do generally need special h/w to do this • eg. radiation counters, radio noise, audio noise, thermal noise in diodes, leaky capacitors, mercury discharge tubes, etc • starting to see such h/w in new CPU's • A true random number generator may produce an output that is biased in some way. Various methods of modifying a bit stream to reduce or eliminate the bias have been developed.

  16. Published Sources • a few published collections of random numbers • Rand Co, in 1955, published 1 million numbers • generated using an electronic roulette wheel • has been used in some cipher designs cf Khafre • earlier Tippett in 1927 published a collection • issues are that: • these are limited • too well-known for most uses

  17. Review Number Theory The Devil said to Daniel Webster: "Set me a task I can't carry out, and I'll give you anything in the world you ask for." Daniel Webster: "Fair enough. Prove that for n greater than 2, the equation an + bn = cn has no non-trivial solution in the integers." They agreed on a three-day period for the labor, and the Devil disappeared. At the end of three days, the Devil presented himself, haggard, jumpy, biting his lip. Daniel Webster said to him, "Well, how did you do at my task? Did you prove the theorem?' "Eh? No . . . no, I haven't proved it." "Then I can have whatever I ask for? Money? The Presidency?' "What? Oh, that—of course. But listen! If we could just prove the following two lemmas—" —The Mathematical Magpie, Clifton Fadiman

  18. Prime Numbers • prime numbers only have divisors of 1 and self • they cannot be written as a product of other numbers • note: 1 is prime, but is generally not of interest • eg. 2,3,5,7 are prime, 4,6,8,9,10 are not • prime numbers are central to number theory • list of prime number less than 200 is: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

  19. Prime Factorisation • to factora number n is to write it as a product of other numbers: n=a x b x c • note that factoring a number is relatively hard compared to multiplying the factors together to generate the number • theprime factorisationof a number n is when its written as a product of primes • eg. 91=7x13 ; 3600=24x32x52

  20. Relatively Prime Numbers & GCD • two numbers a, b are relatively primeif have no common divisors apart from 1 • eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor • conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers • eg. 300=21x31x52 18=21x32hence GCD(18,300)=21x31x50=6

  21. Fermat's Theorem • ap-1 = 1 (mod p) i.e. remainder is 1 • i.e. the number (ap-1 − 1) is an integer multiple of p • where p is prime and gcd(a,p)=1 • also known as Fermat’s Little Theorem • also ap = a (mod p) • i.e., the number (ap − a) is an integer multiple of p • useful in public key and primality testing

  22. Euler Totient Function ø(n) • when doing arithmetic modulo n • complete set of residues is: 0..n-1 • reduced set of residues is those numbers (residues) which are relatively prime to n • eg for n=10, • complete set of residues is {0,1,2,3,4,5,6,7,8,9} • reduced set of residues is {1,3,7,9} • The number of elements in reduced set of residues is called the Euler Totient Function ø(n)

  23. Euler Totient Function ø(n) • to compute ø(n) need to count number of residues to be excluded • in general need prime factorization, but we have some special cases that allow us to quickly calculate ø(n) : • for p (p prime) ø(p) = p-1 • for p.q (p,q prime) ø(pq) =(p-1)x(q-1) • eg. ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12

  24. Euler's Theorem • It is a generalisation of Fermat's Theorem • aø(n) = 1 (mod n) • for any a,n where gcd(a,n)=1 • eg. a=3;n=10; ø(10)=4; hence 34 = 81 = 1 mod 10 a=2;n=11; ø(11)=10; hence 210 = 1024 = 1 mod 11

  25. Primality Testing • often need to find large prime numbers • traditionally sieve using trial division • ie. divide by all numbers (primes) in turn less than the square root of the number • only works for small numbers • alternatively can use statistical primality tests based on properties of primes • for which all primes numbers satisfy property • but some composite numbers, called pseudo-primes, also satisfy the property • can use a slower deterministic primality test

  26. Miller Rabin Algorithm • a test based on Fermat’s Theorem • algorithm is: TEST (n) is: 1. Find integers k, q, k > 0, q odd, so that (n–1)=2kq 2. Select a random integer a, 1<a<n–1 3. if aqmod n = 1then return (“maybe prime"); 4. for j = 0 to k – 1 do 5. if (a2jqmod n = n-1) then return(" maybe prime ") 6. return ("composite")

  27. Probabilistic Considerations • if Miller-Rabin returns “composite” the number is definitely not prime • otherwise is a prime or a pseudo-prime • chance it detects a pseudo-prime is < 1/4 • hence if repeat test with different random a then chance n is prime after t tests is: • Pr(n prime after t tests) = 1-4-t • eg. for t=10 this probability is > 0.99999

  28. Prime Distribution • prime number theorem states that primes occur roughly every ln(n) integers • but can immediately ignore even #’s • so in practice need only test 0.5 ln(n) numbers of size n to locate a prime • note this is only the “average” • sometimes primes are close together • other times are quite far apart

  29. Chinese Remainder Theorem • used to speed up modulo computations, it allows you to perform calculations modulo factors of your modulus, and then combine the answers to get the actual result. • if working modulo a product of numbers • eg. mod M = m1m2..mk • Chinese Remainder theorem lets us work in each moduli mi separately • since computational cost is proportional to size, this is faster than working in the full modulus M

  30. Chinese Remainder Theorem • can implement CRT in several ways • to compute A(mod M) • first compute all ai = A mod miseparately (i.e., don’t use big M, use small mi) • determine constants ci below, where Mi = M/mi • then combine results to get answer using: Final result!

  31. Primitive Roots • from Euler’s theorem have aø(n)mod n=1 • consider am=1 (mod n), GCD(a,n)=1 • must exist for m = ø(n) but may be smaller • once powers reach m, cycle will repeat • if smallest is m = ø(n)then a is called a primitive root • if p is prime, then successive powers of a "generate" the group mod p • these are useful but relatively hard to find

  32. Discrete Logarithms • the inverse problem to exponentiation is to find the discrete logarithmof a number modulo p • that is to find x such that y = gx (mod p) • this is written as x = logg y (mod p) • if g is a primitive root then it always exists, otherwise it may not, eg. x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive powers • whilst exponentiation is relatively easy, finding discrete logarithms is generally a hard problem

  33. Summary • have considered: • prime numbers • Fermat’s and Euler’s Theorems & ø(n) • Primality Testing • Chinese Remainder Theorem • Discrete Logarithms

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