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Applied Cryptography (Public key) Part I

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Let’s first finish “Symmetric Key” before talking about public key 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

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

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

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Placement of Encryption

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when using end-to-end encryption must leave headers in clear so network can correctly route information 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 end-to-end protects data contents over entire path and provides authentication link protects traffic flows from monitoring link protects traffic flows from monitoring

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

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Encryption vs Protocol Level

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

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

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Linear Congruential Generator common iterative technique using: X n+1 = (aX n + 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 function generates a full-period generated sequence should appear random generated sequence should appear random efficient implementation with 32-bit arithmetic 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

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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 X i = E Km [i] Output Feedback Mode X i = E Km [X i-1 ]

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ANSI X9.17 PRG

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Blum Shub Generator based on public key algorithms use least significant bit from iterative equation: x i = x i-1 2 mod n x i = x i-1 2 mod n where n=p.q, and primes p,q=3 mod 4 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

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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 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 problems of bias or uneven distribution in signal have to compensate for this when sample and use have to compensate for this when sample and use best to only use a few noisiest bits from each sample best to only use a few noisiest bits from each sample

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

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Chapter 8 – Introduction to 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 a n + b n = c n 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

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Prime Numbers prime numbers only have divisors of 1 and self they cannot be written as a product of other numbers they cannot be written as a product of other numbers note: 1 is prime, but is generally not of interest 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

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Prime Factorisation to factor a 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 the prime factorisation of a number n is when its written as a product of primes eg. 91=7x13 ; 3600=2 4 x3 2 x5 2 eg. 91=7x13 ; 3600=2 4 x3 2 x5 2

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Relatively Prime Numbers & GCD two numbers a, b are relatively prime if 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 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=2 1 x3 1 x5 2 18=2 1 x3 2 hence GCD(18,300)=2 1 x3 1 x5 0 =6 eg. 300=2 1 x3 1 x5 2 18=2 1 x3 2 hence GCD(18,300)=2 1 x3 1 x5 0 =6

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Fermat's Theorem a p-1 = 1 (mod p) where p is prime and gcd(a,p)=1 where p is prime and gcd(a,p)=1 also known as Fermat’s Little Theorem also a p = p (mod p) useful in public key and primality testing

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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, eg for n=10, complete set of residues is {0,1,2,3,4,5,6,7,8,9} complete set of residues is {0,1,2,3,4,5,6,7,8,9} reduced set of residues is {1,3,7,9} reduced set of residues is {1,3,7,9} number of elements in reduced set of residues is called the Euler Totient Function ø(n)

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Euler Totient Function ø(n) to compute ø(n) need to count number of residues to be excluded in general need prime factorization, but for p (p prime) ø(p) = p-1 for p (p prime) ø(p) = p-1 for p.q (p,q prime) ø(pq) =(p-1)x(q-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

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

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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 ie. divide by all numbers (primes) in turn less than the square root of the number only works for small numbers only works for small numbers alternatively can use statistical primality tests based on properties of primes for which all primes numbers satisfy property for which all primes numbers satisfy property but some composite numbers, called pseudo-primes, also satisfy the property but some composite numbers, called pseudo-primes, also satisfy the property can use a slower deterministic primality test

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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)=2 k q 2. Select a random integer a, 1
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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 Pr(n prime after t tests) = 1-4 -t eg. for t=10 this probability is > 0.99999 eg. for t=10 this probability is > 0.99999

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

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Chinese Remainder Theorem used to speed up modulo computations if working modulo a product of numbers eg. mod M = m 1 m 2..m k eg. mod M = m 1 m 2..m k Chinese Remainder theorem lets us work in each moduli m i separately since computational cost is proportional to size, this is faster than working in the full modulus M

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Chinese Remainder Theorem can implement CRT in several ways to compute A(mod M) first compute all a i = A mod m i separately first compute all a i = A mod m i separately determine constants c i below, where M i = M/m i determine constants c i below, where M i = M/m i then combine results to get answer using: then combine results to get answer using:

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Primitive Roots from Euler’s theorem have a ø(n) mod n=1 consider a m =1 (mod n), GCD(a,n)=1 must exist for m = ø(n) but may be smaller must exist for m = ø(n) but may be smaller once powers reach m, cycle will repeat 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

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Discrete Logarithms the inverse problem to exponentiation is to find the discrete logarithm of a number modulo p that is to find x such that y = g x (mod p) this is written as x = log g y (mod p) if g is a primitive root then it always exists, otherwise it may not, eg. x = log 3 4 mod 13 has no answer x = log 2 3 mod 13 = 4 by trying successive powers whilst exponentiation is relatively easy, finding discrete logarithms is generally a hard problem

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

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