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Elgamal demonstration project on calculators TI-83+ Gerard Tel Utrecht University With results from Jos Roseboom and Meli Samikin

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Workshop Elgamal 2 Overview of the lecture 1.History and background 2.Elgamal (Diffie Hellman) 3.Discrete Log: Pollard rho 4.Experimentation results 5.Structure of Function Graph: Cycles, Tails, Layers 6.Conclusions

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Workshop Elgamal 3 1. History and background , lecture for school teachers about Elgamal , lecture with calculator demo Why Elgamal, not RSA? Functional property easy to show Security: rely on complexity Compare exponentiation and DLog

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Workshop Elgamal 4 Programming Experiences Nuisances: –typing by selecting symbols –no subroutines: inline exponentiation –no local variables Limitation: arithmetic in 14 digits –Limit modulus to 7 digits

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Workshop Elgamal 5 Math: Modular arithmetic Compute modulo prime p (95917) with 0, 1, … p-2, p-1 Generator g of order q (prime) (g is 29609, q is 7993) Rules of algebra are valid (g a ) k = (g k ) a Secure application: p has ~309 digits!!

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Workshop Elgamal 6 Calculator TI-83, 83+, 84+ Grafical, 14 digit Programmable Generally available in VWO (pre- academic school type in the Netherlands) Cost 100 euro (free for me)

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Workshop Elgamal 7 The Elgamal program Ceasar cipher (symmetric) Elgamal parameter and key generation Elgamal encryption and decryption Discrete Logarithm: Pollard Infeasible problem!! But doable for 7 digit modulus

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Workshop Elgamal 8 2. Public Key codes The problem of Key Agreement: A and B are on two sides of a river They want to have common z Oscar is in a boat on the river Oscar must not know z Common parameters: p, q, g (Or: group with hard DLog problem)

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Workshop Elgamal 9 Solution: Diffie-Hellman Alice takes random a, shouts b = g a Bob takes random k, shouts u = g k Alice computes z = u a = (g k ) a Bob computes z = b k = (g a ) k The two numbers are the same The difference in complexity for A&B and O is relevant

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Workshop Elgamal 10 Parameter generation Hoofdmenu, parameters, Maak p,q,r Input limits on q and p Search for prime q from q-limit down Search for prime p from p-limit down among multiples of 2q + 1 Generator: try 100 (p-1)/q, 101 (p-1)/q, …

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Workshop Elgamal 11 What does Oscar hear? Seen: 1.Public b = g a 2.Public u = g k Not computable: 1.Secret a, k 2.Common z This needs discrete logarithm Oscar sees the communication, but not the secrets

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Workshop Elgamal 12 The Elgamal program In class use Program, explanation, slides on website Program extendible Booklet with ideas for experimenting, papers All in Dutch!

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Workshop Elgamal Pollard Rho Algorithm Fixed p (modulus), g, q (order of g); H is set of powers of g Size of H is q Discrete Logarithm problem: –Given y in H –Return x st g x = y Pollard Rho: randomized, √q time

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Workshop Elgamal 14 Pollard Rho: Representation Representation of z: z = y a.g b Two representations of same number reveil log y: If y a.g b = y c.g d, then y = g (b-d)/(c-a) Goal: find 2 representations of one number z (value does not matter)

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Workshop Elgamal 15 Strategy: Birthday Theorem All values z = y a.g b are in H Birthday Theorem: In a random sequence, we expect a collision after √q steps Simulate effect of random sequence by pseudorandom function: z i+1 = f (z i ) (Keep representation of each z i )

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Workshop Elgamal 16 Cycle detection Detect collision by storing previous values: too expensive Floyd cycle detection method: –Develop two sequences: z i and t i –Relation: t i = z 2i –Collision: t i = z i, i.e., z i = z 2i In each round, z “moves” one step and t moves two steps.

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Workshop Elgamal Experimentation results pqxm12345Ave , , , Spring 2006, by Barbara ten Tusscher, Jesse Krijthe, Brigitte Sprenger

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Workshop Elgamal 18 Barbara, Jesse, Brigitte Verify Pollard rho analysis Use various values of p, q, y Clear dependence of time on q Ignoring 80, cor- relation to √q is overly exact. pqav. it

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Workshop Elgamal 19 Dependence on y Run same p, q combination with different inputs y = g x Correspondence to √q again Not to x: the log of small power of g is no easier pqxtime

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Workshop Elgamal 20 Surprise: individual numbers pqx Iterations: equal or have high common factor!

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Workshop Elgamal 21 Observations Average number of iterations coincides well with √q Almost no variation within one row Is this a bug in the program?? –Bad randomization in calculator? –Or general property of Pollard Rho?

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Workshop Elgamal Function graph Function f: z i -> z i+1 defines graph Out-degree 1, cycles with in-trees Length, component, size Graph is the same when algorithm is repeated with the same input Starting point differs As z i = z 2i, i must be multiple of cycle length

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Workshop Elgamal 23 Layers in a component Layer of node: measure distance to cycle in terms of its length l: –Point z in cycle has layer 0 –Point z is in layer 1 if f (l) (z) in cycle –Point z is in layer c if f (c.l) (z) in cycle Lemma: z 0 in layer c gives c.l iter. Is there a dominant component or layer?

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Workshop Elgamal 24 Layers 0 and 1 dominate Probability theory analysis by Meli Samikin Lemma: Pr(layer ≤ 1) = ½ Proof: Assume collision after k steps: z 0 -> z 1 -> … -> … -> z k-1 -> ?? Layer of z 0 is 0 if z k = z 0, Pr = 1/k Layer of z 0 is 1 if z k = z j < k/2, Pr ≈ 1/2

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Workshop Elgamal 25 Dominant Component Lemma: Random z 0 and w 0, Pr(same component) > ½. Proof: First collision after k steps: z 0 -> z 1 -> … -> … -> z k-1 -> ?? w 0 -> w 1 -> … -> … -> w k-1 -> ?? Pr ( z meets other sequence ) = ½. Then, w-sequence may collide into z.

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Workshop Elgamal 26 Experiments: dominance Jos Roseboom: count points in layers of each component ACS Experimentation Project, Fall 2007 Explicitly construct and measure function graphs

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Workshop Elgamal 27 Size of largest component

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Workshop Elgamal 28 Conclusions Elgamal + handcalculators = fun Functional requirements easier to explain than for RSA Security: experiment with DLog Pollard, only randomizes at start Iterations: random variable, but takes only limited values Most often: size of heaviest cycle

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Workshop Elgamal 29 Rabbit Formula Ontsleutelen is: v delen door u a u (a1+a2) is: u a1.u a2 Deel eerst door u a1 en dan door u a2 Team 1: bereken v’ = Dec a1 (u, v) Team 2: bereken x = Dec a2 (u, v’)

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Workshop Elgamal 30 Overzicht van formules Constanten: Priemgetal p, grondtal g Sleutelpaar: Secret a en Public b = g a Encryptie: (u, v) = (g k, x.b k )met b Decryptie: x = v/u a met a Prijsvraag: b = b 1 b 2. Ontsleutelen?

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