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Modern Cryptography New Directions in Cryptography W.Diffie & M.E.Hellman Probabilistic Encryption S.Goldwasser & S.Micali

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By Theoretically – Perfect secrecy [Shannon]. NOT MUCH BESIDES… The notion of a function easy to compute but hard to “inverse” arose... [Purdy] Complexity: NP (completeness) vs. P [Cook, Karp]. Practically – Computers and “Private key security” exist (DES), and are becoming more and more applicable.

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In fact, computers and cryptography go hand in hand from the first computers. (WWII) In fact, there were confidential papers in cryptography (in CESG): Non-secret-encryption [J.H.Ellis ‘70] (with a proof!) ¼RSA [C.C.Cocks ’73] By (hush hush!)

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By (biographical details) In 1972, Whitfiled Diffie, an AI graduate student, developes more than an interest in cryptography. In 1974, at the age of 30, he phones Martin Hellman, assistant professor in Stanford, to discuss issues in crypto. They begin collaborating. In 1975, Diffie thinks of quitting altogether. "I was worried that I wasn't particularly remarkable as a programmer and that my lot in life would get progressively worse if things continued going as they were." Also In 1975, he bares success. "The thing I remember distinctly is that I was sitting in the living room when I thought of it the first time and then I went downstairs to get a Coke and I almost lost it," he says. "I mean, there was this moment when - I was thinking about something. What was it? And then I got it back and didn't forget it."

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New Directions in Cryptography W.Diffie & M.E.Hellman HellmanDiffie We stand today on the brink of a “We stand today on the brink of a revolution in cryptography”

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Emphasis 1.NO definitions, notations, claims, proofs etc. This is an invited paper, so: 2. HOWEVER: clever ideas, clever insights! 3. Practicality. Historical survey.

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So, what do we have in “ conventional cryptographic system ” (block or stream)? S k :{P}!{C}

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“ Conventional Cryptographic System ” Goal: Enciphering and deciphering – “inexpensive”, but any “cryptananlytic operation” is “too complex to be economical”. “We call a task computationally infeasible, if its cost... is finite but impossibly large.” Important desired property- Error propagation: “A small change in the input block produces a major change in the resulting output”.

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“ Conventional Cryptographic System ” Threats: ({S k } is known) Eavesdropping – “Ciphertext only”, “Known plaintext”, “Chosen plaintext”. Injecting – new messages, or combining/repeating. Problems: 3. n users ) (n 2 ) keys. 1. Where does the secure channel comes from? 2. Authentication & Signature.

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Introducing: THE PUBLIC KEY CRYPTOSYSTEM!

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THE PUBLIC KEY CRYPTOSYSTEM! Two families {E k } k, {D k } k of invertible transformations, E k, D k :{M}!{M}, s.t. the following holds: 1. 8 k, E k is the inverse of D k k, 8 m2{M}, E k (m), D k (m), are “easy to compute”. 3. For almost every k, each easily computed algorithm equivalent D k to is computationally infeasible to derive given E k k, it easy to come up with the pair h D k, E k i. Publicize E k, but keep D k to yourself! RANDOMIZED!

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Suggestions 1.(useless) An invertible matrix E, D = E -1. (n 2 vs. n 3, at the time) 2.“One way compiler”. Public Key Distribution System: “Securely exchange a key over an insecure channel”. 3. Merkle. 4. The Diffie-Hellman key exchange.

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The DH Key Exchange Everybody knows: q – a prime, g – a generator for Z * q A Selects x A 2 r Z* q. Sends m A = g x A mod q. Computes K = m B x A mod q. B Selects x B 2 r Z* q. Sends m B = g x B mod q. Computes K = m A x B mod q. K = g x A x B mod q. Secure, if discrete log takes (q 1/2 )

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Signature By public key cryptosystem! A function f is a one-way function if it is easy to compute f(x), but for almost every y it is “computationally infeasible to solve the equation y=f(x).” (“Polynomials offer an elementary example of one-way functions.” “One way functions are easy to devise.”) Just send - h m, D k (m)i. One Way

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One Way Authentication Techniques: 1.Login: user picks PW, but sends f(PW). 2.Login revised: user picks PW, send f T (PW). At time t, user authenticates by sending f T-t (PW) (requires fast enumerations of f). 3. Select x 0 1,x 1 1,x 0 2,x 1 2,…,x 0 N,x 1 N. Compute their images under f: y 0 1, y 1 1, y 0 2, y 1 2,…,y 0 N,y 1 N. Publicize these 2N images. Send the message m = m 1,m 2,…m N and x 1 m 1,x 2 m 2,…,x N m N

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Insights “A cryptosystem which is secure against a known plaintext attack, can be used to produce a OWF”. Choose P 0 arbitrarily. Define: f(x) = S x (P 0 )

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Insights (cont.) Trap-door OWF: a simply computed inverse exists, but given only f it is infeasible to find an inverse. Only possession of a trap-door information allows computing an inverse easily. (e.g. The random string used to produce E,D.) (A quasi-OWF: same definition, without the trap-door information.) Trap-door cipher: resists any cryptanalysis by anyone not in possession of a trap-door information. “A trap-door cryptosystem can be used to produce a public key distribution system”. A enciphers and publicize m, E k (m), B breaks the encryption.

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Insights (cont.) Public Key Cryptosystem ) OW authentication. “Not conversly”. Public Key Cryptosystem ) Public Key Distribution System. “The converse does not appear to hold”. Public Key Cryptosystem ) Trap-door OWF. The converse – the function “must be invertible”

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Connection to Complexity “The cryptanalytic difficulty of a system whose encryption and decryption operations can be done in P time cannot be greater than NP”. Nondeterministically, choose the key (maybe also the message). Verify by encryption / decryption in polytime. “The general cryptanalytic problem is NP-complete.” By Constructing a OWF from the Knapsack Problem.

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The Knapsack Problem Given {a 1, a 2, …, a n }, and x2{0,1} n, computing y=f(x)= i a i x i is easy, yet finding a subset of {a i } i that sums up to a given y is NP-complete. Problems: 1. f cannot be degenerate. 2. f cannot be super-increasing. Is f hard on average? …Probably not. Knapsack based encryption – given `77 [Merkle, Hellman], broken `82 [Shamir] and later others.

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Historical Note From Caesar cipher to WWII. References – a book [~ 1200 pages]: D. Kahn, The Codebreakers, The Story of Secret Writing. Emphasize the following point: “innovation has come primarily from the amateurs”. “We hope this will inspire others to work in this facinating area in which participation has been discouraged in the recent past by a nearly total government monopoly.”

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And what happened to Diffie & Hellman? Diffie didn't finish his degree, left to work in cryptography oriented companies. Works till today. Was awarded doctorate in 1992 (!) by the Swiss Federal IT. Hellman became a prof. in `79 and is currently retired. Both – highly respected, highly awarded.

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After DH: Practical Public Keys Several suggestions, including the knapsack, and McEliece (ECC of invertible matrix and permutation + a random small mistake) – RSA! 1979 – Rabin (RSA with squaring) Mathematical proofs of security: – Blum; Goldwasser & Micali.

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