David Evans CS588: Security and Privacy University of Virginia Computer Science Lecture 12: Non-secret Key Cryptosystems (How Euclid, Fermat and Euler Created E-Commerce) Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers. G. H. Hardy, The Mathematician’s Apology, 1940.

CS588 Spring Applications of RSA Privacy: –Bob encrypts message to Alice using E A –Only Alice knows D A Signatures: –Alice encrypts a message to Alice using D A –Bob decrypts using E A –Knows it was from Alice, since only Alice knows D A Things you use every day: ssh, SSL, DNS,...

CS588 Spring Public-Key Applications: Privacy Alice encrypts message to Bob using Bob’s Private Key Only Bob knows Bob’s Private Key  only Bob can decrypt message Encrypt Decrypt Plaintext Ciphertext Plaintext Alice Bob Bob’s Public Key Bob’s Private Key

CS588 Spring Signatures Bob knows it was from Alice, since only Alice knows Alice’s Private Key Non-repudiation: Alice can’t deny signing message (except by claiming her key was stolen!) Integrity: Bob can’t change message (doesn’t know Alice’s Private Key) Encrypt Decrypt Plaintext Signed Message Plaintext Alice Bob Alice’s Private Key Alice’s Public Key

CS588 Spring Public-Key Cryptography Private procedure: E Public procedure: D Identity: E (D (m)) = D (E (m)) = m Secure: cannot determine E from D But didn’t know how to find suitable E and D

CS588 Spring Properties of E and D Trap-door one way function: 1. D (E (M)) = M 2. E and D are easy to compute. 3.Revealing E doesn’t reveal an easy way to compute D Trap-door one way permutation: also 4. E (D (M)) = M

CS588 Spring RSA E(M) = M e mod n d D(C) = C d mod n pqpq n = pqp, q are prime d d is relatively prime to (p – 1)(q – 1) d ed  1 (mod (p – 1)(q – 1)) red (red things are secret)

CS588 Spring Properties of E and D Trap-door one way function: 1. D (E (M)) = M 2. E and D are easy to compute. 3.Revealing E doesn’t reveal an easy way to compute D Trap-door one way permutation: also 4. E (D (M)) = M

CS588 Spring Property 1: D (E (M)) = M E(M) = M e mod n D(E(M)) = (M e mod n) d mod n = M ed mod n (as in D-H proof) Can we choose e, d and n with this property: M  M ed mod n equivalently: 1  M ed-1 mod n

CS588 Spring Finding e, d and n We are looking for e, d and n such that: M ed-1  1 mod n Euler’s Theorem: for a and n relatively prime: a  (n)  1 mod n Next: –What is  (n) –Proof of Euler’s Theorem –How it works for arbitrary M –Given  (n) how do we find e and d

CS588 Spring Euler’s Totient Function  (n) = number of positive integers less than n that are relatively prime to n If n is prime,  (n) = n – 1 –Proof by contradiction What if n = pq where p and q are prime?

CS588 Spring Totient Products For primes, p and q : n = pq  (n) = numbers < n not relatively prime to pq = pq – 1 ; numbers less than pq – (q – 1) ; size of p, 2p, …, (q – 1)p – (p – 1) ; size of q, 2q, …, (p – 1)q = pq – 1 – (q – 1) – (p – 1) = pq – (p + q) + 1 = (p – 1) (q – 1) =  (p)  (q)

CS588 Spring Fermat’s Little Theorem If n is prime and a is not divisible by n a n-1  1 mod n

CS588 Spring Fermat’s Little Theorem Proof If n is prime and a is not divisible by n : {a mod n, 2a mod n, …, (n-1)a mod n} = {1, 2, …, (n – 1) } Product of all elements in sets: a  2a  …  (n – 1) a  (n – 1)! mod n (n – 1)!a n-1  (n – 1)! mod n a n-1  1 mod n QED.

CS588 Spring Euler’s Theorem For a and n relatively prime: a  (n)  1 mod n Partial Proof: If n is prime,  (n) = n – 1 and a n - 1  1 mod n by Fermat’s Little Theorem What if n is not prime?

CS588 Spring Euler’s Theorem, cont. For a and n relatively prime: a  (n)  1 mod n  (n) = number of numbers < n not relatively prime to n We can write those numbers as: R = { x 1, x 2, …, x  (n) }

CS588 Spring Proving Euler’s Theorem R = { x 1, x 2, …, x  (n) } multiply by a mod n : S = { ax 1 mod n, ax 2 mod n, …, ax  (n) mod n} S is a permutation of R : a is relatively prime to n a is relatively prime to all x i ax i is relatively prime to n –Hence all elements of S are in R. –There are no duplicates in S. If ax i mod n = ax j mod n then i = j. since a is relatively prime to n

CS588 Spring Proving Euler’s Theorem x 1  x 2 …  x  (n) = ax 1 mod n  ax 2 mod n …  ax  (n) mod n  (ax 1  ax 2 …  ax  (n) ) mod n  a  (n)  x 1  x 2 …  x  (n) mod n 1  a  (n) mod nQED.

CS588 Spring Recap We are looking for e, d and n such that: M ed-1  1 mod n Euler’s Theorem: 1  a  (n) mod n for a and n relatively prime If n is prime,  (n) = n – 1. For p and q prime,  (pq) =  (p)  (q) ed – 1 =  (n) = (p-1)(q-1) What if M is not relatively prime to n ? n = pq

CS588 Spring M and n Suppose M and n not relatively prime: gcd (M, n)  1 Since n = pq and p and q are prime: gcd (M, p)  1 OR gcd (M, q)  1 Case 1: M = cp gcd (M, q) = 1 (otherwise M is multiple of both p and q, but M < pq). So, M  (q)  1 mod q (by Euler’s theorem, since M and q are relatively prime)

CS588 Spring M and n, cont Case 1: M = cp gcd (M, q) = 1 (otherwise M is multiple of both p and q, but M < pq). So, M  (q)  1 mod q (by Euler’s theorem, since M and q are relatively prime) M  (q)  1 mod q (M  (q) )  (p)  1 mod q M  (q)  (p)  1 mod q M  (n)  1 mod q

CS588 Spring M and n M  (n)  1 mod q M  (n) = 1 + k q for some k M = cp recall gcd (M, p)  1 M  M  (n) = (1 + k q)cp M  (n) + 1 = cp + k qcp = M + k cn M  (n) + 1  M mod n

CS588 Spring Where’s ED? ed – 1 =  (n) = (p-1)(q-1) So, we need to choose e and d: ed =  (n) + 1 = n – (p + q) Pick random d, relatively prime to  (n) gcd (d,  (n)) = 1 Since d is relatively prime to  (n) it has a multiplicative inverse e : de  1 mod  (n)

CS588 Spring Identity de  1 mod  (n) So, d * e = (k *  (n)) + 1 for some k. Hence, M ed-1 mod n = M k *  (n) mod n

CS588 Spring D (E (M)) = M M ed-1 mod n = M k *  (n) mod n Euler says 1  M  (n) mod n. So 1  M k *  (n) mod n 1  M ed-1 mod n M  M ed mod n QED.

CS588 Spring Properties of E and D Trap-door one way function: 1. D (E (M)) = M 2. E and D are easy to compute. 3.Revealing E doesn’t reveal an easy way to compute D Trap-door one way permutation: also 4. E (D (M)) = M 

Movie Break Adam Glaser and Portman Wills CS588 Fall 2001 PS4

CS588 Spring Questionable Statements in RSA Paper: Finalists 1) "The reader is urged to find a way to "break" the system. Once the method has withstood all attacks for a sufficient length of time it may be used with a reasonable amount of confidence." The authors appear to advocating the same method of validation that they called "fruitless" earlier in the paper (referring to the NBS certification). 2) RSA seems to gloss over the whole PKI issue. They suggest either a single authority to hold all the keys or publishing a book to all the users. I'd also like to point out that RSA like to "excessively" to quotation marks for no apparent "purpose." 1. The problem is mentioned on page 4 and 6 of the paper: The trusted distribution of the public portion of the key. If one were to modify the public keys in transport or to attack a central repository, then it would be impossible to be sure of the authenticity of the keys. (The suggestion of having a telephone book is not even possibly applicable due to the need to securely deliver it to all users from trusted central source). This can be seen as present problem with the loss of keys by Microsoft (resulting in forced revocation) and the limited trust one can put in Verisign due to limited checks done. 2. The assumption in conclusion that a protocol is secure due to lack of success in attacks for some period of time. The recent attacks upon SHA/MD5 show that even 10 years could be insufficient time to prove security.

CS588 Spring Only Two Submissions This is pathetic! There will be a Short Quiz in class Tuesday –Closed book, closed notes –Covers material in RSA paper and new paper handed out today –Andrew and Aleks are exempt

CS588 Spring Two “Questionable” Statements in RSA Paper 1.“The need for a courier between every pair of users has thus been replaced by the requirement for a single secure meeting between each user and the public file manager when the user joins the system.” (p. 6)

CS588 Spring Two “Questionable” Statements in RSA Paper 2.“(The NBS scheme (DES) is probably somewhat faster if special-purposed hardware encryption devices are used; our scheme may be faster on a general-purpose computer since multiprecision arithmetic operations are simpler to implement than complicated bit manipulations.)” (p. 4)

CS588 Spring Who really invented RSA? General Communications Headquarters, Cheltenham (formed from Bletchley Park after WWII) 1969 – James Ellis asked to work on key distribution problem Secure telephone conversations by adding “noise” to line Late 1969 – idea for PK, but function

CS588 Spring RSA & Diffie-Hellman Asks Clifford Cocks, Cambridge mathematics graduate, for help He discovers RSA (four years early) Then (with Malcolm Williamson) discovered Diffie-Hellman Kept secret until 1997! NSA claims they had it even earlier

CS588 Spring Charge Reread the parts of RSA paper you didn’t understand the first time Work on your project! Short Quiz on RSA material and Encrypted Searches paper in class Tuesday –Closed-book, closed-notes, open-T-shirt Next time: RSA Properties 2, 3 and 4