# CS Computer and Network Security: Applied Cryptography

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1 CS Computer and Network Security: Applied Cryptography Professor Patrick Traynor Spring 2016

2 Reminders Project Ideas are due on Tuesday. Where are we with these? Assignment #2 is posted. Let s get to work! 2

3 Key Distribution/Agreement Key Distribution is the process where we assign and transfer keys to a participant Out of band (e.g., passwords, simple) During authentication (e.g., Kerberos) As part of communication (e.g., skip-encryption) Key Agreement is the process whereby two parties negotiate a key 2 or more participants Typically, key distribution/agreement this occurs in conjunction with or after authentication. However, many applications can pre-load keys 3

4 Diffie-Hellman Key Agreement The DH paper really started the modern age of cryptography, and indirectly the security community Negotiate a secret over an insecure media E.g., in the clear (seems impossible) Idea: participants exchange intractable puzzles that can be solved easily with additional information. Mathematics are very deep Working in multiplicative group G Use the hardness of computing discrete logarithms in finite field to make secure Things like RSA are variants that exploit similar properties 4

5 Diffie-Hellman Protocol For two participants p 1 and p 2 Setup: We pick a prime number p and a base g (<p) This information is public E.g., p=13, g=4 Step 1: Each principal picks a private value x (<p-1) Step 2: Each principal generates and communicates a new value y = g x mod p Step 3: Each principal generates the secret shared key z z = y x mod p Perform a neighbor exchange. 5

6 Attacks on Diffie-Hellman This is key agreement, not authentication. You really don t know anything about who you have exchanged keys with The man in the middle A B Alice and Bob think they are talking directly to each other, but Mallory is actually performing two separate exchanges You need to have an authenticated DH exchange The parties sign the exchanges (more or less) See Schneier for a intuitive description 6

7 Public Key Cryptography Public Key cryptography Each key pair consists of a public and private component: k + (public key), k - (private key) D(E(p, k + ),k )=p D(E(p, k ),k + )=p Public keys are distributed (typically) through public key certificates Anyone can communicate secretly with you if they have your certificate E.g., SSL-base web commerce 7

8 RSA (Rivest, Shamir, Adelman) A dominant public key algorithm The algorithm itself is conceptually simple Why it is secure is very deep (number theory) Use properties of exponentiation modulo a product of large primes "A method for obtaining Digital Signatures and Public Key Cryptosystems, Communications of the ACM, Feb., (2) pages

9 RSA Key Generation Pick two large primes p and q Calculate n = pq Pick e such that it is relatively prime to phi(n) = (q-1)(p-1) Euler s Totient Function d -1 = e (mod phi(n)) de mod phi(n) = 1 or 1. p=3, q=11 2. n = 3*11 = phi(n) = (2*10) = e = 7 GCD(20,7) = 1 5. Euclid s Algorithm d -1 = e (mod phi(n)) 1 = 7d mod 20 d = 3 9

10 RSA Encryption/Decryption Public key k + is {e,n} and private key k - is {d,n} Encryption and Decryption E(k+,P) : ciphertext = plaintext e mod n D(k-,C) : plaintext = ciphertext d mod n Example Public key (7,33), Private Key (3,33) Data 4 (encoding of actual data) E({7,33},4) = 47 mod 33 = mod 33 = 16 D({3,33},16) = 163 mod 33 = 4096 mod 33 = 4 10

11 Encryption using private key Encryption and Decryption E.g., E(k -,P) : ciphertext = plaintext d mod n D(k +,C) : plaintext = ciphertext e mod n E({3,45},4) = 43 mod 33 = 64 mod 33 = 31 D({7,45},19) = 317 mod 33 = 27,512,614,111 mod 33 = 4 Q: Why encrypt with private key? 11

12 Digital Signatures Models physical signatures in digital world Association between private key and document and indirectly identity and document. Asserts that document is authentic and non-reputable To sign a document Given document d, private key k- Signature S(d) = E( k-, h(d) ) Validation Given document d, signature S(d), public key k+ Validate D(k+, S(d)) = H(d) 12

13 Cryptanalysis and Protocol Analysis Cryptographic Algorithms Complex mathematical concepts May be flawed What approaches are used to prove correct/find flaws? Cryptographic Protocols Complex composition of algorithms and messages May be flawed What approaches are used to prove correct/find flaws? 13

14 RSA Exponent Problems Small Private Exponent Speeds decryption time However, Known Attacks Exist on Small Private Keys Due to Mike Wiener, can recover private key Result: If N is 1024 bits, d of private key must be at least 256 bits Some workarounds are known (e.g., based on Chinese Remainder Theorem), but not proven secure Small Public Exponent Speed signature verification time Smallest possible value is 3, but recommend Can recover M encrypted with multiple, small public keys Can recover private key from small public + bits of private 14

15 Timing Attacks Use the timing behavior of system to extract secret Suppose a smartcard stores your private key By precisely measuring the time it takes to perform private key ops, we can recover the key Due to Kocher At most 2n operations required, where n is the number of bits in the key Attack summary Adversary asks smartcard to generate signatures on several messages Recover one bit at a time starting with least significant Compare times to those measured offline Solution: blinding 15

16 Power Analysis Attacks Also, Discovered by Kocher Power usage is higher than normal in these computations Measure the timing of high power consumption Simple Power Analysis Direct interpretation of power measurements Reveals instructions executions Some crypto ops may be sensitive to data, e.g., DES S- boxes Differential Power Analysis Statistical analysis of power data correlations Solution: Must change the code! 16

17 Power and Timing What is the threat model in power/timing attacks? How does this conflict with the trust model? What is the vulnerability? 17

18 A Protocol Story Needham-Schroeder Public Key Authentication Protocol Defined in 1978 Assumed Correct Many years without a flaw being discovered Proven Correct BAN Logic So, It s Correct, Right? 18

19 Needham-Schroeder Public Key Protocol in detail: Nonce Message a.1: A B : A,B, {N A, A} PKB A initiates protocol with fresh value for B Message a.2: B A : B,A, {N A, N B } PKA B demonstrates knowledge of N A and challenges A Message a.3: A B : A,B, {N B } PKB A demonstrates knowledge of N B A and B are the only ones who can read N A and N B 19

20 Gavin Lowe Attack An active intruder X participates... Message a.1: A X : A,X, {N A, A} PKX Message b.1: X(A) B : A,B, {N A, A} PKB X as A initiates protocol with fresh value for B Message b.2: B X(A) : B,A, {N A, N B } PKA Message a.2: X A : X,A, {N A, N B } PKA X asks A to demonstrates knowledge of N B Message a.3: A X : A,X, {N B } PKX A tells X N B; thanks A! Message b.3: X(A) B : A,B, {N B } PKB X completes the protocol as A 20

21 What Happened? X can get A to act as an oracle for nonces Hey A, what s the N B in this message from any B? A assumes that any message encrypted for it is legit Bad idea X can enable multiple protocol executions to be interleaved Should be part of the threat model? 21

22 The Fix It s Trivial (find it) Message a.1: A --> B : A,B, {N A, A} PKB A initiates protocol with fresh value for B Message a.2: B --> A : B,A, {N A, N B, B} PKA B demonstrates knowledge of N A and challenges A Message a.3: A --> B : A,B, {N B} PKB A demonstrates knowledge of N B 22

23 Impact on Protocol Analysis Protocol Analysis Took a Black Eye BAN Logic Is Insufficient BAN Logic Is Misleading Protocol Analysis Became a Hot Topic Lowe s FDR Meadow s NRL Analyzer Millen s Interrogator Rubin s Non-monotonic protocols In the end, could find known flaws, but... attacker model is too complex 23

24 Dolev-Yao Result Strong attacker model Attacker intercepts every message Attacker can cause one of a set of operators to be applied at any time Operators for modifying, generating any kind of message Attacker can apply any operator except other s decryption Theoretical Results Polynomial Time for One Session Undecidable for Multiple Sessions Moral: Analysis is Difficult Because Attacker Can Exploit Interactions of Multiple Sessions End Result: Manual Induction and Expert Analysis are the main approaches 24

25 Review: secret vs. public key crypto Secret key cryptography Symmetric keys, where A single key (k) is used is used for E and D D( E( p, k ), k ) = p All (intended) receivers have access to key Note: Management of keys determines who has access to encrypted data E.g., password encrypted Also known as symmetric key cryptography Public key cryptography Each key pair consists of a public and private component: k+ (public key), k- (private key) D( E(p, k+), k- ) = p D( E(p, k-), k+ ) = p Public keys are distributed (typically) through public key certificates Anyone can communicate secretly with you if they have your certificate E.g., SSL-based web commerce 25

26 The symmetric/asymmetric key tradeoff Symmetric (shared) key systems Efficient (Many MB/sec throughput) Difficult key management Kerberos Key agreement protocols Asymmetric (public) key systems Slow algorithms (so far ) Easy (easier) key management PKI - public key infrastructures Webs of trust (PGP) 26

27 Crypto... the end? This marks the end of our coverage of cryptographic building blocks for the semester. There are open problems in crypto (e.g., quantumresistant systems) However, we have plenty of basic constructions to help us build secure systems. You will see crypto pop up throughout the semester, but those with more interest should take the crypto course here at UF. 27

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