# Introduction to Computer Security

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1 Introduction to Computer Security Hash Functions and Digital Signatures Pavel Laskov Wilhelm Schickard Institute for Computer Science

2 Integrity objective in a wide sense Reliability Transmission errors Corruption of stored data Security Manipulation of data in transmission Manipulation of stored data

3 Integrity checking: a general framework Compute a for the original message D = h(m), such that. P(D is corrupt) P(M is corrupt) To check the integrity of a message M at a later time, compute D = h(m ) and verify that D = D.

4 Hash functions Hash function converts large, variable size input into small, fixed size output. Applications: Efficient search (hash tables) Indexing of variable size data Finding duplicate entries Finding similar entries keys John Smith Lisa Smith Sam Doe Sandra Dee hash function Requirements: Efficiency: less that log 2 n comparisons Determinism: always maps the same input to the same output Small probability of collisions Uniformity: equal probability of output values hashes : 15

5 Hash function design Fixed length (numeric) keys: Division method: h(k) = k mod m Multiplication method: h(k) = m(ka mod 1) Variable length keys (e.g. strings): Convert a string into a fixed number (e.g. add up all characters) Compute a hash of a fixed number Hash function reduces the dimension of the key set: Collisions are unavoidable!

6 A simple message application Hashing algorithm Alice Bob Hashing algorithm Transmit equal?

7 Insecurity of message Alice Eve Bob Hashing algorithm A A B B B A A B B B equal? B

8 authentication code (MAC) Hashing algorithm Alice Bob Hashing algorithm Transmit Shared secret Shared secret equal?

9 Secure hash function requirements Compression: h reduces M to a fixed size. For any M, h(m) is easy to compute. Preimage resistance: For any value D, it is computationally infeasible to find M such that D = h(m). Second preimage resistance: For any values D and M such that D = h(m), it is computationally infeasible to find M = M such that D = h(m ). Collision resistance: It is computationally infeasible to find any pair M 1, M 2 such that h(m 1 ) = h(m 2 ).

10 A birthday paradox How many people must be in a room for someone to have the same birthday as you?

11 A birthday paradox How many people must be in a room for someone to have the same birthday as you? (365)

12 A birthday paradox How many people must be in a room for someone to have the same birthday as you? (365) How many people must be in a room for some people to have the same birthday?

13 A birthday paradox How many people must be in a room for someone to have the same birthday as you? (365) How many people must be in a room for some people to have the same birthday? (23)

14 A birthday paradox How many people must be in a room for someone to have the same birthday as you? (365) How many people must be in a room for some people to have the same birthday? (23) Probability that n people have different birthdays is: ( P(n) = ) ( 1 2 ) (... 1 n 1 ) ) ) ) (e (e (e n = e ( n ) e n2 Solving for n and equating to 0.5, we obtain: n = 2 ln = 22.54

15 Brute force attacks For an ideal hash function with the output size n, it should take 2 n operations to stage a second-preimage attack, 2 n/2 operation to stage a collision attack. A cryptographic strength of a hash function strongly depends on its output size.

16 A simple hash function x 1 x 2... x n IV Fix an initialization value IV. Compute intermediate states s 1 = IV + x 1 s i = s i 1 + x i, i = 2,..., n Output the final state s n as a hash value

17 Insecurity of a simple hash function Suppose the D is known. The IV can be found by sending the word (0, 0,..., 0). Then the message (IV, D, 0,..., 0) will produce the hash value D: the second preimage property is broken!

18 General design of a hash function x 1 x 2 x n IV f s 1 f s 2... f s n Iterated application of a compression function s i f (x i, s i 1 ) s 0 is initialized to some fixed IV Collision resistance of f implies collision resistance of a hash function (Merkle s principle).

19 MD5 algorithm Pad a message to a the length 448 mod 512, add message length as a 64-bit value. Initialize 4 32-bit registers A, B, C, D with pre-defined values. Divide each 512-bit block in 16 words w of length 32; for i = , do: compute A + f i (B, C, D) add M i = w gi add K i = sin(i + 1) 2 32 shift left by s i positions Add B and save in B, C B, D C, A D Mi Ki A B C D <<< s F A B C D Proceed to the next block using the state A, B, C, D.

20 Insecurity of MD5 [1996]: H. Dobbertin demonstrated a collision of the compression function of MD5. [2004]: X. Wang and H. Yu showed a modular differential attack on the complete MD5 hash function. [2007]: M. Stevens, A. Lenstra and B. de Weger demonstrated a chosen prefix attack: given a prefix P, find suffixes S 1 and S 2 such that MD5 hashes of P S 1 and P S 2 are the same. [2008]: A. Sotirov et al. generated a rogue X.509 certificate to demonstrate practical consequences of MD5 vulnerability.

21 SHA-1 algorithm 160-bit output instead of 128 in MD5; 80 rounds per 512-bit block. Similar initialization and overall scheme. 16 initial words w i are extended into the total of 80 words, one per round: w i = (w i 3 w i 8 w i 14 w i 16 ) 1 Fixed constants and shifts per round. Best known collision attack requires 2 69 operations, compared to 2 80 by brute force. A B C D E F <<< 5 <<< 30 A B C D E W t K t

22 Authentication The reciever must verify the claimed identity of a sender. The sender cannot deny having sent a message. The reciever cannot have created the message himself.

23 Symmetric key signatures A,K A (B,P) Alice TTP K B (A,P,K T (A,P)) Bob Identity of A is proved to T by K A. The fact of A s sending a message is proved by K T (A, P). B cannot forge having received a message from A because he does not know K T.

24 Public key signatures Alice s computer Bob s computer P Alice s private key, K r A Alice s public key, K u A P D K r A (P) E K u A (D K r A (P)) Identity of A is proved by B s being able to encrypt a message with K u A. The fact of A s sending a message is proved by the existence of a message decrypted by K r A. B cannot forge having received a message from A because he cannot produce D K r A (M).

25 Digest signatures: DSS Alice s private key, K r A Alice s plaintext message M SHA-1 algorithm H ElGamal algorithm D K r A (H) Sent to Bob For efficiency reasons, public-key decryption is applied to a short of the plaintext message. ElGamal public-key encryption/decryption algorithm is used.

26 Digest signature application Private key Signature algorithm Alice Bob Hashing algorithm Transmit Public key Public key equal?

27 Digest signature attack Private key Signature algorithm Eve Bob Hashing algorithm Transmit Public key Public key equal?

28 Public key certificates A certificate is a binding between an entity name and its public key. Certificates are issued by a certification authority (CA), a trusted third party. A certificate is generated locally on a computer. To grant a certificate its validity, a CA signes it with its private key. Since the CA s public key is well known everybody can verify the validity of a certificate.

29 The structure of a X.509 certificate

30 Signature computation The signature in the X.509 certificate is computed as: where CA<<A>>= K CA [V, SN, AI, CA, T A, A, K + A ], V : X.509 version SN : certificate serial number AI : Algorithm ID CA : certificate authority name T A : validity timespan of the certificate K + A A : subject name : subject public key

31 Certificate chains D D F B F C E G A C E G If a client trusts a root CA, it uses a top-down trust chain to verify a target certificate: D<<B>> B<<C>> If a client trusts a local CA, e.g., G, is must traverse the certificate hierarchy: G<<F>> F<<D>> D<<B>> B<<C>>

32 Rogue certificate authority An attacker obtains a legitimate (end) certificate G D He creates a rogue certificate H and signs it with G as a CA E F G H

33 Rogue certificate authority An attacker obtains a legitimate (end) certificate G D He creates a rogue certificate H and signs it with G as a CA The use of certificates as CA is restricted by the extension field basicconstraints E F G H

34 Summary Integrity of data can be enforced by computing cryptografic hash functions (one-way, collision-resistant). Authentication and non-repudiation objectives are attained by digital signatures that combine public key cryptography with secure hashing. Binding of digital signatures to entities is achieved by putting the relevant information in X.509 certificate issued by a trusted certification authority (CA).

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