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1 Message Authentication Codes Reading: Chapter 4 of Katz & Lindell 1

2 Message authentication • m Bob receives a message from Alice, he wants to know  he message was (Data origin authentication) whether t e. really sent by Alic  e message has been modified. (Data integrity) whether t h • Two solutions: (MAC)  message authenti Alice attaches a code cation to the message (using a symmetri c key to compute the MAC). hes Or a u signat e attac sh  to the message (using an re gnature asymmetric key to compute the si ). 2

3 Basic idea of MAC Message authentication protocol: • . 1. Alice and Bob share a secret key k s a tag : ( ) 2. To send a message , Alice compute MAC m m t = k , to Bob. and sends mt ) ( ′′ ′ ′ 3. On receiv , Bob checks whether ( ). , ing = MAC m m t t ( ) k therwise, he reje If so, he accepts the message; o cts it. code (MAC The tag is called a . t • ) message authentication mputat • ionally infeasible to forge a Security requirement: co k x MAC x he key . valid pair ( , ( )) without knowing t k 3

4 MAC Scheme (formal definition) , ): A MAC scheme is a triple ( , • Gen Mac Vrfy n Key generation algorithm: On input 1 , outputs  n {0,1} . a key k ← u ut p Tag generation algorithm : On in Mac  k a key and a message outputs a tag . We write ( ). ∈← , t t m M Mac Mac m k * () ln MM or {0,1} e space. Assume M {0,1} .) is the messag ( = = m Vrfy Verification algorithm k : On input a key , a  essage , and m Vrfy t ning ) outputs 1 (mea a tag , algorithm id val = Vrfy m t inv id al ( , ) 0 or 1. ). or 0 ( k Vrfy  Gen Mac is deterministic. , are probabilistic algorithms. • ∈∈ uirement: for every and , k K mM req Correctness , ( ) 1. Vrfy m Mac m = ) ( kk 4

5 is deterministic): Canon (used when ication ical verif Mac • 1 if ( ) t Mac m =  k ( ,) Vrfy m t =  k 0 otherwise  () ln , the scheme is said to {0,1} If be M • = a MAC scheme for messag fixed-len es of le gt h ln ngth . () Fixed-length MAC schemes are easier to construct. • * • nstructed from {0,1} can be co = General MAC schemes for M chemes. h s fixed-lengt 5

6 Chosen-Message Attacks on MAC Schemes n ): Experiment MAC-Fo rge ( , Π A n 1. A key kG ← (1 ) is generated. n oracle access to ( ). A Mac ⋅ 2. The adversary is given inpu t 1 and k A for messages of its choice. may ask the oracle to compute tags Let be the set of all queries has made to the oracle. QA 3. event ually outputs a pair ( , ). A mt f e ( tries to g a valid pair of message and tag. or ) A ∉ = = 4. MAC-For m Q Vrfy m ge ( ) 1 A ( succeeds) if and ( , ) 1. n t Π k A , • Remarks:  age attacker. Adversary: an adaptive chosen-mess  forge tia ry . : an existen Forgery l 6

7 MAC security: existential unforgeability under an adaptive chosen-message attack Gen Mac Vrfy , , existentially Definition: A MAC scheme ( ) is e y ) secur adaptive chosen-message atta unforgeable ck under an mpl (or si A if for all polynomial-time adversaries , there exists a negligible function negl such that = ≤ negl n n ( ) 1 ( ) Pr MAC-Forge  A , Π  or ac M ) ( ⋅ nn k  =←≤ Vrfy A k negl n 1 : P r 1 {0 , 1} ( ) ( ) ) ( ku  where th e output of , ( , ), satisfies . A mt m Q ∉ 7

8 Strong MAC security If a MAC scheme is secure, the probabil ity is negligible that • . m Q mt A can forge a valid ( , ) with ∉ ′ ifferent A tt valid tag d However, it may be possible for to forge a ≠ • fo , where is the tag retur ∈ r some message m t mQ ned by the oracle on . ure strongly sec . • If no adversary is able to do so, the MAC scheme is To formally define strong security, mo a • dify the experiment s follows: ′  the oracle on ( , ) : , is the tag returned by Q . ∈ = Let m mt m Q t { } ′  ∉ succeeds if an only if it outputs A a valid pair ( , ) mt Q . cure". ⇔ is deterministic, then "secure" MAC • If " strongly se 8

9 Constructing secure MAC schemes • F Let be a pseudorandom function. • F hemes in several steps. We will use to construct secure MAC sc  h n gt en h t fixed-leng MAC schemes for messages of l Secure n ln ⋅  Secure MAC schemes for messages of le ngth fixed-length ( )  th g messages Secure MAC schemes for arbitrary-len • n s a multiple of . For simplicity, assume message length i .) do padding to make this assumption true ys lwa (We can a 9

10 n = M {0,1} Secure MAC schemes for • F Let be a pseudorandom function. • n length : Fixed-length MAC scheme for messages of nn  ← k Key generation: On input 1 , {0,1} . u nn ∈ ∈ km  t {0,1} and message {0,1} , ration: On inpu Tag gene = t Fm output the tag : ( ). k = Fm t 1 if ( )  k  = mt Vrfy mt Verification: On input ( , ), ( , ) :  k e rwis 0 othe  • Theorem: Such a MAC scheme is secure. 10

11 Basic CBC-MAC • F Let be a pseudorandom function. • Basic CBC-MAC works as follows: n  ← k Key generation: {0,1} . u n nq ⋅  ∈∈ km {0,1} an Tag generation: For key {0,1} , d message  = mmm m q se as ( , , ) // blocks // − par q 1 n m −= IV apply CBC to with 0 , i.e., let n ≤≤ i q t Fm t = = ⊕ t : 0 and : ) for 1 ( ki i i − 01 − t output as the tag q  nical Verification: can o • Theorem: For any length function , bas fixe d l ic CBC-MAC is secure ength ) (. nln ⋅ for messages of l 11

12 Remarks It is important that ( ) is fixed, or t he scheme would be insecure. t IV • 0 • Also, the scheme variab insecure . if message length is would be le ′ m m t Mac m t Mac m  = =  ( and : ) ( : Suppose ). 123 4 kk ′′  . Let be such that m tm m ⊕= 44 4 ′ ′  =  ) . Then ( t m Mac m mm k 1 4 23 n = 0 IV 12

13 CBC-MAC for arbitrary-length messages A FIPS and ISO standard. • There are several variants of CBC- MAC. • One variant of CBC-MAC: Prepend the message with its length (as an -bit string) m mn • an d then compute basic CBC-MAC on the result. Remarks: • .  There is a limitation on m mm is appended to the end of . It would be insecure if  13

14 14

15 Another variant of CBC-MAC n kk Generate two keys , {0,1} . •← 12 u • m To authenticate a message , let the tag be = m tF : basic-CBC-MAC ( ) . ( ) kk 21 kk • One may use only one key and generate , from : kk 2 1 = (1) and : (2) = : kF k F 12 kk 15

16 h) Security of CBC-MAC (for arbitrary lengt CBC-MAC is secure if is a pseudorando ction. Theo m fun F : rem In practice, block ciphers (such as DES , AES) are used. • 16

17 Authenticated Encryption To ensure both secrecy and integrity 17

18 Unforgeable encryption • n ( ) : Experiment Enc-Forge Π , A n Gen  (1 ) to obtain a key . k Run n Enc ss to oracle  ⋅ ( ), A The adversary is given 1 and acce k and outputs a ciphertext c . ()  . Let be the set of all messages that = D e m cc : QA Let k on. has asked the oracle for encrypti A ( succeeds) if and only if The output of the experiment is 1  ) an is a valid m essage ( ∈ m mQ d . m ∉ M An encryption scheme ( , • ) is Π= Definition: unforgeab l Gen Enc Dec , e ( ) 1 A n negl n ( ). if for every , Pr Enc-Forge = ≤  A Π ,  18

19 Authenticated encryption scheme • d on: Definiti icate hent ut a n A symmetric-key encryption scheme is a forgeable. un e and encryption scheme if it is CCA-secur ption scheme from a • ncry authenticated e We will construct an trongly secure MAC scheme. CPA-secure encryption scheme and a s • Three natural ways:  Encrypt and authenticate (insecure)  e) cur e Authenticate then encrypt (ins  Encrypt then authenticate (secure) • sage, an encryption key, In the following slides, let be a mes mk E a MAC key. k and M 19

20 Encrypt and Authenticate m pendently. That is, • Sender: encrypt and authenticate inde ct the ciphertext is where , t Mac () c Enc m ←← (), m kk EM • ct Receiver: given ciphertext , , do ( ), and then check if ( , ) 1. ←= m t Vrfy c Dec m kk EM • Security: re, since might leak info about . Not necessarily EAV-secu tm  Mac hen the scheme If is deterministic (e.g., CBC-MAC), t  is not CPA- secure . 20

21 Authenticate then Encrypt ncrypt and the tag. Sender: authenticate first and then e • mm : Thus, the ciphertext is computed as c ( ), ( ) t Mac m c Enc m t ←←  kk ME Receiver: given ciphert ext , do • c ( ) and then check if ( , ) 1. Dec  ←= m t Vrfy c m t kk EM A potential attack: • Suppose PKCS#5 padding is used. Suppos e the receiver does:  p if the adding is incorrect then return a "bad padding" error the tag is incorrect retur n a "bad mac" error. elsei f th en t he en The padding attack can be conducted to r i ecover . t re mt   21

22 Encrypt then Authenticate m • Sender: encrypt first and then authenticate the result. Thus, where the ciphertext is , t c ( ), ( ) m t Mac c ←← Enc c kk EM ( ). ( , ) 1 then f Receiver: on receiving , , i ← = c Dec m c t rfy • tV c kk ME If the encryption scheme is and the M AC CPA-secure em: or The • scheme is strongly secure thenticate , then the encryption-then-au ction te cons yields authe an tru ntica scheme. d e ncrypti on y secure MAC CPA-secure encryption + strongl CCA-secure and unforgeabl ncry e e ption ⇒ 22

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