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Security Models: Dolev-Yao, Semantic Security, Probabilistic Encryption and ZKIP

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Dolev-Yao For distributed systems and networks, we often should assume that there are adversaries –Everywhere in the network –Adversary may: eavesdrop, manipulate, inject, alter, duplicate, reroute, etc… –Adversary may control a large number of network nodes that are geographically separated Dolev-Yao Threat Model: –A very powerful adversarial model that is widely accepted as the standard by which cryptographic protocols should be evaluated –Eve, the adversary, can: u Obtain any message passing through the network u Act as a legitimate user of the network (i.e. can initiate a conversation with any other user) u Can become the receiver to any sender u Can send messages to any entity by impersonating any other entity

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Dolev-Yao, pg. 2 This seems very powerful, but not entirely so… Under Dolev-Yao: –Any message sent via the network is considered to have been sent by Eve –Thus, any message received “might” have been manipulated by Eve –Eve can control how things are sent What is not possible: –Eve cannot guess a random number which is chosen as part of a security protocol –Without knowledge of a key, Eve cannot figure out a plaintext from a ciphertext, nor can she create ciphertexts from a plaintext. –Eve can’t solve the private-key pairing of a public key –Eve cannot control the “memory” of a computing device of a legitimate user (i.e. Eve can only play with the communication)

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Strong Security Definitions Generally, when discussing the crypto algorithms in this class, we have considered a weak confidentiality model, in which our enemy was a passive eavesdropper For real applications, however, we should consider an active adversary also– they may modify a ciphertext or calculate a plaintext and send the result to a user to get an oracle service Oracle service: A principal is used as an oracle when the principal performs a cryptographic operation inadvertantly for the attacker We should anticipate that Eve is an active adversary who is clever, and can set up Oracle services

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Strong Security Definitions, pg. 2 Further, in many applications, the plaintext messages contain easy-to-guess information (e.g. a beginning of an , or some fields of some type of form, or commands to a receiver) This is problematic: –To guess, Eve need only encrypt the plaintext herself and see if the result matches a received ciphertext –This is a problem with textbook-style crypto.

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Returning to Attack Models from Day 1 Chosen Plaintext Attack (CPA): An attacker chooses plaintext messages and gets encryption assistance to obtain the corresponding ciphertext messages. The task for the adversary is to weaken the cryptosystem using the plaintext-ciphertext pairs. Chosen Ciphertext Attack (CCA): An attacker chooses ciphertext messages and gets decryption assistance to obtain the corresponding plaintext messages. Objective is to weaken the cryptosystem. The attacker is successful if he can retrieve some secret plaintext information from a target ciphertext which is given to the attacker after the decryption assistance is stopped. Adaptive Chosen Ciphertext Attack (CCA2): A CCA where the decryption assistance will be available forever, except for the target ciphertext.

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Attack models further discussed Note 1: Encryption assistance of a public key system is always available since the public key is always available… In otherwords, CPA can always be mounted against a public key system. Hence all public key systems *should* resist CPA!!! Note 2: Attackers may exploit the nice homomorphic properties of public key systems to make up a ciphertext via some clever calculations. –If the attacker is assisted by a decryption service, then manipulations may enable him to obtain some plaintext information –One must be careful whether one is “used” to provide a decryption oracle –Where do we see this problem? Answer: challenge response and nonces!!!

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What does it mean to be secure? For a PK encryption algorithm, some starting points: –It means that recovering the private key from the public key should be hard –With high probability, a message should not be recovered from seeing its encrypted form –No useful information can be computed from the encrypted form of a message –We do not want the adversary to be able to compute useful facts from just seeing messages (i.e. that two messages are of identical content)

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Polynomial Time Indistinguishability Polynomial Indistinguishability: An encryption scheme is polynomial time indistiguishable if no adversary can find two messages whose encryptions he can distinguish between –I.e. if ciphertexts are like messages in an envelope, we can’t tell two envelopes apart –Or, it is impossible (in polynomial in k-bits) to find two messages m 0 and m 1 such that a polynomial-time algorithm can distinguish between c 0 and c 1 Note: Any encryption algorithm in which the encryption algorithm is deterministic immediately fails –i.e. given f, m 0 and m 1, and c coming from either m 0 or m 1, it is easy to decide which one it came from (just calculate f(m 0 ) and f(m 1 )

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Semantic Security Consider two games, and let h:M {0,1}* where M is a message space. E.g. h() may be any function that finds information about m from M (e.g. does m have an “e” in it?) Game 1: We tell Eve that we are about to choose m and ask her to guess h(m) Game 2: We tell Eve a=E(m) for some m, and ask her to guess h(m) In first game: the adversary only knows that a message m is about to be chosen In second game: the adversary sees a ciphertext. Semantic Security: The probability of winning Game 1is the same as the probability of winning Game 2. –i.e. the adversary should not gain any advantage or information from seeing the ciphertext Again, this implies that E() must not be deterministic, otherwise Eve would calculate E(0…0), E(0…1), … E(1…1), see which matched c, and then calculate h(m) on her own

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SRA Mental Poker Suppose Alice lives in one town and Bob lives in another, they would like to play poker over the phone. To do this, though, they would like to have some digital way of accomplishing this –Cards are encoded into messages so that the card game can be played in communications –Cards must be dealt fairly: u 1) Deal must distribute all hands with equal probability u 2) Alice and Bob must know the cards in their own hand, but not in another’s hands u 3) Alice and Bob must be viewed as potential cheaters who cannot be relied upon u 4) Alice and Bob should both be able to verify that a preceding game was played fairly

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SRA Mental Poker, pg. 2 The idea behind this was originally proposed by Shamir, Rivest, and Adleman Makes use of commutative ciphers (e.g. RSA): A message can be doubly encrypted by Alice and Bob and the result does not depend on the order: E A (E B (M)) = E B (E A (M)) Protocol: Suppose (for simplicity) we have three cards M1, M2, M3 –Alice encrypts the three cards as C i =E A (M i ). She sends Bob these three ciphertexts in a random order –Bob picks at random one ciphertext (C), he double encrypts to get CC= EB(C), and also picks another C’ and sends both to Alice. –Alice decrypts both CC and C’. Decryption of C’ is her hand, decryption of CC is C’’ which is sent to Bob –Bob decrypts C’’ and thereby obtains his hand.

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SRA Mental Poker, pg. 3 Suppose that the encryption algorithm is strong– this will not mean that the poker game is really strong. In fact, the fact that an adversary (given a plaintext) without the correct encryption key cannot create a valid ciphertext, will not mean “secure”. The SRA protocol: –1) Alice and Bob do obtain a hand (1 card) with equal probabilities because Alice shuffles in step 1. Note: Alice wants to shuffle uniformly, otherwise Bob would have an advantage since he selects. –2) Each of the two parties knows his own hand after double decryption, but does not know the other hand. –3) The protocol does not rely on any party to be honest.

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SRA Mental Poker, pg. 4 Property 4 (Fairness) is harder than it looks... Look at the Shamir proposal, which used a variation of RSA, where both parties share the same N but keep their encryption+decryption exponent private until after the game is over to verify the game. Let N be shared RSA modulus, and exponents (e A, d A ) and (e B, d B ). Here E X (M) = M eX. Before completing the game, both parties keep their encryption and decryption exponents secret. Thus, no one can create a valid ciphertext which has been created by the other party. Also, neither party can decrypt a ciphertext which has been created by the other party. Thus, the cryptosystem is “kind of strong”. After the game finishes, both parties can disclose their encryption and decryption exponents to the other party, and they can verify that no cheating was performed anywhere… so (4) works

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SRA Mental Poker, pg. 4 But strong is not strong enough– an attacker’s inability to create a valid ciphertext from a given plaintext without the correct encryption key, or decrypt without the correct decryption key Why? Lipton’s attack shows why… the cryptosystem is unable to hide certain a priori information in the plaintext… A number a is a quadratic residue modulo N if gcd(a,N) =1 and there is an x such that x 2 = a mod N Note that e and d must be odd (since (N) is even) M is a quadratic residue modulo N iff the ciphertext is also a quadratic residue modulo N –C = M e = (x 2 ) e = (x e ) 2 mod N –Knowing factorization of N (which both parties do) allows for easy determination of whether something is a quadratic residue (calculate Legendre symbols) –Thus, if some card a quadratic residue, then a party who knows this fact will have an advantage in the game… (unless all cards are quadratic residues) Thus the mental poker game is not secure against “chosen plaintext attack” (where adversary cleverly chooses the plaintext to represent the card)

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Enter Probabilistic Encryption In order to have semantic (or indistinguishability) security, we must abandon the trapdoor and deterministic model of public key crypto Probabilistic encryption: –Still assumes the existence of a trapdoor –We begin (and end) by securely encrypting single bits

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Probabilistic Encodings A powerful approach to secret dissemination uses probabilistic encryption (similar idea in Wyner codes) Let f(x) be a trapdoor function and G(x) a hardcore predicate for f(x) (i.e. hard to calculate G(x) from just f(x) ) A probabilistic encryption procedure is:

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Dissemination: Probabilistic Encodings (Wyner) Applying this idea: –Take G(x)=m to be the parity function of x 1. Alice wants to send m=0 or 1 to Bob 2. She chooses a random x of length N such that G(x)=m 3. Transmits x to Bob Assuming p AB =0, then Bob recovers m by calculating G(x) For large N, the probability of an odd amount of bit errors is More generally, apply ECC across m’s to handle p AB not 0 This method needs p AB < p E !

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Non-Malleable Cryptography Model Non-malleable cryptography strengthens the notions of (public key) cryptography: it should not be possible for Eve to modify a plaintext message in a meaningfully controllable manner via modifying the plaintext –The usual Dolev explanation: Suppose you have two companies making a bid for a contract. If company A sends in a bid for X dollars by sending E A (X), and B can intercept, modify this to produce E A (Y) in a meaningful manner, and forward then B will have a bidding advantage. We have seen such a problem before: –One-time pads and Vernam ciphers: it is possible to modify select bits –We saw this type of weakness in WEP

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Malleability Attacks In a malleability attack, Eve’s objective is, given a ciphertext C, not to learn something about the plaintext M, but instead to wreak havoc upon the eventual decoding –Eve needs to create a relationship C C’ that results in a meaningful relationship M M’ Problem: Most conventional cryptographic algorithms are the result of trapdoor functions –Partial information oracles exist for these public key schemes (e.g. the math that lets one learn parity can be the basis for conducting a malleability attack) Example: How to double a plaintext in RSA encryption –Take C=M e mod N and then produce C’ = C2 e mod N

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Byzantine: What comes after Dolev-Yao The Dolev-Yao model is the foundation of security analysis for active adversary scenarios, but does not capture everything that an adversary can do: –It does not involve entity compromise In situations involving many participants (e.g. distributed computing or peer- to-peer), it is natural to ask what can happen if a legitimate entity becomes compromised A Byzantine failure is one where a node/entity fails to operate properly, but continues to operate (as opposed to fail-stop failures) Example Byzantine Failures: –A node may lie about connectivity –Flood network with false traffic –Falsely describe opinions of another node (e.g. P2P) –Capture a strategic subset of devices and collude

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How to Cope with Byzantine Behavior General Strategies for coping with Byzantine : –1) Need to assure there is at least reliable information then issue becomes discerning between good and bad information u Statistical robustness, modeling and outlier filtering u Ensure that there are multiple sources of information –2) Failure detection u Setup protocols to identify who is not trustworthy –3) Resource reservation and fairness u Make sure that every entity gets its fair share of resources, in spite of being bad… prevents greedy behavior –4) Multipath flows and service replication –5) Distribute information and require cooperation to prevent information exposure u Threshold cryptography

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Case Study: Robust Flooding Flooding: Each router R that receives a packet from neighbor N forwards the packet to each neighbor. –Flooding looks like it has the properties we want –A packet from S will reach D, provided S and D are connected by at least one correctly functioning path, regardless of arbitrary Byzantine behavior of routers and links not on that path, provided the network has infinite capacity. Why “provided the network has infinite capacity?” –Prevent Byzantine nodes from flooding network with bogus (though maybe cryptographically legitimate) traffic Reality: Networks have finite capacity, so we need fair allocation of resources –Computation in routers require devices to have CPU rates better than link line rates –Memory in the routers each device sets up internal queues for each potential source along with authenticated sequence numbering… then queues are fairly processed in round-robin manner –Bandwidth on the links cycle through each buffer, and transmit packets from queues in a fair manner

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Zero Knowledge Interactive Proofs The basic idea: 1. Alice says she has a key to the door in the cave but does not want to show Bob 2. Alice proves she has the key by entering the cave, choosing a random direction to go, and waits near the door 3. Bob then comes to cave entrance and asks Alice to come out a “random side” of the cave (left/right) 4. If Alice has the key, then she can always come out the correct side 5. If Alice does not have the key, she can only succeed 50% of the time The proof: repeat this process many times

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ZKIP: Mathematical Version Let n=pq, and y be a square mod n. It turns out that finding square roots mod n is equivalent to factoring n Suppose Peggy, the prover, wants to prove that she knows a square root s of y but she does not want to reveal s. 1) Peggy chooses random numbers r 1 and r 2 with r 1 r 2 = s mod n 2) Peggy calculates x 1 = (r 1 ) 2, x 2 =(r 2 ) 2 mod n and sends these to Victor, the verifier 3) Victor checks that x 1 x 2 = y mod n. He chooses either x 1 or x 2 and asks Peggy to produce a square root of it 4) The process is repeated several times until Victor is happy

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ZKIP, pg. 2 Clearly, if Peggy knows s, there is no problem. If Peggy does not, then she can still send x 1 and x 2 If she knows a square root of x 1 and a square root of x 2, then she knows a square root of y ≡ x 1 x 2. Therefore, for at least one of them, she does not know a square root. At least half the time, Victor is going to ask her for a square root she doesn’t know. Since computing square roots is hard, she is not able to produce the desired answer, and therefore Victor finds out that she doesn’t know s. Suppose, however, that Peggy predicts correctly that Victor will ask for a square root of x2. Then she could have chosen a random r 2, compute x 2 ≡ r 2 2 (mod n), and let x 1 ≡ yx 2 −1 (mod n). She would then send x 1 and x 2 to Victor, and everything works. This method gives Peggy a 50% chance of fooling Victor on any given round, but it requires her to guess which number Victor will request each time. As soon as she fails, Victor will find out that she doesn’t know s.

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