Security Definitions in Computational Cryptography

Anupam DattaCMU

Fall 2009

18739A: Foundations of Security and Privacy

Cryptographic Concepts Signature scheme Symmetric encryption scheme

Signature Scheme Key generation algorithm

Input: security parameter n Output: a private signing & public verification key pair

Algorithm to sign data Algorithm to verify signature Correctness:

Message signed with a signing key verifies with the corresponding verification key

verify(m,sign(m,sk(A)), pk(A)) = ok Symbolic Security:

A signature cannot be produced without access to the private signing key

UF-CMA Security

A

sign(mi, sk(C))

mi

UF-CMA security: PPT attackers A negligible function f n0 security parameters n ≥ n0 Prob [m ≠mi | A plays by the rules] <= f(n)

Csign(m, sk(C))

Symmetric Encryption Scheme Key generation algorithm

Input: security parameter n Output: a key that is used for encryption and

decryption Algorithm to encrypt a message Algorithm to decrypt a ciphertext Correctness:

Decrypting a ciphertext obtained by encrypting message m with the corresponding key k returns m

dec(enc(m,k),k) = m

What is a secure encryption scheme? List of possible properties

Given a list of message, ciphertext pairs, it should not be possible to recover the key

Given ciphertext, it should not be possible recover plaintext

Given ciphertext, it should not be possible to recover 1st bit of plaintext

All of the above, but what else?

Given ciphertext, adversary should have no information about underlying plaintext (not true because of apriori information)

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IND-EAV security definition(eavesdropping attacks)

CA

enc(k, mb)

m0, m1

d

k, b

IND-EAV security: PPT attackers A negligible function f n0 security parameters n ≥ n0 Prob [d = b | A plays by the rules] <= ½ + f(n)

Example General sends an encrypted message where

the plaintext is either “attack” or “don’t attack”.

Adversary should not be able to figure out what the plaintext is although she knows that it is one of these two values.

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IND-CPA security definition (chosen-plaintext attacks)

CA

enc(k, mb)

m0, m1

d

k, b

IND-CPA security: PPT attackers A negligible function f n0 security parameters n ≥ n0 Prob [d = b | A plays by the rules] <= ½ + f(n)

enc(k, mi)

mi

enc(k, mi)

mi

Example US Navy cryptanalysts received a ciphertext

containing the word “AF” that they believed corresponded to “Midway island” (May, 1942)

Concluded that Japan was planning to attack Midway island, but could not convince top brass

Sent out a message saying Midway island was low on water supply

Japanese intercepted this message and sent out a message saying “AF” was running low on water supply

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IND-CCA secure encryption (chosen-ciphertext attacks)

CA

enc(k, mb)

m0, m1

d

k, b

IND-CCA security: PPT attackers A negligible function f n0 security parameters n ≥ n0 Prob [d = b | A plays by the rules] <= ½ + f(n)

enc(k, mi) or dec(k,ci)

mi or ci

A cannot submit enc(k,mb) to the decryption oracle

mi or ci

enc(k, mi) or dec(k,ci)

Example (public-key version) Network protocols Q1 and Q2

QI C B: enc(pk(B), secret, Q1)

Q2A B: enc(pk(B),nonce, Q2)B A: nonce

Adversary A has access to B’s decryption oracle, but should still not be able to learn additional information about C’s secret (e.g., cannot tell whether it is “attack” or “don’t attack”)

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Questions?