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### Security Definitions in Computational Cryptography

18739A: Foundations of Security and Privacy

Anupam Datta

CMU

Fall 2009

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

mi

sign(mi, sk(C))

C

A

sign(m, sk(C))

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

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)

IND-EAV security definition(eavesdropping attacks)

k, b

m0, m1

enc(k, mb)

C

A

d

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.

IND-CPA security definition (chosen-plaintext attacks)

mi

k, b

enc(k, mi)

m0, m1

enc(k, mb)

C

A

mi

enc(k, mi)

d

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)

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

IND-CCA secure encryption(chosen-ciphertext attacks)

mi or ci

k, b

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

m0, m1

enc(k, mb)

C

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

A

mi or ci

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

d

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)

Example (public-key version)

- Network protocols Q1 and Q2
- QI

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

- Q2

A 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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