CMSC 414 Computer and Network Security Lecture 9

CMSC 414Computer and Network Security

Lecture 9

Jonathan Katz

Non-malleable public-key enc. RSA-based: OAEP Diffie-Hellman based

PKCS #1 v2.1 (e.g., OAEP)

G

m || 0…0

H

r

m1 m2

emod Nc =

Status of OAEP Can be proven secure against chosen-ciphertext

attacks based on the RSA assumption and the assumption that the hash functions G, H are “truly random”

Other alternatives There exist variants of El Gamal encryption that

can be proven secure against chosen-ciphertext attacks based on the DDH assumption (and no unrealistic assumptions regarding hash functions)– Factor of ~2 less efficient than El Gamal

Hybrid encryption When using hybrid encryption, if both

components are secure against chosen-ciphertext attacks then the combination is also secure against chosen-ciphertext attacks

Recommendations Always use authenticated encryption in the

private-key setting– E.g., encrypt-then-authenticate

Always use a public-key encryption scheme secure against chosen-ciphertext attacks!– E.g., RSA PKCS #1 v2.1

When using hybrid encryption, combine them!

Signature schemes

Basic idea A signer publishes a public key pk

– As usual, we assume everyone has a correct copy of pk

To sign a message m, the signer uses its private key to generate a signature

Anyone can verify that is a valid signature on m with respect to the signer’s public key pk– Since only the signer knows the corresponding private

key, we take this to mean the signer has “certified” m

Security: no one should be able to generate a valid signature other than the legitimate signer

Typical application Software company wants to periodically release

patches of its software– Doesn’t want a malicious adversary to be able to

change even a single bit of the legitimate patch

Solution: – Bundle a copy of the company’s public key with initial

copy of the software– Software patches signed (with a version number)– Do not accept patch unless it comes with a valid

signature (and increasing version number)

Signatures vs. MACs Could MACs work in the previous example?

– Computing one signature vs. multiple MACs– Managing one key vs. multiple keys– Public verifiability– Transferability– Non-repudiation

Not obtainedby MACs!

Functional definition Key generation algorithm: randomized algorithm

that outputs (pk, sk) Signing algorithm:

– Takes a private key and a message, and outputs a signature; Signsk(m)

Verification algorithm:– Takes a public key, a message, and a signature and

outputs a decision bit; b = Vrfypk(m, )

Correctness: for all (pk, sk), Vrfypk(m, Signsk(m)) = 1

Security? Analogous to MACs

– Except that adversary is given the signer’s public key

(pk, sk) generated at random; adversary given pk Adversary given 1 = Signsk(m1), …,

n = Signsk(mn) for m1, …, mn of its choice

Attacker “breaks” the scheme if it outputs a forgery; i.e., (m, ) with:

• m ≠ mi for all i

• Vrfypk(m, ) = 1

“Textbook RSA” signatures Public key (N, e); private key (N, d) To sign message m ZN

*, compute = md mod N

To verify signature on message m, check whether e = m mod N

Correctness holds…

…what about security?

Security of textbook RSA sigs? Textbook RSA signatures are not secure

– Easy to forge a signature on a random message– Easy to forge a signature on a chosen message, given

two signatures of the adversary’s choice

Hashed RSA Public key (N, e); private key (N, d) To sign message m, compute = H(m)d mod N To verify signature on a message m, check

whether e = H(m) mod N Why does this prevent previous attacks?

Note: has the added advantage of handling long messages “for free”

Security of hashed RSA Hashed RSA signatures can be proven secure

based on the hardness of the RSA problem, if the hash is modeled as a random function– Proof in CMSC456

Variants of hashed RSA have been standardized, and are used in practice

DSA/DSS signatures Another popular signature scheme, based on the

hardness of the discrete logarithm problem– Introduced by NIST in 1992– US government standard

I will not cover the details, but you should know that it exists

Hash-and-sign Say we have a secure signature scheme for “short”

messages (e.g., hashed RSA, DSS, …)– How to extend it for longer messages?

Hash and sign– Hash message to short “digest”; sign the digest

Used extensively in practice

H SignM H(M)

sk

Crypto pitfalls and recommendations

Crypto pitfalls? Crypto deceptively simply

– Why does it so often fail?

Important to distinguish various issues:1. Bad cryptography, bad implementations, bad design,

etc.2. Even good cryptography can often be ‘circumvented’

by adversaries operating ‘outside the model’3. Even the best cryptography only shifts the weakest

point of failure to elsewhere in your system4. Systems are complex

Avoid the first; be aware of 2-4

Cryptography is not a “magic bullet”

Crypto is difficult to get right– Must be implemented correctly– Must be integrated from the beginning, not added on

“after the fact”– Need expertise; “a little knowledge can be a dangerous

thing…”– Can’t be secured by Q/A, only (at best) through

penetration testing and dedicated review of the code by security experts

Cryptography is not a “magic bullet”

Crypto alone cannot solve all security problems– Key management; social engineering; insider attacks– Need to develop appropriate threat/trust models for the

system as a whole

Defense in depth– Need for review, detection, and recovery– Security as a process, not a product

Human factors Crypto needs to be easy to use both for end-users

and administrators Important to educate users about appropriate

security practices

General recommendations Use only standardized algorithms and protocols

– No security through obscurity!

Don’t implement your own crypto– If your system cannot use “off-the-shelf” crypto

components, re-think your system– If you really need something new, have it designed

and/or evaluated by an expert

Don’t use the same key for multiple purposes– E.g., encryption and MAC; or RSA encryption and

signatures

Use sufficient entropy when choosing keys