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CMSC 414 Computer and Network Security Lecture 9 Jonathan Katz

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

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PKCS #1 v2.1 (e.g., OAEP) G m || 0 … 0 H r m1m1 m2m2 e mod N c =

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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”

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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

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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

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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!

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Signature schemes

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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

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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)

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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 obtained by MACs!

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Functional definition Key generation algorithm: randomized algorithm that outputs (pk, sk) Signing algorithm: –Takes a private key and a message, and outputs a signature; Sign sk (m) Verification algorithm: –Takes a public key, a message, and a signature and outputs a decision bit; b = Vrfy pk (m, ) Correctness: for all (pk, sk), Vrfy pk (m, Sign sk (m)) = 1

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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 = Sign sk (m 1 ), …, n = Sign sk (m n ) for m 1, …, m n of its choice Attacker “breaks” the scheme if it outputs a forgery; i.e., (m, ) with: m ≠ m i for all i Vrfy pk (m, ) = 1

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“Textbook RSA” signatures Public key (N, e); private key (N, d) To sign message m Z N *, compute = m d mod N To verify signature on message m, check whether e = m mod N Correctness holds… …what about security?

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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

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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”

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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

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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

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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 HSign M H(M) sk

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Crypto pitfalls and recommendations

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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 system 4.Systems are complex Avoid the first; be aware of 2-4

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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

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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

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Human factors Crypto needs to be easy to use both for end-users and administrators Important to educate users about appropriate security practices

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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

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