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RSA: More about attacks

RSA: More about attacks. Need to take care with the implementation, e.g.: Do not take p or q very small. Difference of p and q should not be very small. More subtle, e.g.: Thm: If 3d < n^{ ¼ } and q<p<2q, then d can be found in polynomial time. More info: Section 6.2.

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RSA: More about attacks

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  1. RSA: More about attacks • Need to take care with the implementation, e.g.: • Do not take p or q very small. • Difference of p and q should not be very small. • More subtle, e.g.: • Thm: If 3d < n^{ ¼ } and q<p<2q, then d can be found in polynomial time. • More info: Section 6.2

  2. RSA: More about attacks Public-key cryptosystem: can it have perfect secrecy ? RSA is insecure against a chosen-ciphertext attack (we’ll do soon). Is RSA insecure against a known-plaintext attack ? Is RSA insecure against a chosen-plaintext attack ?

  3. RSA: Protocol Failures Secure system can still be used in an insecure (careless) way. This is called a protocol failure. Examples: Exercises 16 and 17 (Chapter 6, page 194)

  4. RSA: Insecurity against Chosen-Ciphertext Eve wants to decrypt y (a ciphertext). She can choose another ciphertext ŷ \neq y that she can use to decrypt y. Choose a random x_0 and compute y_0 = (x_0)^e mod n. Let ŷ = y_0y mod n. Eve gets the decryption of ŷ. How to find y ?

  5. RSA: Timing Attacks In 1995, Paul Kocher (an undergraduate at Standford), discovered that it is possible to determine d (the decryption exponent) by carefully timing the computation times for a sequence of decryptions. Moral of the story: a new type of attack can break a system that is though to be secure… Good news for RSA: it is possible to thwart the timing attack.

  6. Other Public-key Cryptosystems • RSA is the “standard” but there are other public-key cryptosystems. E.g. one by Rabin and one by ElGamal. • All three cryptosystems: • thought to be secure • can be used for digital signatures • slow • Hence: used to encrypt a session key, then use a (secure) private key cryptosystem

  7. Other Public-key Cryptosystems • The Rabin cryptosystem: • based on the difficulty of finding square roots mod a composite number (problem equivalent to factoring) • provably secure (unlike RSA; assuming factoring is computationally infeasible, the Rabin cryptosystem is secure) • 4 possible plaintexts for each ciphertext • [RSA: conjectured to be as secure as factoring.] • The ElGamal cryptosystem: • based on the difficulty of computing discrete logarithms in a finite field • used in many cryptographic protocols

  8. Public-key Cryptosystems • Outline of a general public-key cryptosystem: • components: a set M of messages, a set K of keys, for each key k2K, an encryption function Ek and a decryption function Dk (usually functions from M to M) • requirements: • Ek(Dk(m)) = Dk(Ek(m)) for all m, k. • Ek(m) and Dk(m) are easy to compute for all m, k. • figuring out Dk from Ek is computationally infeasible for almost all k2K • given k2K, finding Ek and Dk is easy

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