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Digital Signatures

Digital Signatures. CSIS 5857: Encoding and Encryption. Need for Digital Signature. Authentication Problem : How can recipient be sure that message was sent by particular person ?. “Give Darth a $10,000 raise -- Alice”. E. Masquerading as Alice. Digital Signatures.

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Digital Signatures

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  1. Digital Signatures CSIS 5857: Encoding and Encryption

  2. Need for Digital Signature • Authentication Problem: How can recipient be sure that message was sent by particular person? “Give Darth a $10,000 raise-- Alice” E Masquerading as Alice

  3. Digital Signatures • Based on some signing algorithm • Algorithm applied to message (like message digest) • Message and signature sent to recipient • Recipient uses algorithm to verify signature • Must involve “secret knowledge” known only to signer • Otherwise, adversary could “forge” signature “I can’t create this”

  4. Public Keys and Digital Signatures • Signing algorithm involves private key • Public/private key pair generated by sender • Opposite of public key encryption • Sender stores private key, gives public key to recipient • Private key used to sign message • Public key used to verify signature

  5. Signing Message Digests • Problem: • Creating signature from entire message as costly as encrypting entire message with public key algorithm • Solution: • Sender creates message digest (much shorter than message) • Sender creates signature from digest • Recipient creates same message digest from received message • Recipient verifies signature based on message digest

  6. Digital Signatures and Confidentiality • Sender: • Signs message with sender private key • Encrypts message with recipient public key • Recipient • Decrypts message with recipient private key • Verifies signature with sender public key Authentication Confidentiality

  7. RSA Digital Signature Scheme • Encryption/Decryption: • Encryption by sender: C = Pemod n • Decryption by recipient: P = Cdmod n = Pde mod n • Digital signature just reverses order • Key pair generated in same way • Public key: n, e • Private key: d • Works since d  e = e  d • Signature by sender: S = Md mod n • Verification by recipient: M = Semod n = Mde mod n

  8. RSA Digital Signature Scheme • Recipient has sender’s public key • Sent message M and signature S generated from M • Uses key to “decrypt” signature S and compare to M

  9. Attacks on Digital Signatures • Known Message Attack • Adversary has intercepted several messages and their corresponding signatures. • Goal: Create fake message M´ and legitimate corresponding signature from those previous messages • Chosen Message Attack • Adversary has ability to make sender sign messages that adversary chooses (“We like kittens”) • Goal: Choose those messages to make it possible to create fake message M´ and legitimate corresponding signature

  10. Known Message Attack on RSA • Based on multiplicative property of RSA • Darth intercepts message pairs (M1, S1) and (M2, S2) • Computes M´ = M1M2 • Corresponding signature: S´ = S1S2 • Idea: S´ = S1S2 = (M1dM2d) mod n = (M1M2)d mod n = M´d mod n • Darth now has fake message M´ and matching signature S´ without having to know Alice’s private key!

  11. Known Message Attack on RSA • Problem for Darth:Fake message M´ = M1M2 almost certain to be meaningless • Darth can’t control messages M1,M2 • Bob will treat as noise and ignore ???

  12. Chosen Message Attack on RSA • Darth chooses messages M1,M2 such that: • M1,M2 appear harmless (and can convince sender to sign) • M1M2 has advantage to Darth • Most serious attack on most digital signature schemes! Alice creates signatures using her private key Darth asks Alice to sign these Darth sends fake message and signature to Bob

  13. Chosen Message Attack on RSA Solution: • Create message digesth(M ) • h(M ) must be preimage resistant hash function • Create signature from h(M ) instead of from M Why is this effective? • Darth has a fake message M´ • Darth can compute its digest h(M´) • Darth can find two digests h(M1),h(M2) such that h(M´) = h(M1)  h(M2) • Darth cannot find messages M1,M2 with the desired digests h(M1),h(M2) !

  14. Digital Signature Standard • NIST standard (FIPS 186) • Algorithms: • SHA-512 hashing • Schnorr public key encryption scheme (similar to ElGamal)

  15. DSS Components • Global public key components (PUG) • p : Large prime (between 512 and 1024 bits) • q : prime divisor of p -1 (approx. 160 bits) • g = h(p-1)/qmod p where h is some integer < p -1 such thath(p-1)/qmod p > 1 • Sender’s private key (PRa) • Random integer < q • Sender’s public key (PUa) • PUa = gPRa mod p

  16. Signing a Message • Generate random one-time key k < q • Compute components of message: • r = (gk mod p) mod q • s = [k -1 (H(M) + PUa)] mod q • Signature = (r, s) • Efficiency: only modular exponentiation is gk mod p which can be computed in advance

  17. Verifying a Message • w = s -1 mod q • u1 = [H(M) w] mod q • u2 = (rw) mod q • v = [(gu1 PUau2) mod p) mod q • Verifiedifv= r

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