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Digital Signatures

Learn about digital signatures, shared keys, classical and public key signatures, RSA, attack methods, key storage, escrow, revocation, and key points for effective cryptosystems. Explore key concepts in digital signatures and their significance in computer security.

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Digital Signatures

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  1. Digital Signatures CSSE 490 Computer Security Mark Ardis, Rose-Hulman Institute April 12, 2004

  2. Digital Signature • Construct that authenticated origin, contents of message in a manner provable to a disinterested third party (“judge”) • Sender cannot deny having sent message (service is “nonrepudiation”) • Limited to technical proofs • Inability to deny one’s cryptographic key was used to sign • One could claim the cryptographic key was stolen or compromised • Legal proofs, etc., probably required; not dealt with here

  3. Shared Key • Alice, Bob share key k • Alice sends m || { m }k to Bob • Is this a digital signature?

  4. Classical Digital Signatures • Require trusted third party • Alice, Bob each share keys with trusted party Cathy • To resolve dispute, judge gets { m }kAlice , { m }kBob , and has Cathy decipher them; if messages matched, contract was signed { m }kAlice Bob Alice { m }kAlice Bob Cathy { m }kBob Cathy Bob

  5. Public Key Digital Signatures • Alice’s keys are dAlice, eAlice • Alice sends Bob m || { m }dAlice • In case of dispute, judge computes { { m }dAlice }eAlice • and if it is m, Alice signed message • She’s the only one who knows dAlice!

  6. RSA Digital Signatures • Use private key to encipher message • Protocol for use is critical • Key points: • Never sign random documents, and when signing, always sign hash and never document • Mathematical properties can be turned against signer • Sign message first, then encipher • Changing public keys causes forgery

  7. Properties of modulo arithmetic 1/3 P1: ((a mod p)•(b mod p)) mod p = (a•b) mod p Proof: a = j•p+x, b = k•p+y for x,y < p a•b = (j•p+x) • (k•p+y) = (...)•p + x•y (a•b) mod p = (x•y) mod p

  8. Properties of modulo arithmetic 2/3 P2: a mod p = b mod p  aZ mod p = bZ mod p Proof: a = j•p+x, b = k•p+x for x < p aZ = (j•p+x)Z = (...)•p+xz bZ = (k•p+x)Z = (...)•p+xz

  9. Properties of modulo arithmetic 3/3 P3: fz• gz = (f • g)z Therefore: ((a mod p)z •(b mod p)z) mod p = ((a mod p) • (b mod p))z mod p = (a • b)z mod p

  10. Attack #1 • Want to claim agreement on m • Find m1, m2 such that: • m1 • m2 mod nB = m mod nB • Obtain signed versions of m1, m2 • a1 = m1dB mod nB • a2 = m2dB mod nB • Produce a1 • a2 mod nB = mdB mod nB

  11. Attack #2 • Suppose Alice sends a signed message by enciphering first, then signing: c = (meB mod nB)dA mod nA • Bob finds another public key r•eB, such that Mr = m c = (Mr•eB mod nB)dA mod nA

  12. Storing Keys • Multi-user or networked systems: attackers may defeat access control mechanisms • Encipher file containing key • Attacker can monitor keystrokes to decipher files • Key will be resident in memory that attacker may be able to read • Use physical devices like “smart card” • Key never enters system • Card can be stolen, so have 2 devices combine bits to make single key

  13. Key Escrow • Key escrow system allows authorized third party to recover key • Useful when keys belong to roles, such as system operator, rather than individuals • Business: recovery of backup keys • Law enforcement: recovery of keys that authorized parties require access to • Goal: provide this without weakening cryptosystem • Very controversial

  14. Components • User security component • Does the encipherment, decipherment • Supports the key escrow component • Key escrow component • Manages storage, use of data recovery keys • Data recovery component • Does key recovery

  15. Key Revocation • Certificates invalidated before expiration • Usually due to compromised key • May be due to change in circumstance (e.g., someone leaving company) • Problems • Entity revoking certificate authorized to do so • Revocation information circulates to everyone fast enough • Network delays, infrastructure problems may delay information

  16. CRLs • Certificate revocation list lists certificates that are revoked • PGP: signers can revoke signatures; owners can revoke certificates, or allow others to do so

  17. Key Points • Key management critical to effective use of cryptosystems • Different levels of keys (session vs. interchange) • Keys need infrastructure to identify holders, allow revoking • Key escrowing complicates infrastructure • Digital signatures provide integrity of origin and content Much easier with public key cryptosystems than with classical cryptosystems

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