![CSE 597E Fall 2001 PennState University15
Global Public-Key Components
p A prime number of L bits where L is a multiple of 64 and 512 ≤ L ≤ 1024
q A 160-bit prime factor of p-1
g = h(p-1)/q
mod p, where h is any integer with 1<h< p-1, such that (h(p-1)/q
mod
p)>1
User’s Private Key
x A random or pseudorandom integer with 0<x<q
User’s Public Key
y = gx
mod p
User’s Per-Message Secret Number
k A random or pseudorandom integer with 0<k<q
Signing
r = (gk
mod p) mod q s = [k-1
(H(M) = xr)] mod q
Signature = (r, s)
Verifying
w = (s’)-1
mod q
u1
= [H(M’)w] mod q u2
= (r’)w mod q v = [(gu1
yu2
) mod p] mod q
Test: v = r’
The Digital Signature Algorithm (DSA)](https://image.slidesharecdn.com/digitalsignatureschemes-170302084559/85/Digital-signature-schemes-15-320.jpg)
Digital signature schemes
- 1. CSE 597E Fall 2001 PennState University 1 Digital Signature Schemes Presented By: Munaiza Matin
- 2. CSE 597E Fall 2001 PennState University2 Introduction Cryptography – art & science of preventing users from unauthorized or illegal actions towards information, networking resources and services. Cryptographic transformation – conversion of input data into output data using a cryptographic key. Cryptosystem – forward and inverse cryptographic transformation pair
- 3. CSE 597E Fall 2001 PennState University3 A Cryptosystem Input data Forward Cryptographic Transformation Inverse Cryptographic Transformation Key Key Output data Input data Sender Receiver
- 4. CSE 597E Fall 2001 PennState University4 Types of Cryptosystems Private key cryptosystem – a private key is shared between the two communicating parties which must be kept secret between themselves. Public key cryptosystem – the sender and receiver do not share the same key and one key can be public and the other can be private
- 5. CSE 597E Fall 2001 PennState University5 Types of Cryptosystems Forward Cryptographic Transformation Inverse Cryptographic Transformation Key Key Output data Input data Sender Receiver Input data Share private key A Private Key Cryptosystem
- 6. CSE 597E Fall 2001 PennState University6 Types of Cryptosystems Forward Cryptographic Transformation Inverse Cryptographic Transformation 1st Key 2nd Key Output data Input data Sender Receiver Input data Do not share the same key information and one key may be public A Public Key Cryptosystem
- 7. CSE 597E Fall 2001 PennState University7 Digital Signatures Encryption, message authentication and digital signatures are all tools of modern cryptography. A signature is a technique for non- repudiation based on the public key cryptography. The creator of a message can attach a code, the signature, which guarantees the source and integrity of the message.
- 8. CSE 597E Fall 2001 PennState University8 Properties of Signatures Similar to handwritten signatures, digital signatures must fulfill the following: Must not be forgeable Recipients must be able to verify them Signers must not be able to repudiate them later In addition, digital signatures cannot be constant and must be a function of the entire document it signs
- 9. CSE 597E Fall 2001 PennState University9 Types of Signatures Direct digital signature – involves only the communicating parties Assumed that receiver knows public key of sender. Signature may be formed by (1) encrypting entire message with sender’s private key or (2) encrypting hash code of message with sender’s private key. Further encryption of entire message + signature with receiver’s public key or shared private key ensures confidentiality.
- 10. CSE 597E Fall 2001 PennState University10 Types of Signatures Problems with direct signatures: Validity of scheme depends on the security of the sender’s private key ⇒ sender may later deny sending a certain message. Private key may actually be stolen from X at time T, so timestamp may not help.
- 11. CSE 597E Fall 2001 PennState University11 Types of Signatures Arbitrated digital signature – involves a trusted third party or arbiter Every signed message from sender, X, to receiver, Y, goes to an arbiter, A, first. A subjects message + signature to number of tests to check origin & content A dates the message and sends it to Y with indication that it has been verified to its satisfaction
- 12. CSE 597E Fall 2001 PennState University12 Basic Mechanism of Signature Schemes A key generation algorithm to randomly select a public key pair. A signature algorithm that takes message + private key as input and generates a signature for the message as output A signature verification algorithm that takes signature + public key as input and generates information bit according to whether signature is consistent as output.
- 13. CSE 597E Fall 2001 PennState University13 Digital Signature Standards NIST FIPS 186 Digital Signature Standard (DSS) El Gamal RSA Digital Signature - ISO 9796 - ANSI X9.31 - CCITT X.509
- 14. CSE 597E Fall 2001 PennState University14 DSS Public-key technique. User applies the Secure Hash Algorithm (SHA) to the message to produce message digest. User’s private key is applied to message digest using DSA to generate signature.
- 15. CSE 597E Fall 2001 PennState University15 Global Public-Key Components p A prime number of L bits where L is a multiple of 64 and 512 ≤ L ≤ 1024 q A 160-bit prime factor of p-1 g = h(p-1)/q mod p, where h is any integer with 1<h< p-1, such that (h(p-1)/q mod p)>1 User’s Private Key x A random or pseudorandom integer with 0<x<q User’s Public Key y = gx mod p User’s Per-Message Secret Number k A random or pseudorandom integer with 0<k<q Signing r = (gk mod p) mod q s = [k-1 (H(M) = xr)] mod q Signature = (r, s) Verifying w = (s’)-1 mod q u1 = [H(M’)w] mod q u2 = (r’)w mod q v = [(gu1 yu2 ) mod p] mod q Test: v = r’ The Digital Signature Algorithm (DSA)
- 16. CSE 597E Fall 2001 PennState University16 DSS DSA - M = message to be signed - H(M) = hash of M using SHA - M’, r’, s’ = received versions of M, r, s
- 17. CSE 597E Fall 2001 PennState University17 El Gamal Signature Scheme A variant of the DSA. Based on the assumption that computing discrete logarithms over a finite field with a large prime is difficult. Assumes that it is computationally infeasible for anyone other than signer to find a message M and an integer pair (r, s) such that aM = yr rs (mod p).
- 18. 18 El Gamal Signature Scheme Parameters of El Gamal p A large prime number such that p-1 has a large prime factor x The private key information of a user where x<p a A primitive element of the finite field for the prime p y = ax mod p (p,a,y) The public key information
- 19. 19 El Gamal Signature Scheme Step 1 Randomly choose an integer k such that (k, p-1) = 1, 1<k<p-1, and k has not been used to sign a previous message Step 2 Calculate r = ak (mod p) Step 3 Find s such that M = xr + ks (mod (p-1)) Step 4 Collect the pair (r, s) as the digital signature on the message M Since, M = xr + ks (mod (p-1)) ⇒ aM = a(xr+ks) = axr aks = yr rs (mod p) ⇒ Given M and (r, s), the receiver or 3rd party can verify the signature by checking whether aM = yr rs (mod p) holds or not.
- 20. CSE 597E Fall 2001 PennState University20 RSA Digital Signature Scheme Based on the difficulty of factoring large numbers. Given M, RSA digital signature can be produced by encrypting either M itself or a digest of M using the private signature key s. Signature, S = ws mod n, where w is message to be signed or message digest and n = pq (p and q are large primes). Verification: w = Sv mod n, where (v, n) is the public verification key.
- 21. CSE 597E Fall 2001 PennState University21 Conclusions Digital signatures are an effective mechanism used for authenticity and non- repudiation of messages. Several signature schemes exist, but DSS is probably the most popular. Digital signatures may be expanded to be used as digital pseudonyms which would prevent authorities from figuring out a sender’s identity, for example by cross- matching
- 22. CSE 597E Fall 2001 PennState University 22 Thank you!