Cryptography and Network Security:
Principles and Practice
Eighth Edition
Chapter 13
Digital Signatures
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Figure 13.1 Simplified Depiction of
Essential Elements of Digital
Signature Process
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Digital Signature Properties
• It must verify the author and the date and time of the
signature
• It must authenticate the contents at the time of the
signature
• It must be verifiable by third parties to resolve disputes
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Attacks
• Key-only attack
– C only knows A’s public key
• Known message attack
– C is given access to a set of messages and their signatures
• Generic chosen message attack
– C chooses a list of messages before attempting to break A’s
signature scheme, independent of A’s public key; C then obtains
from A valid signatures for the chosen messages
• Directed chosen message attack
– Similar to the generic attack, except that the list of messages to be
signed is chosen after C knows A’s public key but before any
signatures are seen
• Adaptive chosen message attack
– C may request from A signatures of messages that depend on
previously obtained message-signature pairs
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Forgeries
• Total break
– C determines A’s private key
• Universal forgery
– C finds an efficient signing algorithm that provides an
equivalent way of constructing signatures on arbitrary
messages
• Selective forgery
– C forges a signature for a particular message chosen
by C
• Existential forgery
– C forges a signature for at least one message; C has
no control over the message
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Digital Signature Requirements
• The signature must be a bit pattern that depends on the
message being signed
• The signature must use some information unique to the sender
to prevent both forgery and denial
• It must be relatively easy to produce the digital signature
• It must be relatively easy to recognize and verify the digital
signature
• It must be computationally infeasible to forge a digital signature,
either by constructing a new message for an existing digital
signature or by constructing a fraudulent digital signature for a
given message
• It must be practical to retain a copy of the digital signature in
storage
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Direct Digital Signature
• Refers to a digital signature scheme that involves only the communicating
parties
– It is assumed that the destination knows the public key of the source
• Confidentiality can be provided by encrypting the entire message plus
signature with a shared secret key
– It is important to perform the signature function first and then an outer
confidentiality function
– In case of dispute some third party must view the message and its
signature
• The validity of the scheme depends on the security of the sender’s private key
– If a sender later wishes to deny sending a particular message, the sender
can claim that the private key was lost or stolen and that someone else
forged his or her signature
– One way to thwart or at least weaken this ploy is to require every signed
message to include a timestamp and to require prompt reporting of
compromised keys to a central authority
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ElGamal Digital Signature
• Scheme involves the use of the private key for encryption
and the public key for decryption
• Global elements are a prime number q and a, which is a
primitive root of q
• Use private key for encryption (signing)
• Uses public key for decryption (verification)
• Each user generates their key
– Chooses a secret key (number): 1 < xA < q-1
– Compute their public key: yA = a
xA mod q
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Schnorr Digital Signature
• Scheme is based on discrete logarithms
• Minimizes the message-dependent amount of computation
required to generate a signature
– Multiplying a 2n-bit integer with an n-bit integer
• Main work can be done during the idle time of the
processor
• Based on using a prime modulus p, with p – 1 having a
prime factor q of appropriate size
– Typically p is a 1024-bit number, and q is a 160-bit
number
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N I S T Digital Signature Algorithm
• Published by N I S T as Federal Information Processing
Standard F I P S 186
• Makes use of the Secure Hash Algorithm (S H A)
• The latest version, F I P S 186-3, also incorporates digital
signature algorithms based on R S A and on elliptic curve
cryptography
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Figure 13.2 Two Approaches to
Digital Signatures
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Figure 13.3 The Digital Signature
Algorithm (D S A)
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Figure 13.4 D S A Signing and Verifying
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Elliptic Curve Digital Signature
Algorithm (E C D S A)
• Four elements are involved:
– All those participating in the digital signature scheme use
the same global domain parameters, which define an elliptic
curve and a point of origin on the curve
– A signer must first generate a public, private key pair
– A hash value is generated for the message to be signed;
using the private key, the domain parameters, and the hash
value, a signature is generated
– To verify the signature, the verifier uses as input the signer’s
public key, the domain parameters, and the integer s; the
output is a value v that is compared to r ; the signature is
verified if the v = r
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Figure 13.5 E C D S A Signing and
Verifying
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R S A-P S S
• R S A Probabilistic Signature Scheme
• Included in the 2009 version of F I P S 186
• Latest of the R S A schemes and the one that R S A Laboratories
recommends as the most secure of the R S A schemes
• For all schemes developed prior to P S S it has not been possible
to develop a mathematical proof that the signature scheme is as
secure as the underlying R S A encryption/decryption primitive
• The PSS approach was first proposed by Bellare and Rogaway
• This approach, unlike the other R S A-based schemes,
introduces a randomization process that enables the security of
the method to be shown to be closely related to the security of
the R S A algorithm itself
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Mask Generation Function (M G F)
• Typically based on a secure cryptographic hash function
such as S H A-1
– Is intended to be a cryptographically secure way of
generating a message digest, or hash, of variable
length based on an underlying cryptographic hash
function that produces a fixed-length output
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Figure 13.6 R S A-P S S Encoding
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Figure 13.7 R S A-P S S E M Verification
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Summary
• Present an overview of the digital signature process
• Understand the ElGamal digital signature scheme
• Understand the Schnorr digital signature scheme
• Understand the N I S T digital signature scheme
• Compare and contrast the N I S T digital signature scheme
with the ElGamal and Schnorr digital signature schemes
• Understand the elliptic curve digital signature scheme
• Understand the R S A-P S S digital signature scheme
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