Cryptography V: Digital Signatures · Signature mechanism A signature mechanism for principal A is...

Cryptography V:Digital Signatures

Computer Security Lecture 12

David Aspinall

School of InformaticsUniversity of Edinburgh

19th February 2009

Outline

Basics

Constructing signature schemes

Security of signature schemes

ElGamal

Summary

Outline

Basics

ElGamal

Summary

É Digital signatures allow a principal tocryptographically bind (a representation of) itsidentity to a piece of information.

É Signatures can help establish security propertiessuch as:

É authenticationÉ accountability/non-repudiationÉ unforgeabilityÉ integrityÉ verifiability by independent, public or 3rd party

É Digital signatures are the asymmetric analogue ofMACs, with a crucial difference. (Exercise:what?)

É Note: electronic signatures are a more generalnotion.

É Signatures can help establish security propertiessuch as:

É authenticationÉ accountability/non-repudiationÉ unforgeabilityÉ integrityÉ verifiability by independent, public or 3rd party

É Signatures can help establish security propertiessuch as:É authentication

É accountability/non-repudiationÉ unforgeabilityÉ integrityÉ verifiability by independent, public or 3rd party

É Signatures can help establish security propertiessuch as:É authenticationÉ accountability/non-repudiation

É unforgeabilityÉ integrityÉ verifiability by independent, public or 3rd party

É Signatures can help establish security propertiessuch as:É authenticationÉ accountability/non-repudiationÉ unforgeability

É integrityÉ verifiability by independent, public or 3rd party

É Signatures can help establish security propertiessuch as:É authenticationÉ accountability/non-repudiationÉ unforgeabilityÉ integrity

É verifiability by independent, public or 3rd partyÉ Digital signatures are the asymmetric analogue of

MACs, with a crucial difference. (Exercise:what?)É Note: electronic signatures are a more general

notion.

É Signatures can help establish security propertiessuch as:É authenticationÉ accountability/non-repudiationÉ unforgeabilityÉ integrityÉ verifiability by independent, public or 3rd party

Handwritten versus Digital Signatures

ink binds to paper cryptographically bound to data

verifier needs signature verifier needs public key

signatures always same depends on document

copies apparent copies indistinguishable

signer saw document computer added signature

have legal impact may have legal impact

Signature mechanism

A signature mechanism for principal A is given by:É A message space M of messages for signingÉ A set S of signatures (e.g. strings {0,1}n)É A secret signing function SA :M→ SÉ A public verification function VA :M× S → Bool

satisfying the correctness and security properties:1. VA(m, s) = true if and only if SA(m) = s.2. For any principal other than A, it is computationally

infeasible to find for any m ∈M, an s ∈ S such thatVA(m, s) = true.

Usually use a public algorithm yielding key-indexedfamilies {Ss | s ∈ K} of signing and verification functions{Vv | v ∈ K}. Principal advertises v.Remark: nobody has proved a signature mechanismsatisfying 2 exists, although there are good candidates.

Signature mechanism

Usually use a public algorithm yielding key-indexedfamilies {Ss | s ∈ K} of signing and verification functions{Vv | v ∈ K}. Principal advertises v.

Remark: nobody has proved a signature mechanismsatisfying 2 exists, although there are good candidates.

Signature mechanism

Using a signature scheme

É To sign a message the signer A

1. Computes s = SA(m).2. Sends the pair (m, s).

É To verify that a signature s on a message m wascreated by A, another principal, the verifier:

1. Obtains the verification function VA for A.2. Computes u = VA(m, s)3. Accepts the signature if u = true, rejects it if

u = false.

É To sign a message the signer A

1. Computes s = SA(m).2. Sends the pair (m, s).

u = false.

É To sign a message the signer A1. Computes s = SA(m).

2. Sends the pair (m, s).É To verify that a signature s on a message m was

created by A, another principal, the verifier:

u = false.

É To sign a message the signer A1. Computes s = SA(m).2. Sends the pair (m, s).

u = false.

1. Obtains the verification function VA for A.

2. Computes u = VA(m, s)3. Accepts the signature if u = true, rejects it if

u = false.

1. Obtains the verification function VA for A.2. Computes u = VA(m, s)

3. Accepts the signature if u = true, rejects it ifu = false.

u = false.

Outline

Basics

ElGamal

Summary

Digital signatures with a TTPÉ Given a trusted third party, it is possible to use

symmetric cryptography techniques.

É Let secure Sam S be the TTP, who shares a key witheach principal.

É For A to send a signed contract M to B, S acts as anintermediary.

Message 1. A→ S: {M}Kas

Message 2. S→ B: {M}Kbs

(like Wide Mouthed Frog key exchange protocol, Mshould include time-stamps and names).

É If A and B disagree about a signature, a judge Judycan verify the contracts also using S:

Message 1. J→ S: {M}Kas ,{M}Kbs

Message 2. S→ J: {yes or no}Kjs

symmetric cryptography techniques.É Let secure Sam S be the TTP, who shares a key with

each principal.

É For A to send a signed contract M to B, S acts as anintermediary.

each principal.É For A to send a signed contract M to B, S acts as an

intermediary.

each principal.É For A to send a signed contract M to B, S acts as an

intermediary.

Digital signatures from PK encryptionÉ Suppose we have a public-key encryption scheme

with M = C, and (d,e) a key-pair. Then because Ee

and Dd are both permutations on M, we have that:

Dd(Ee(m)) = Ee(Dd(m)) = m for all m ∈M

A public-key scheme of this type is calledreversible.

É RSA is reversible, but not every PK scheme is.É We can define a digital signature scheme by

reversing encryption and decryption:

É Message space M, signature space C (=M).É the signing function SA = DdÉ the verification function VA is defined by

VA(m, s) =

true if Ee(s) = m,false otherwise.

É However, this scheme is somewhat too simple. . .

É RSA is reversible, but not every PK scheme is.

É We can define a digital signature scheme byreversing encryption and decryption:

VA(m, s) =

reversing encryption and decryption:

VA(m, s) =

reversing encryption and decryption:É Message space M, signature space C (=M).

É the signing function SA = DdÉ the verification function VA is defined by

VA(m, s) =

reversing encryption and decryption:É Message space M, signature space C (=M).É the signing function SA = Dd

É the verification function VA is defined by

VA(m, s) =

reversing encryption and decryption:É Message space M, signature space C (=M).É the signing function SA = DdÉ the verification function VA is defined by

VA(m, s) =

reversing encryption and decryption:É Message space M, signature space C (=M).É the signing function SA = DdÉ the verification function VA is defined by

VA(m, s) =

Existential forgery

É The previous scheme is too simple becausesignatures are forgeable: a principal B can generatea random s ∈ S as a signature, apply the publicencryption function to get a message m = Ee(s),and transmit (m, s).

É Obviously this verifies! It is an example ofexistential forgery.

É The message m is not likely to be of B’s choosing(and probably garbage).

É But this ability violates property 2 given earlier.

Existential forgery

Signatures with redundancy

É A fix to reduce likelihood of existential forgery is totake M′ ⊂M to be messages with a specialredundant structure, which is publicly known e.g.,messages padded to an even length, surroundedwith a fixed bit pattern.

É This format is easily recognized by the verifier:

VA(s) =

true if Ee(s) ∈M′,false otherwise.

É Now A only transmits the signature s, since themessage m = Ee(s) can be recovered by theverification function.

É This property is message recovery, the scheme iscalled a signature scheme with recovery.

É Existential forgery is less likely.

VA(s) =

Signatures and hash functions

É In practice, usually the signing function isconstructed by first making a hash of the inputdocument, and signing that. Reasons:

1. efficiency: signature is on smaller text2. avoid attacks on cipher system

É Signer: computes and transmits (m, s) wheres = SA(h(m)).

É Verifier: computes h(m) and verifies VA(h(m), s).É When the original message is required as an input

to the verification function, this is known as adigital signature scheme with appendix.

É The hash function must satisfy appropriateproperties (see Hash Functions lecture).

1. efficiency: signature is on smaller text

2. avoid attacks on cipher systemÉ Signer: computes and transmits (m, s) where

s = SA(h(m)).É Verifier: computes h(m) and verifies VA(h(m), s).É When the original message is required as an input

É Verifier: computes h(m) and verifies VA(h(m), s).

É When the original message is required as an inputto the verification function, this is known as adigital signature scheme with appendix.

Outline

Basics

ElGamal

Summary

Attacks on signature schemes [HAC]É An adversary seeks to forge signatures.

Possibilities:

1. Total break. Adversary can compute the privatekey or find an equivalent signing function.

2. Selective forgery. Adversary can create a validsignature for some chosen message, without usingthe signer.

3. Existential forgery. Adversary can create a validsignature for at least one message, without explicitchoice of the message. May involve signer.

É The adversary may have different knowledgelevels. For PK schemes:

1. Key-only attack: adversary only knows PK.2. Known-message attack: adversary has

signatures for a known set of messages not chosenby him.

3. Chosen-message attack: adversary can obtainsignatures for messages of his choosing. Messagesmay be determined in advance or in adaptive way,using signer as oracle.

Possibilities:1. Total break. Adversary can compute the private

key or find an equivalent signing function.

2. Selective forgery. Adversary can create a validsignature for some chosen message, without usingthe signer.

key or find an equivalent signing function.2. Selective forgery. Adversary can create a valid

signature for some chosen message, without usingthe signer.

1. Key-only attack: adversary only knows PK.

2. Known-message attack: adversary hassignatures for a known set of messages not chosenby him.

Outline

Basics

ElGamal

Summary

ElGamal signaturesÉ Setup as encryption: p an appropriate prime, g a

generator of Z∗p

, and the private signing key, d arandom integer with 1 ≤ d ≤ p− 2.

É The public verification key is (p,g,gd mod p).É To sign a message m, 0 ≤m ≤ p, the signer picks a

random secret number r with 1 ≤ r ≤ p− 2 andgcd(r,p− 1) = 1, and computes:

Sd(m) = (e, s) where e = gr mod pde + rs ≡m (mod p− 1).

É The verification function checks that 1 ≤ e ≤ p− 1,and an equation:

V(p,g,gd)(m, (e, s)) =

true if (gd)ees ≡ gm (mod p),

false otherwise.

É Verification works because for a correct signature,

(gd)ees ≡ gde+rs ≡ gm (mod p).

generator of Z∗p

É The public verification key is (p,g,gd mod p).

É To sign a message m, 0 ≤m ≤ p, the signer picks arandom secret number r with 1 ≤ r ≤ p− 2 andgcd(r,p− 1) = 1, and computes:

V(p,g,gd)(m, (e, s)) =

false otherwise.

generator of Z∗p

V(p,g,gd)(m, (e, s)) =

false otherwise.

generator of Z∗p

V(p,g,gd)(m, (e, s)) =

false otherwise.

generator of Z∗p

V(p,g,gd)(m, (e, s)) =

false otherwise.

Outline

Basics

ElGamal

Summary

From ElGamal to DSAÉ The Digital Signature Algorithm is part of the NIST

Digitial Signature Standard [FIPS-186].

É Based on ElGamal, but with improved efficiency.É The first digital signature scheme to be recognized

by any government.É Based on two primes: p, which is 512–1024 bits

long, and q, which is a 160-bit prime factor of p− 1.A signature signs a SHA-1 hash value of a message.(In fact, ElGamal signing should be used with ahash function to prevent existential forgery)

É Security of both ElGamal and DSA schemes relieson the intractability of the DLP.

É Comparison with RSA signature scheme: keygeneration is faster; signature generation is aboutthe same; DSA verification is slower. Verification isthe most common operation in general.

Digitial Signature Standard [FIPS-186].É Based on ElGamal, but with improved efficiency.

É The first digital signature scheme to be recognizedby any government.

É Based on two primes: p, which is 512–1024 bitslong, and q, which is a 160-bit prime factor of p− 1.A signature signs a SHA-1 hash value of a message.(In fact, ElGamal signing should be used with ahash function to prevent existential forgery)

Digitial Signature Standard [FIPS-186].É Based on ElGamal, but with improved efficiency.É The first digital signature scheme to be recognized

by any government.

É Based on two primes: p, which is 512–1024 bitslong, and q, which is a 160-bit prime factor of p− 1.A signature signs a SHA-1 hash value of a message.(In fact, ElGamal signing should be used with ahash function to prevent existential forgery)

Outline

Basics

ElGamal

Summary

Summary: Digital Signature SchemesÉ RSA, ElGamal, DSA already described. There are

several variants of ElGamal, including schemes withmessage recovery.

É Notice difference between randomized anddeterministic schemes.

É Schemes for one-time signatures (e.g., Rabin,Merkle), require a fresh public key for each use.É Typically more efficient than RSA/ElGamal methods.É But tedious for multiple documents

É E-cash protocols use blind signature schemesthat prevent the signer (e.g., a bank) linking asigned message (e.g., the cash) with the user.

É For real world security guarantees:É obtaining correct public key is vital;É non-repudiation supposes that private key has

not been stolen;É we may require secure time stamps.

References

Alfred J. Menezes, Paul C. Van Oorschot, and Scott A.Vanstone, editors. Handbook of Applied Cryptography.CRC Press Series on Discrete Mathematics and ItsApplications. CRC Press, 1997.Online version athttp://www.cacr.math.uwaterloo.ca/hac.Digital signatures covered in Section 1.6 and Chapter 11.

Nigel Smart. Cryptography: An Introduction.McGraw-Hill, 2003. Third edition online:http://www.cs.bris.ac.uk/~nigel/Crypto_Book/

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