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Boneh-Franklin Identity Based Encryption Scheme

Parshuram BudhathokiDepartment of Mathematics

Florida Atlantic University

28 March, 2013

03/28/2013Graduate Student Seminar, Department of

Mathematics, FAU

Motivation:

Alice wants to send a message to Bob.

Securely

Alice Bob


Mathematics, FAU

Motivation:

Alice Bob


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Alice Bob

Private Key Cryptography

AES DES

Motivation:


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Alice Bob

Private Key Cryptography

Limitation: The Key-Distribution Problem. Key Storage and Secrecy. Problem in Open Systems.

Motivation:


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Motivation:

Alice Bob

Public Key Cryptography

In 1976, Whitfield Diffie and Martin Hellman


Mathematics, FAU

Motivation:

Alice Bob


Public Key Private Key


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Motivation:

Alice Bob


Public Key

AAAAB3NzaC1yc2EAAAABJQAAAQB/nAmOjTmezNUDKYvEeIRf2YnwM9/uUG1d0BYsc8/tRtx+RGi7N2lUbp728MXGwdnL9od4cItzky/zVdLZE2cycOa18xBK9cOWmcKS0A8FYBxEQWJ/q9YVUgZbFKfYGaGQxsER+A0w/fX8ALuk78ktP31K69LcQgxIsl7rNzxsoOQKJ/CIxOGMMxczYTiEoLvQhapFQMs3FL96didKr/QbrfB1WT6s3838SEaXfgZvLef1YB2xmfhbT9OXFE3FXvh2UPBfN+ffE7iiayQf/2XR+8j4N4bW30DiPtOQLGUrH1y5X/rpNZNlWW2+jGIxqZtgWg7lTy3mXy5x836Sj/6L


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Motivation:

Alice


Before starting communication:o Alice has to get Bob’s Public key o She has to verify that this Public Key is correct one.o So, she needs chain of certificates.


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Motivation:

Alice Bob

In 1984 Adi Shamir suggested

Identity Based Cryptography

Public Key 1. email id : [email protected]. phone : 561297-0bob3. Address : 777 Glades Road


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mailto:[email protected]

Motivation:

Alice Bob

In 1984 Adi Shamir suggested


Public Key 1. email id : [email protected]. phone : 561297-0bob3. Address : 777 Glades Road

In 2001 Dan Boneh and Matthew Franklin proposed an encryption scheme.


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Outline

• Identity Based Cryptography• Pairing• Hash functions• Bilinear Diffie - Hellman problem.• BF encryption scheme.


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1. Setup 2. Extract

Encryption Scheme

4. Encrypt5. Decrypt


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Encryption Scheme:

1. Setup

Public Parameter

Master Key

2. Extract

Security Parameter

Identity, Master Key, params

Private Key

3. EncryptMessage and params

Trust Authority

Ciphertext

4. Decrypt Private Key, Ciphertext, and params Message

params


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Secure ID-based encryption scheme:

Adversary

GAME

Challenger uses Setup algorithm to generates params and Master key

Challenger


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Secure ID-based encryption scheme:

Adversary

GAME

params

Master Key

Challenger


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Secure Id-based encryption scheme:

Challenger Adversary

GAMEparams Master Key

• The Adversary issues m queries- extraction query for < Idi >- decryption query <Idi , Ci >


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Adversary

GAMEparams

• The Adversary picks M0 , M1 and a public key ID

• The Challenger picks a random b ∈ { 0, 1 } and sends C = Encrypt( params , ID, Mb )

Challenger

Master Key


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Adversary

GAMEparamsMaster Key

• The Adversary issues m additional queries - extraction query < Idi >- decryption query < Idi , Ci >

Challenger


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Adversary

GAMEparamsMaster Key

• The Adversary outputs b’ • The Adversary wins if b = b’

|P ( the adversary wins ) – 1/2| should be negligible.

• Semantic security against an adaptive chosen ciphertext and Id attackIND-ID-CCA

Challenger


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Domain

G1

Domain

G2

Range

V

P

Q

e

e(P,Q)

Domain

G

Domain

G

Pairing

Asymmetric

Symmetric


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1) Bilinearity : ∀ P, Q , R ∈ G we have

e(P+R, Q)= e(P,Q) e(R,Q)and

e(P, R+Q)= e(P,R) e(P,Q)

2) Non-degeneracy : There exists P, Q ∈ G such that e(P,Q) ≠1.

3) e is efficiently computable.

Pairing

Let (G,+) and (V, ∙ ) denote cyclic groups of prime order q ,P ∈ G, a generator of G and a pairing e: G x G V is a map which satisfies the following properties:


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Important property of bilinearity:∀ P, R ∈ G and any integer n we have

e(nP, R) = e(P + P + … + P, R ) = e(P, R) e(P, R) … e(P, R)= e(P, R)n

= e( P, nR)

Pairing


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Hash Functions:

H

Domain

x

Range

Fixed size

H(x)

Any size

No Inverse


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Hash Function:

• One way transformation

• Input := Any size, Output:= Fixed size

• H(x1 ) ≠ H(x2) for x1 ≠ x2 , Collision free


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Bilinear Diffie-Hellman Problem:

Let G1 and G2 be two groups of prime order q. Let e: G1 × G1 G2

be a pairing and let P be a generator of G1 . The BDH problem in <G1 , G2 , e > is a computation ofe(P, P ) abc , by using <P, aP, bP, cP > for some a, b, c ∈ Z*

q


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Alice

Identity Based Encryption Scheme :


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params : <G1 , G2 ,P, Ppub , e, H1 , H2 , … >

Alice

Public



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Alice Bob




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BobTrust Authority [email protected]




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BobTrust Authority




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Bob




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Bob

Any One




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params : <G1 , G2 ,P, Ppub , e, n, H1 , H2 >

Bob



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Setup Master Key:= s ∈ Z*

qk ∈ Z+

Trust Authority

BF Identity based encryption scheme :

Ppub = sP

Assume H1 : {0,1}* G1* and H2 : G2 {0,1}n

Message space = {0,1}n

Ciphertext space = G1* × {0, 1}n

params : <G1 , G2 ,P, Ppub ,q, e, n, H1 , H2 >


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BF Identity Based Encryption Scheme :

Encrypt

Alice

To encrypt message M 1. Compute QID = H1 ( ID)2. choose random r ∈ Z*

q

3. Ciphertext C := < rP , M ⨁ H2 ( grID ) >

Where gID = e( QID , Ppub ) ∈ G2*

params : <G1 , G2 ,P, Ppub ,q, e, n, H1 , H2 >


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Alice

params : <G1 , G2 ,P, Ppub , e, q, n, H1 , H2 >

Bob

C := < rP , M ⨁ H2 ( grID ) >


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params : <G1 , G2 ,P, Ppub , e, n, q, H1 , H2 >

Bob

C

Trust Authority

[email protected]


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Trust Authority

[email protected]

Extract

After getting ID ∈{0,1}*

1. Compute QID = H1 ( ID ) ∈G1*

2. Private Key = dID = s QID


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Trust Authority

Extract

[email protected]

Bob03/28/2013

Graduate Student Seminar, Department of Mathematics, FAU



Bob

Decrypt

Let C = <U, V>, then by using private key dID :

V ⨁ H2 ( e(dID , U) = M

[email protected]

C := < rP , M ⨁ H2 ( grID ) >


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Bob

Correctness of Decryption

H2 ( e(dID , U) = H2 ( e ( s H1 ( ID ) , rP))

= H2 ( e(H1 (ID) , P)sr )

= H2 ( e( H1 (ID) , sP)r )

= H2 ( (gID )r )

V ⨁ H2 ( e(dID , U) = M ⨁ H2 ( (gID )r ) ⨁ H2 ( (gID )r )

= M

[email protected]

C := < rP , M ⨁ H2 ( grID ) >


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