The Dual Receiver Cryptosystem The Dual Receiver Cryptosystem and its Applicationsand its Applications

Presented by Brijesh Shetty

OverviewOverview

Dual Receiver Cryptosystem – Concept

Interesting Applications Combined Cryptosystem Useful Puzzle Solving

Dual Receiver Cryptosystem

Encryption Scheme

Ciphertext can be decrypted by two independent receivers!

Bilinear Diffie Hellman Assumption (based on elliptic curves)

Elliptic Curve based Discrete Log problem

Given Y = k . P and Y,P(i.e P added to itself k times)

Find k ???? (P,P)--- g (Y,P)--- h By definition of Bilinear Curve, we get

h = gk [since (aP,bQ)=(P,Q)ab](Y,P)=(kP,P)=(P,P)k=gk

“ Key Escrow ” (in the context of Dual receiver)

An arrangement where keys needed to decrypt encrypted data must be held in escrow by a third party.

Eg. Govt. agencies can use it to decrypt messages which they suspect to be relevant to national security.

Dual Receiver Cryptosystem

Ciphertext C

Message m

Encrypt using public keys of

B and C

B

C

A

Decrypts to m

C can also decrypt!C does not learn about the private keys of B or A !!

Dual Receiver Cryptosystem- The Scheme

Some Definitions

(Semantically secure) Dual Receiver Cryptosystem scheme

Definitions (Randomised algorithms)

Key Generation algorithm K(k) = (e,d) & (f,g)

Encryption algorithm E e,f (m) = c

Definitions (contd..)

Decryption Algorithm D Dd,f (c) = m

Recovery Algorithm RRe,g (c) = m

Dual Receiver Cryptosystem- The Scheme

Some Definitions

(Semantically secure) Dual Receiver Cryptosystem scheme

Semantically secure Dual Receiver Cryptosystem

B

C

A

(x, xP)

(y, yP)

Message mRandom r

(u1,u2,u3)private

Hx is a hash fn associated with public key xP

Semantically secure Dual Receiver Cryptosystem

B

C

A

(x, xP)

(y, yP)

Message mRandom r u1 = rP

u2 = yPu3 = m+ Hx(<xP,yP>r)

Decryption

B<u1,u2>x = <rP,yP>x

= <xP,yP>r

= <P,P>xyr

U3 + Hx(<xP,yP>r) = m

Recovery (Second Receiver)

C

<u1,xP>y = <rP,xP>y

= <xP,yP>r

= <P,P>xyr

U3 + Hx(<xP,yP>r) = m

Dual Receiver Cryptosystem- The Scheme

Some Definitions

(Semantically secure) Dual Receiver Cryptosystem scheme

OverviewOverview

Dual Receiver Cryptosystem – Concept

Interesting Applications Combined Cryptosystem Useful Puzzle Solving

Combined Cryptosystem

We combine using a single key x

Dual Receiver Encryption

Signature

Signature (in Combined scheme)

Same key x . Hash I:{0,1}n -> G1

A B

Sign the hash

Message m

σ = x . I(m)

Verification..

BVerify

<P, σ > = <xP, I(m)>

B has m, σ

If they are same both must be equal <P,I(m)>x

Combined Cryptosystem

What is so special?

Dual receiver encryption facilitates escrow of the decryption capability & non escrow of the signature capability using the same key!!

The security of either of the schemes is not compromised

OverviewOverview

Dual Receiver Cryptosystem – Concept

Interesting Applications Combined Cryptosystem Useful Security Puzzles

Useful Security Puzzles

Application Areas When Server wants to rate-limit the

clients (against DOS attacks) Lighten the server’s computational

burden

Example : File Server

File Server (Security Puzzle)

Client

Abcde……

FileServer

¤¥§~¶…….

KsEke,Ka(Ks)

(C1,C2)[ ]STORING FILE

File Server … (Request File)¤¥§~¶……. (C1,C2)[ ]

C1, PaComputeDPa,Ke(C1)

Random pC1 = Eke,Ka(p)

u1 = Ks+ G(p)u2 = F(p,Ks,C1,u1)

C2 = [u1,u2]

G, F are hashes

TD1 G(TD1)+u1 = mCheck

u2=F(p,m,c,u1)

Client

Decryption..Computation intensive

XOR and hashDual Receiver Encrytpion

Thank you