Signcryption Parshuram Budhathoki Department of Mathematical Sciences Florida Atlantic University April 18, 2013

Motivation: Confidentiality : Integrity : Authentication : Nonrepudiation : Keeping information secret from all other than those who are authorized to see it. Ensuring that the information has not been altered by unauthorized entities. The assurance that the communicating party is the one that it claims to be. Preventing the denial of previous commitments or actions.

Motivation: Confidential and Authenticate Traditional Method “Signature- Then- Seal”

Motivation: Confidentiality : How do we get these things in modern cryptography ? Signature Scheme Encryption Scheme Nonrepudiation : Integrity : Authentication :

Signature Scheme RSA Schnorr DSS Others … Motivation: How do we get these things in modern cryptography ?

Encryption Scheme RSA ElGamal Others … Motivation: How do we get these things in modern cryptography ?

In 1997 Yulian Zheng proposed a separate primitive called Signcryption. Is it possible to deliver messages of varying length in a secure and authenticated way with an expense less than that required by “Signature-Then-Encryption ” ? Motivation:

Outline : Why Signcryption ? Signcryption Shortening ElGamal-Based Signatures. Secure Signcryption Scheme. Signcryption Scheme by Y. Zheng.

Why Signcryption ? Cost of Signcryption < Cost of Signature + Cost of Encryption 1.Computational cost 2.Communication overhead

1.Computational cost We estimate computational cost by counting the number of operations involved : Private key encryption and decryption Hashing addition Multiplication Division Exponentiation Why Signcryption ?

2. Communication overhead In addition to computational cost, digital signature and encryption based on public key cryptography also require extra bits to be appended to a message. We call these extra redundant bits the communication overhead involved. Why Signcryption ?

Signcryption Gen Security Parameter key-pair Public Private SC Private key sender, Message, R ID C= SC Private Key ( Message, R ID ) DSCDSC Private Key ( C, S ID ) Private key receiver, C, S ID

Shortening ElGamal-Based Signatures: Key Generation: Choose x randomly from {1, …, p-1} Public key = g x Private key = x Let p is a large prime, q is a large prime factor of p-1 and g is an integer from {1, …, p-1}. Let h: {1,..., p-1} x {0,1} * {1,..., p-1} be a hash function.

Sign : Choose y randomly from {1, …, p-1} r = h( g y mod p, M) s = y / ( r + x ) mod p, where M = message Signature = Let p is a large prime, q is a large prime factor of p-1 and g is an integer from {1, …, p-1}. Let h: {1,..., p-1} x {0,1} * {1,..., p-1} be a hash function. Shortening ElGamal-Based Signatures:

Let p is a large prime, q is a large prime factor of p-1 and g is an integer from {1, …, p-1}. Let h: {1,..., p-1} x {0,1} * {1,..., p-1} be a hash function. Verify : Compute k = ( g x ∙ g r ) s mod p Accept if r = h( k, m) Shortening ElGamal-Based Signatures:

Secure Scheme: Secure Signature Scheme : Unforgeable under adaptively chosen message attack. Secure Encryption Scheme : Indistinguishable against adaptively chosen cipher attack.

Unforgeable : It is computationally infeasible for an adaptive attacker to create a signcrypted text. Non-repudiation : It is computationally feasible for a third party to settle a dispute between signer and receiver where signer denies the fact that he/she is the originator of a signcrypted text. Confidential: It is computationally infeasible for an adaptive attacker to gain any partial information on the contents of a signcrypted text. Secure Signcryption Scheme:

YZ- Signcryption Scheme: Gen : Using this algorithm sender and receiver choose their key-pair. Let x, y from [1,..., q-1] are sender’s and receiver’s private keys and S= g x and R= g y are their respective public keys. p : a large prime ( public ) q : a large prime factor of p-1 ( public ) g : a ( random ) integer in [1,..., p-1] with order q mod p ( public ) h : a one-way hash function ( public )

YZ- Signcryption Scheme: p : a large prime ( public ) q : a large prime factor of p-1 ( public ) g : a ( random ) integer in [1,..., p-1] with order q mod p ( public ) h : a one-way hash function ( public ) S : Sender’s public key R : Receiver’s public key SC : Pick r randomly from [1,..., q-1] 1. compute k = R r mod p. Split k into k 1 and k 2 of appropriate length. 2. n= h(M, k 2 ), where M=message 3. s= r/( n + x ) mod q 4. c = E_k 1 ( M ), where E := Encryption in AES Signcrypted text =

YZ- Signcryption Scheme: DSC : Recover k from n, s, g, p, S and R: 1. k = ( S ∙ g h ) s ・ y mod p 2. Split k into k 1 and k 2 3. M = D_k 1 ( c ), Where D := Decryption in AES 4. Accept M as a valid message if h(M, k 2 ) = n p : a large prime ( public ) q : a large prime factor of p-1 ( public ) g : a ( random ) integer in [1,..., p-1] with order q mod p ( public ) h : a one-way hash function ( public ) S : Sender’s public key R : Receiver’s public key NOTE : D_k 1 ( E_k 1 (M)) = M

Security parameters |p|, |q|, |h(. )| Saving in comp. cost Saving in comm. overhead 768, 152, 8050%76.8% 1024, 160, 8050%81.01% 2048, 192, 9650%87.7% 4096, 256, 12850%91.0% 8192, 320, 16050%94.0% 10240, 320, 16050%96.0% Cost of Signcryption vs. Cost of Sign-Then- Encryption

Question ? Thank You !!!