Authentication and Digital Signatures CSCI 5857: Encoding and Encryption

Outline Authentication Digital signature concepts RSA digital signature scheme Attacks on digital signatures The Digital Signature Standard

3 Need for Authentication Authentication Problem: How can recipient be sure that message was sent by particular person? Masquerading as Alice “Give Darth a $10,000 raise -- Alice” E

4 Authentication Terminology: –Claimant: Entity desiring to prove their identity (real or fraudulent ) –Verifier: Entity checking identity of claimant

5 Digital Signatures Based on some signing algorithm –Algorithm applied to message (like message digest) –Message and signature sent to recipient –Recipient uses related algorithm to verify signature Must involve “secret knowledge” known only to signer –Otherwise, adversary could “forge” signature “I can’t create this”

6 Public Keys and Digital Signatures Signing algorithm involves private key –Public/private key pair generated by sender Opposite of public key encryption –Sender stores private key, gives public key to recipient Private key used to sign message Public key used to verify signature

7 Digital Signatures and Confidentiality Sender: –Signs message with sender private key –Encrypts message with recipient public key Recipient –Decrypts message with recipient private key –Verifies signature with sender public key Authentication Confidentiality

8 RSA Digital Signature Scheme Encryption/Decryption: –Encryption by sender: C = P e mod n –Decryption by recipient: P = C d mod n = P d  e mod n Digital signature just reverses order –Key pair generated in same way Public key: n, e Private key: d –Signature by sender: S = M d mod n –Verification by recipient: M = S e mod n = M d  e mod n –Works since d  e = e  d

9 RSA Digital Signature Scheme Recipient has sender’s public key Sent message M and signature S generated from M Uses key to “decrypt” signature S and compare to M

10 Attacks on Digital Signatures Known Message Attack –Adversary has intercepted several messages and their corresponding signatures. –Goal: Create fake message M´ and legitimate corresponding signature from those previous messages Chosen Message Attack –Adversary has ability to make sender sign messages that adversary chooses (“We like kittens”) –Goal: Choose those messages to make it possible to create fake message M´ and legitimate corresponding signature

11 Known Message Attack on RSA Based on multiplicative property of RSA –Darth intercepts message pairs (M 1, S 1 ) and (M 2, S 2 ) Computes M´ = M 1  M 2 Corresponding signature: S´ = S 1  S 2 –Idea: S´ = S 1  S 2 = (M 1 d  M 2 d ) mod n = (M 1  M 2 ) d mod n = M´ d mod n Darth now has fake message M´ and matching signature S´ without having to know Alice’s private key!

12 Known Message Attack on RSA Problem for Darth: Fake message M´ = M 1  M 2 almost certain to be meaningless –Darth can’t control messages M 1, M 2 –Bob will treat as noise and ignore M1M1 “Buy low” M2M2 “Sell high” M1  M2M1  M2 “9485h1342nf” ???

13 Chosen Message Attack on RSA Darth chooses messages M 1, M 2 such that: –M 1, M 2 appear harmless (and can convince sender to sign) –M 1  M 2 has advantage to Darth M1M1 “We like kittens” S1S1 M2M2 “YSU rules!” S2S2 M1  M2M1  M2 “Give Darth a raise” S1  S2S1  S2 Darth asks Alice to sign these Alice creates signatures using her private key Darth sends fake message and signature to Bob

14 Signing Message Digests Sender creates message digest Sender creates signature from digest –Much more efficient than signing entire message Recipient creates same message digest from received message Recipient verifies signature based on message digest

15 Chosen Message Attack on RSA Signing message digest h(M ) instead of message M provides resistance to multiplicative attacks –h(M ) must be preimage resistant hash function Why is this effective? Darth has a fake message M´ Can compute its digest h(M´ ) Can find digests h(M 1 ), h(M 2 ) such that h(M´ ) = h(M 1 )  h(M 2 ) Darth cannot find messages M 1, M 2 with the desired digests h(M 1 ), h(M 2 ) !

16 Digital Signature Standard NIST standard (FIPS 186) Algorithms: –SHA-512 hashing –Schnorr public key encryption scheme (similar to ElGamal)

17 DSS Components Global public key components (PU G ) –p : Large prime (between 512 and 1024 bits) –q : prime divisor of p -1 (approx. 160 bits) –g = h (p-1)/q mod p where h is some integer 1 Sender’s private key (PR a ) –Random integer < q Sender’s public key (PU a ) –PU a = g PRa mod p

18 Signing a Message Generate random one-time key k < q Compute components of message: –r = (g k mod p) mod q –s = [k -1 (H(M) + PU a )] mod q Signature = (r, s) Efficiency: only modular exponentiation is g k mod p which can be computed in advance

19 Verifying a Message w = s -1 mod q u 1 = [H(M) w] mod q u 2 = (r w) mod q v = [(g u1 PU a u2 ) mod p) mod q Verified if v = r