PUBLIC-KEY CRYPTOGRAPHY AND RSA – Chapter 9 PUBLIC-KEY CRYPTOGRAPHY AND RSA – Chapter 9 Principles Applications Requirements RSA Algorithm Description Security

PUBLIC-KEY CRYPTOGRAPHY (PKC) – A New Idea Historically – Symmetric-Key (one key) substitution (confusion) permutation (diffusion) More Recently – Asymmetric-Key (two keys)

MISCONCEPTIONS PKC vs Symmetric Encryption MISCONCEPTIONS PKC vs Symmetric Encryption PKC more secure than symmetric encryp. WRONG!! PKC more useful than symmetric encryp. WRONG!! – PKC costly PKC doesn’t need complicated protocol WRONG!!

PKC - USES PKC - USES Key Management Signature

PKC – SIX INGREDIENTS PKC – SIX INGREDIENTS Plaintext – input to encryp. algorithm output from decryp. algorithm Encryp. Algorithm – acts on plaintext - controlled by public or private key Public and Private Key - one for encryption - one for decryption Ciphertext – output from encryp. algorithm input to decryp. algorithm Decryp. Algorithm – acts on ciphertext - controlled by public or private key

PKC – STEPS PKC – STEPS 1.Each user generates two related keys - PUBLIC and PRIVATE 2. Each user makes: public key  PUBLIC private key  PRIVATE access  ALL public keys 3. BOB: Encr ( plaintext,PUBLIC Alice )  ciphertext ALICE 4. ALICE: Decr ( ciphertext,PRIVATE Alice )

PKC for a) ENCRYPTION b) AUTHENTICATION

At ANY TIME, ANY Private/Public key pair can be changed. Public key should be made public IMMEDIATELY KEYS EASILY UPDATED

Symmetric-Key: One SECRET KEY Asymmetric-Key (PKC): One PRIVATE KEY One PUBLIC KEY CIPHER TERMINOLOGY CIPHER TERMINOLOGY

CONFIDENTIALITY

AUTHENTICATION (source) (Integrity/Signature)

CONFIDENTIALITY and AUTHENTICATION

Encryp./Decryp. Sender encrypts with RECIPIENT’S PUBLIC key. Applied to ALL of message. Digital Signature Sender signs with SENDER’S PRIVATE key. Applied to ALL or PART of message. Key Exchange Uses one or more PRIVATE keys. Several approaches APPLICATIONS OF PKC APPLICATIONS OF PKC

Table 9.2 APPLICATIONS OF PKC APPLICATIONS OF PKC

Every value has an inverse Y = F(X)  X = F -1 (Y) Y = F(X) - easy X = F -1 (Y) - infeasible easy – polynomial time (poly in message length) infeasible - > poly time (e.g. exp. in message length) ONE-WAY FUNCTION ONE-WAY FUNCTION

Y = f k (X) - easy if k and X known X = f k -1 (Y) - easy if k and Y known X = f k -1 (Y) - infeasible if only Y known TRAP-DOOR ONE-WAY FUNCTION (e.g. PKC) TRAP-DOOR ONE-WAY FUNCTION (e.g. PKC)

Brute-Force Attack  Use LARGE keys But, PKC COMPLEXITY GROWS fast with key size So, PKC TOO COMPLEX encryp/decryp PKC only for key management and signature PKC – THE PROBLEM OF KEY SIZE PKC – THE PROBLEM OF KEY SIZE

PKC: 1960’s (NSA) 1970 Ellis – CESG 1976 Diffie and Hellman RSA: 1973 Cocks – CESG 1977 Rivest, Shamir, Adleman - MIT RSA ALGORITHM RSA ALGORITHM

Plaintext and Ciphertext integers between 0 and n-1 i.e. k bits, 2 k < n <2 k+1 Encryption: C = M e mod n Decryption: M = C d mod n = (M e ) d mod n = M ed mod n RSA RSA

Sender knows n,e Receiver knows n,d  PUBLIC key, KU = {e,n}  PRIVATE key, KR = {d} RSA (continued) RSA (continued)

1. There exists e,d,n s.t. M ed = M mod n 2. Easy to calculate M e and C d given {M,e} or {C,d}, resp. 3. Infeasible to find d given {e,n} PKC REQUIREMENTS OF RSA PKC REQUIREMENTS OF RSA

p = 17, q = 11 n = p.q = 187 mod p = 17, {1,6,6 2,6 3,6 4,6 5,6 6,6 7,6 8,6 9,6 10,6 11,6 12,6 13,6 14,6 15 } = {1,6,2,12,4,7,8,14,16,11,15,5,13,10,9,3} Mod p = 11 {1,2,4,8,5,10,9,7,3,6} EXAMPLE EXAMPLE

57 = (6,2), 57 2 = (2,4), 57 3 = (12,8), 57 4 = (4,5) EXAMPLE

We want number, g, between 1 and 186 s.t. g mod 17 = 6, g mod 11 = 2 Use CRT: g = mod 187 = 57 EXAMPLE Chinese Remainder Theorem EXAMPLE Chinese Remainder Theorem

EXAMPLE RSA COMPUTATION

Brute-Force Attacks – try all possible private keys. Mathematical Attacks - all equivalent to factoring n. Timing Attacks - depend on running time of decryption algorithm. SECURITY OF RSA SECURITY OF RSA

Table 9.3 Progress in Factorisation Progress in Factorisation

MIPS-years NEEDED TO FACTOR

For Decryption: Constant exponentiation time Random delay Blinding Generate random r C’ = Cr e M’ = C’ d M = M’r -1 TIMING ATTACKS ON RSA - countermeasures