Security & Cryptography

Security & Cryptography

Protection vs Security • The protection mechanisms (ACLs, etc) discussed earlier assist us in preventing unauthorized access and use of computer resources • what happens if an intruder bypasses the protection mechanisms? • Cryptography can be used so that an intruder is unable to understand or use information obtained without authorization

Cryptography Terminology • Plaintext (or cleartext) • is the intelligible message • Ciphertext • is the unintelligible message • Encryption and decryption • Are the processes to convert between plaintext and ciphertext • Key • Is the parameter used in an encryption/decryption algorithm

Cryptography Terminology • Cryptosystem • A system for encryption/decryption of information • Symmetric cryptosystem • use the same key for both encryption and decryption • Asymmetric cryptosystem • use the different keys for encryption and decryption • Cryptology • the designing & breaking of cryptosystems • Cryptography • the practice of using cryptosystems for confidentiallity of information • Cryptoanalysis • the breaking cryptosystems

Basic Structure of a Cryptosystem Eve Plaintext M Side Information Break Bob Alice Plaintext M Plaintext M Encrypt Decrypt Ciphertext C Encryption Key Ke Decryption Key Kd

Basic Attacks to Cryptosystems • Cryptosystem attacks are classified based on the amount of side information available to an intruder • Attack classification • ciphertext-only • intruder only has access to the ciphertext • known-plaintext • intruder has access to the ciphertext and considerable amount of plaintext • chosen-plaintext • intruder has access to a chosen plaintext and its corresponding ciphertext

Design Principles for Cryptosystems • Shannon’s principles • Diffusion principle • spread the correlations and dependencies among key and words over the text as much as possible in order to maximize the length of plaintext needed to break the system • Confusion principle • change a piece of information so that ciphertext has no obvious relationship with plaintext • Computational Intractability principle • “every” algorithm for determining a key needed to break cryptosystem is “believed” to require exhaustive search of a very large search space

A Taxonomy of Cryptosystems • Conventional systems • Modern systems • private key systems • public key systems

Conventional Cryptosystems • Conventional cryptosystems are based on substitution ciphers • Caesar’s cipher • E(M) = (M + k) modulo 26 • where M is a letter and k=3 is the key • Simple substitution cipher • E(M) = Key[M] • where Key is an arbitrary permutation of a single alphabet • Vigenere cipher • choose N simple substitution ciphers and encrypt the jth letter using the (j mod N) substitution cipher • One-time pad • encrypt by Xoring message with a key, whose size equals the size of the message

DES • The Data Encryption Standard (DES) is a modern private-key cryptosystem • It is a block cipher that uses two basic operations • permutation, • and substitution • It breaks a message in 64-bit blocks and encrypts/decrypts each block individually • It uses a 56-bit secret key, which is expanded to 64-bits using parity bits

DES • Has three stages • plaintext block undergoes an initial permutation IP • permuted block undergoes for 16 times a complex transformation • A block at the ith iteration is broken into two 32-bit blocks Li & Ri • transformed block undergoes the inverse IP’ of the permutation IP at the 1st stage • DES transformation in the ith iteration, i=1,2,…,16 • K i= Phi(Key, i) 48-bit key of ith iteration • L i = Ri-1 • R i = L i xor F (Ri-1 , K i )

DES • Function F does the following • expands R i into a 48-bits quantity E(R i) by permuting and duplicating some bits of R • Xors E(R i) with K i and partitions the result into eight 6-bit blocks Q1, Q2,…,Q8 • passes each Q j 6-bit block through a separate 6-to-4 bit substitution box • concatenates all transformed 4-bit Q j blocks and then permutes them

DES • Decryption is done by executing the three stages in reverse order and each time using the inverse function/operation • permute cipher text using IP’ • undo the 16 transformations, for i=16,15,…,1, using the same keys K1, K2, …, K16 • R i-1 = R i • L i-1 = R i xor f ( L i , K i ) • permute transformed ciphertext with IP • For added security, block chaining can be used • each plaintext block is Xored with the ciphertext of the previous plaintext block • triple encryption (DES does not form a group) • Rijdael: new private key standard

Public-Key Cryptosystems • Private key cryptosystems requires a secure mechanism for distributing the private keys to communicating parties • Diffie and Hellman proposed public key cryptosystems • public key systems make the encryption key publicly available and keep the decryption key secret • public key systems are based on the computational intractability principle (using problems such as factoring primes, discrete logarithm, knapsack, etc)

Public Key Cryptosystems • public key systems satisfy the following • DSK(EPK(M)) = M for every message M • The encryption and decryption functions E and D are computationally efficient • Knowledge of E, D, and PK (public key) does not compromise SK (secret key) • DPK(ESK(M)) = M for every message M, if message singing/verification is desired

Trapdoor One-Way Functions • One-way functions F • F is invertible and easy to compute • inverting F is computationally intractable, ie given y finding x such that y=F(x) is believed to be computationally infeasible • Trapdoor one-way functions F • y=F(x) can be solved efficiently provided some secret information for F is available • Diffie and Hellman suggested that one way to implement public key systems is to use trapdoor one-way functions

Number Theory Background • GCD Recursion Theorem & the Extended Euclid’s algorithm

Number Theory Background • Euler’s phi function, Euler’s and Fermat’s Theorems

Number Theory Background • The Chinese Remainder Theorem • Origins • Sun-Tsu, circa 100 A.D. considered the problem of finding those integers x that leave remainders 2, 3, and 2 when divided by 3, 5, and 7 respectively (which are of the form x=23+105k). • Its essence

Number Theory Background • A corollary of the Chinese Remainder Theorem states that

RSA • Rivest, Shamir, and Adleman introduced the RSA public-key cryptosystem based on Diffie and Hellman • RSA works as follows

RSA • RSA’s encryption function is • EPK(M) = Me mod n where PK=(e,n) • RSA’s decryption function is • DSK(M) = Md mod n where SK=(d,n) • these two encryption/decryption functions satisfy • DSK(EPK(M)) = M • DPK(ESK(M)) = M • can be computed efficiently given PK or SK • knowledge of PK does not compromise SK

RSA • Correctness of RSA is based on • Fermat’s theorem and on the Chinese Remainder Theorem • Example values for RSA • choose p=5 and q=11 • set n=55 and N=40 • choose d=23 • compute e=7 using the extended Euclid algorithm • encrypt M=8 to 2 using “repeated squaring”

RSA • A more realistic example set of values for RSA (courtesy of Prof. Stephens) • n = 2419753086 4197530864 2125371358 0246913580 2471460971 7 • p = 1555555555 5555555555 560261 • q = 1555555555 5555555555 560497 • e = 512896171 • d = 1955459782 2571725357 3495557871 3933814929 3601459917 1 • sqrt(n) approximately = 1555555555 5555555555 560378 • number of positive integers < n that are relative prime to n is equal to phi(n) • phi(n) = 2419753086 4197530864 2125340246 9135802469 1360348896 0

Authentication • Objective • verify the identity of communicating entities • Authentication services • interactive communication (synchronous) • one-way communication (asynchronous) • signed communication (verifiable conversation by third party) • Potential threats • altering messages • replaying old messages • denial of service • interference with ongoing communication • impersonation

Interactive Communication Protocols • Require an authoritative Authentication Server (AS) for securely distributing conversation keys • Each user registers its secret key with the AS, which is shared only between the AS and the user, and their public key if any • Requirements – use case • Alice wants to communicate with Bob so that • the message is intelligible to Bob, but not Eve • it should be evident that the message was sent by Alice, and that is not a replay of an older message from Alice

Interactive Communication with Private Key Systems • Alice wants to converse with Bob • Denning-Sacco’s modification to handle compromised conversation keys • A message is not a reply attack if |LocalClock-T|<LocalClock’s disrepancy from AS’s clock plus the estimated maximum network delay

Interactive Communication with Public Key Systems • Alice wants to communicate with Bob

One-Way Communication with Private Key Systems • Alice wants to email message M to Bob • Bob should be able to authenticate integrity of Alice’s message even if Alice is not currently available • Eve should not be able to impersonate Alice Protocol is succeptible to playback attacks

One-Way Communication with Public Key Systems • Alice wants to email message M to Bob

Digital Signatures • Must satisfy the following • a user can not forge signatures • sender of signed message can not deny the validity of his signature • receipient can not modify the signature of a signed message

Digital Signatures using Private Key Systems • Alice wants to sign a message to be sent to Bob

Digital Signatures using Public Key Systems • Alice wants to sign a message to be sent to Bob

Kerberos • An authentication system for an open network computing environment where user’s machines are under their complete control and can not be trusted to identify users to network services • Consists of • Client (C) • Kerberos Server (K) • Ticket Granting Server (TGS) • Server (S) • User (U)

Kerberos Phase I: Getting the Initial Ticket • User provides the Client machine his/her identity • Client sends to Kerberos server K the msg • Kerberos server K • Client upon receipt of msg

Kerberos, Phase II: Getting a Server Ticket • User/Client wants to use a network service S • Ticket Granting Server TGS • Client upon receiving msg from TGS

Kerberos, Phase III: Requesting a Service • Client requests service from server S • Service server S upon receipt of the msg

Security & Cryptography