Chapter 10 Key Management;

Other Public Key Cryptosystems

Key Management

• public-key encryption helps address key distribution problems

• have two aspects of this:– distribution of public keys– use of public-key encryption to distribute

secret keys

Distribution of Public Keys

• can be considered as using one of:– Public announcement of public keys– Publicly available directory– Public-key authority– Public-key certificates

Public Announcement

• users distribute public keys to recipients or broadcast to community at large– eg. append PGP keys to email messages or

post to news groups or email list• major weakness is forgery

– anyone can create a key claiming to be someone else and broadcast it

– until forgery is discovered can masquerade as claimed user

Publicly Available Directory

• can obtain greater security by registering keys with a public directory

• directory must be trusted with properties:– contains {name, public-key} entries– participants register securely with directory– participants can replace key at any time– directory is periodically published– directory can be accessed electronically

• still vulnerable to tampering or forgery

Public-Key Authority

• improve security by tightening control over distribution of keys from directory

• has properties of directory• and requires users to know public key for

the directory• then users interact with directory to obtain

any desired public key securely– does require real-time access to directory

when keys are needed

Public-Key Authority

Public-Key Certificates

• certificates allow key exchange without real-time access to public-key authority

• a certificate binds identity to public key – usually with other info such as period of

validity, rights of use etc• with all contents signed by a trusted

Public-Key or Certificate Authority (CA)• can be verified by anyone who knows the

public-key authorities public-key

Public-Key Certificates

Public-Key Distribution of Secret Keys

• use previous methods to obtain public-key• can use for secrecy or authentication• but public-key algorithms are slow• so usually want to use private-key

encryption to protect message contents• hence need a session key• have several alternatives for negotiating a

suitable session

Simple Secret Key Distribution

• proposed by Merkle in 1979– A generates a new temporary public key pair– A sends B the public key and their identity– B generates a session key K sends it to A

encrypted using the supplied public key– A decrypts the session key and both use

• problem is that an opponent can intercept and impersonate both halves of protocol

Simple Secret Key Distribution

Public-Key Distribution of Secret Keys

• if have securely exchanged public-keys:

Diffie-Hellman Key Exchange

• first public-key type scheme proposed • by Diffie & Hellman in 1976 along with the

exposition of public key concepts– note: now know that James Ellis (UK CESG)

secretly proposed the concept in 1970 • is a practical method for public exchange

of a secret key• used in a number of commercial products

Diffie-Hellman Key Exchange• a public-key distribution scheme

– cannot be used to exchange an arbitrary message – rather it can establish a common key – known only to the two participants

• value of key depends on the participants (and their private and public key information)

• based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) - easy

• security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard

Diffie-Hellman Example • users Alice & Bob who wish to swap keys:• agree on prime q=353 and α=3• select random secret keys:

– A chooses xA=97, B chooses xB=233• compute public keys:

– yA=397 mod 353 = 40 (Alice)– yB=3233 mod 353 = 248 (Bob)

• compute shared session key as:KAB= yB

xA mod 353 = 24897 = 160 (Alice)

KAB= yAxB mod 353 = 40233 = 160 (Bob)

ElGamal Public Key Cryptosystem • security relies on the difficulty of computing discrete

logarithms (similar to factoring) – hard• User Alice wants to send a message m to Bob• Bob: q - prime number - a primitive root of q X - private key = X mod q – public key• Alice:

– Downloads (, q, )– Chooses a secret random integer k– Encryption: r k mod q; t km mod q– Send (r, t)=(k, km) to Alice

• Bob:– Decryption: t/rX = m

Elliptic Curve Cryptography• majority of public-key crypto (RSA, D-H) use

either integer or polynomial arithmetic with very large numbers/polynomials

• imposes a significant load in storing and processing keys and messages

• an alternative is to use elliptic curves• offers same security with smaller bit sizes• Certain conventional systems with a 4096-bit

key size can be replaced by 313-bit elliptic curve system.

Elliptic Curves Over Real Number• an elliptic curve is defined by an equation in two

variables x & y, with coefficients• consider a cubic elliptic curve E(a, b) of

– y2 = x3 + ax + b – 4a3 + 27b2 0 to form a group– where x,y,a,b are all real numbers– O: the point at infinity (or be called as zero point)

• have addition operation for elliptic curve– geometrically sum of P+Q is reflection of the line L

through P and Q intersecting E– If P=Q then L is the tangent line of P

Elliptic Curves Over Real Number

• O: point at infinity (or be called as zero point)• O: additive identity

– O = -O – P+O=P

• P=(x, y), -P=(x, -y); P+(-P)=O• P=(x, 0), 2P=O, 3P=P, 4P=O, …• P+Q+R=O P, Q, R are collinear• Associative: (P+Q)+R=P+(Q+R)• Commutative: P+Q=Q+P

Elliptic Curves Over Real Number

E(a, b)= y2 = x3 + ax + b P1=(x1, y1), P2=(x2, y2), P1+P2=P3=(x3, y3) x3=m2-x1-x2

y3=m(x1-x3)-y1

if P1P2 then m=(y2-y1)/(x2-x1) if P1=P2 then m=(3x1

2+ a)/2y1

If m= then P3=O

Finite Elliptic Curves

• Elliptic curve cryptography uses curves whose variables & coefficients are finite

• have two families commonly used:– prime curves Ep(a,b) defined over Zp

• use integers modulo a prime• best in software

– binary curves E2m(a,b) defined over GF(2n)• use polynomials with binary coefficients• best in hardware

Elliptic Curves over Zp

prime curves Ep(a,b) defined over Zp – y2 mod p = (x3 + ax + b) mod p – a,b: nonnegative integer and less than p– (4a3 + 27b2) mod p 0 to form a group– P(x, y): x and y are nonnegative integers and

less than p– (x, y) Ep(a,b) (x, -y) Ep(a,b)

Elliptic Curves over Zp

prime curves Ep(a,b): y2 = (x3 + ax + b) – For each x (0x<p), calculate (x3 + ax + b) mod p– if (x3 + ax + b) is a square mod p then there are two

square roots: y and –y. i.e. (x, y)Ep&(x, -y)Ep

Ex: E23(1,1): y2 = (x3 + x + 1) mod 23

The number of points NEp is bounded by

For large p, N is approximately equal to p+1 points.

ppNpp 2121

Elliptic Curves Over Zp

Ep(a, b): y2 = x3 + ax + b (mod p), p5 is prime

P1=(x1, y1), P2=(x2, y2), P1+P2=P3=(x3, y3)

x3=(m2-x1-x2 ) mod p y3=(m(x1-x3)-y1) mod p if P1P2 then m=((y2-y1)/(x2-x1)) mod p if P1=P2 then m=((3x1

2+ a)/2y1) mod p

If m= then P3=O

Elliptic Curves Over Zp

E23(1, 1): y2 = x3 + x + 1 (mod 23)P1=(3, 10), P2=(9, 7),P1+P2=(m2-x1-x2 , m(x1-x3)-y1)=(17, 20) m=(y2-y1)/(x2-x1)=7-10/9-3=-1/2=11

P1=(3, 10), P2=(3, 10),P1+P2=(m2-x1-x2 , m(x1-x3)-y1)=(7, 12) m=(3x1

2+ a)/2y1=(3*32 + 1)/2*10=1/4=6

Elliptic Curve Cryptography• ECC addition is analog of modulo multiply• ECC multiplication is analog of modulo

exponentiation

Elliptic Curve Cryptography• ECC

– given k, P, find Q=kP EASY– given Q, P, find k HARD– known as the elliptic curve logarithm problem

• RSA– Given two primes p, q, find N=p*q EASY– Given N, find p, q HARD– known as factoring problem

• Diffle-Hellman– Given p, k, find q=pk EASY– Given q, p find k HARD– known as discrete logarithm problem

ECC Diffie-Hellman

• can do key exchange analogous to D-H• users select a suitable curve Ep(a,b)

• select a base point (Generator) G=(x1,y1) with large order n s.t. nG=O

• A & B select private keys nA<n, nB<n

• compute public keys: PA=nA×G, PB=nB×G

• compute shared key: K=nA×PB, K=nB×PA

– same since K=nA×nB×G

ECC Diffie-Hellman

E211(0, -4): y2 = x3 - 4 (mod 211)G=(2, 2) 240G=O nA=121, nB=203

compute public keys: PA=nA×G=(115, 48)

PB=nB×G=(130, 203)

compute shared key: K=nA×PB =(161, 69)

K=nB×PA =(161, 69)

ECC Encryption/Decryption• several alternatives, will consider simplest• must first encode any message M as a point on the

elliptic curve Pm

• select suitable curve & point G as in D-H• each user chooses private key nA<n

• and computes public key PA=nA*G

• to encrypt Pm : Cm={kG, Pm+k*Pb}, k random

• to decrypt Cm : Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm

ElGamal using ECC Encryption/Decryption

E11(1, 6): y2 = x3 + x + 6 (mod 11) G=(2, 7)M=(10, 9)

nA=7,PA=nA*G=(7,2)

to encrypt M : k=3 Cm={kG,M+k*PA}={(8,3), (10,2)}

To decrypt Cm : M+kPA–nA(kG)=(10,2)–(3,5)=(10,2)+(3,6)=(10,9)

Menezes-Vanstone ECC Encryption/Decryption

• M =(m1, m2)• select suitable curve & point G as in D-H• each user chooses private key nA<n• and computes public key PA=nA*G• to encrypt M : Cm={kG, c1, c2}, k random kPA=(x, y)=k*nA*G c1 =x*m1 mod p c2 =y*m2 mod p• to decrypt Cm :

kG*nA =(x, y) (x-1, y-1)(c1*x-1, c2*y-1)= (m1, m2) = M

Menezes-Vanstone ECC Encryption/Decryption

E11(1, 6): y2 = x3 + x + 6 (mod 11); G=(2, 7); M=(9, 1)

nA=7,PA=nA*G=(7,2)to encrypt M : k=6 Cm={kG, c1, c2}= {(7, 9), 6, 3} kG=6(2, 7)=(7, 9) kPA=6(7, 2)=(8, 3) c1 =x*m1 mod p = 8*9 mod 11=6 c2 =y*m2 mod p = 3*1 mod 11=3to decrypt Cm : kG*nA =7(7,9)=(8,3) (x-1,y-1)=(7,4) (c1*x-1, c2*y-1)=(6*7, 4*3)=(9, 1)

ECC Security

• relies on elliptic curve logarithm problem• fastest method is “Pollard rho method”• compared to factoring, can use much

smaller key sizes than with RSA etc• for equivalent key lengths computations

are roughly equivalent• hence for similar security ECC offers

significant computational advantages