Cryptology

Marion ScheepersBoise State University

Boise , Idaho

Sept 24 – 26, 2002

DAAD 2002

Information Security

Information Security = Safeguarding information during storage or transmission.

Some objectives of information security:•Confidentiality•Data integrity•Authentication•Non-repudiation

Cryptology = The mathematics of information security

The idea:

Use mathematics to secure the information Security is mathematically compromised when some

specific mathematical problem is solved No known method solves the mathematical problem

in a reasonable amount of time

Cryptographic Primitives

Encryption schemas Digital signatures schemas Hash functions Random number generators

Crypto-system evaluation criteria

Security levels [How many bit operations are needed to defeat the security objective?]

Performance[How many bit operations are needed to carry out the security objective?]

Ease of implementation[How “easy” is it to create the crypto-system based on existing technology?]

Some Number Theory

Theorem [Euclidean Algorithm] :

For all natural numbers a and b, there are

integers m and n such that gcd(a,b) = ma + nb.

Theorem:

ax mod b =1 has a solution if, and only if,

gcd(a,b) = 1.

Euler’s phi function

Definition phi(n) = |{a< n: gcd(a,n) = 1}|.

Theorem [Euler - Fermat] For any a <n with gcd(a,n) = 1, we have aphi(n) mod n = 1.

Chinese Remainder Theorem

Theorem Let m and n be numbers with gcd(m,n) =1. Then for all a<m and b<n there are a unique x<mn with a = x mod m and b = x mod n.

Dirichlet’s Theorem

Theorem If gcd(a,b) = 1, then the set {an + b: n=1, 2, 3, …}

contains infinitely many prime numbers.

Riemann Hypothesis (RH): For all e>0: pi(x) = li(x)+O(x(1/2 + e))

Extended Riemann Hypothesis (ERH):For all e>0: pi(x,n,a) = li(x)/phi(n)+O(x(1/2 + e))

Miller-Rabin Primality test

Theorem If n is an odd prime, write n-1 = r 2s where r is odd. For any a with gcd(a,n) = 1, 1. either ar mod n = 1, or else

2. there is a j = 0, 1, .., s-1 with a(r 2^j) mod n = n-1

Theorem For n > 9 odd and composite, at most phi(n)/4 positive numbers a < n satisfy clauses 1 or 2 modulo n.

Primitive roots

Definition: g < n is a primitive root of n if gcd(g,n) = 1 and there is for each i<n with gcd(i,n) = 1, a k with i = gk mod n.

Theorem [Gauss] n has a primitive root if, and only if, n = 1, 2, 4, pe or 2pe where p is some odd prime.

Theorem For any M>0 the set {p: p a prime number and M < least primitive root of p < p-M} is infinite.

Theorem [Burgess] For each e>0 there is a p(e) such that for each prime p > p(e) the least primitive root of p < p(1/4 + e)

Computational Complexity I

For f and g sequences of natural numbers: f = O(g): there are an N and C with: for all n>N, f(n) <

Cg(n). f = o(g): lim(f(n)/g(n)) = 0. length(n) = 1 + [log2(n)]. Ln(g,c) = O( e(c (ln(n)^g) (ln(ln(n))^ (1-g)))NOTE: Ln(0,c) = O((ln(n))c) is “polynomial in length of n” Ln(1,c) = O(e(c ln(n))) is “exponential in length of n” Ln(g,c) for 0 < g < 1 is “subexponential and

superpolynomial in length of n”

Computational Complexity 2

Computation Bit operations

m+n O(max{ln(m),ln(n)})

mn O(ln(m)ln(n))

Solve(mu+nv = b) O(ln(m)ln(n))

Solve(a*x mod m =1) O(ln(m)2)

b^n mod m O(ln(n) ln(m)2)

RSA on a finite group G

fn(a) = an

Theorem. If gcd(n,|G|) = 1, then fn is one-to-one. Moreover, if m is a solution for nx mod |G| = 1, then fm is an inverse for fn.

RSA Keys

Public keys: The group G and the function fn.

Coding function: F maps messages in ordinary text into the group G.

Private key: The function fm.

Alice encrypts to Bob

Look up Bob’s public key (G,fn) and coding function F.

With M = F(message), compute E = fn(M). [Then E is the encrypted version of M]

Send E to Bob.

Bob decrypts

Compute D = fm(E).

Compute the inverse of D under the coding function F.

RSA signatures

Public keys: The group G and the function fn.Private key: fm.

Bob signs an item, a, known to Alice: s = fm(a)

Alice checks if s is Bob’s signature on a: b = fn(s) If b = a, then Alice accepts s as Bob’s signature on

a.

Classical RSA

To construct the public and private keys: Choose two prime numbers p and q. Put R = pq. And put G = {a<R: gcd(a,R) = 1} with

operation multiplication mod R. NOTE: |G| = phi(R). Choose n so that gcd(n,phi(R)) = 1. This n specifies

the encryption function fn. Compute m = 1/n mod phi(R). This specifies the

private key fm. Coding Function: ASCIIPad

Example

RSA key generation efficiency

Selection of prime numbers. prime numbers must be large. Issue 1: How to test if a large number is prime?

Issue 2: What is the probability that a large number be prime?

Computing R. O(ln(p)ln(q)). Polynomial time. Selecting n. n must be odd. Issue: Probability(gcd(n,phi(R)) = 1)?

phi(phi(R))/phi(R). Computing m. O(ln(phi(R))2). Polynomial time.

RSA operations - efficiency

Coding computation: polynomial time.

Encryption: fn(M) = Mn mod R. O(ln(n) ln(R)2) polynomial time.

Decryption: fm(E) = Em mod R. O(ln(m)ln(R)2) ~ O(ln(R)3) – polynomial time.

RSA security

Use the factorization of R to compute phi(R) in polynomial time in the length of R. From phi(R) and fn compute fm in polynomial time in the length of phi(R).

Issue: How hard is it to factor R? The best factoring algorithms (Lenstra’s Number Field Sieve) has super polynomial time LR(1/3,c) for some constant c. This is subexponential.

Fermat factoring [Stopped here: Sept 24] Pollard p-1 method

RSA signature issues

If enough small primes have been signed, these can be used to construct forged signatures on specific messages.Example

If items signed are first padded, and n is small, certain signatures can be forged.

El-Gamal crypto system

(G,*): a finite group, g: element of large order.

Private key: a natural number N < |<g>|.

Public keys: G, g and y (=gN)

Coding function: A bijective function F which maps messages in ordinary text into G.

Alice encrypts for Bob

Compute     M:= F(message) Choose a random natural number R<|<g>| Compute:     z:= yR and d:=gR in G. Compute E:=M*z in G. Send (E,d) to Bob.

Bob Decrypts

Compute x = dN.

Compute the inverse, X, of x in G.

Compute D:= X*E in G.

Compute the inverse of F’s value on D.

Signature Schema

Items to be signed: elements of {0, 1, …, |G|-1}.

Given: Bijection f from G to {0, 1, … , |G|-1}.

The public signature key is (G,g,y,f)

Bob signs m

Choose a random number k in {1, …, |G|-1}

Compute t = gk

Compute a solution, s, to the equation m = N f(t) + sk mod |<g>|

The signature is (t,s)

Alice verifies Bob’s signature

Verify that both y and s are less than |<g>|.

Then verify if yf(t)ts (= g Nf(t) + sk) = gm

If both are true: Accept (t,s) as signature of the owner of public key (G,g,y) on m.

Classical El-Gamal

Choose a large prime number p. Put G = {1,2, …, p-1} with operation

multiplication mod p. NOTE: |G| = p-1. Choose a primitive root g of p. Choose N in {1,2,…,p-1} randomly. This

specifies the private key. Compute y = gN mod p Coding Function: ASCIIPad

El-Gamal key generation

Selection of prime numbers. Issue 1: How to test if a large number is prime? Issue 2: probability that a large number is prime?

Selection of a primitive root. Theoretical estimate: There are primitive roots < p¼ + o(1). Exhaustive search can be exponential time (O(e(¼+o(1)) ln(p))). ERH implies: Least primitive root is O(ln(p)6). Issue: How to test if g is a primitive root of p?

Selecting N. N must be large, but not easy to guess. Issue: What is a good way to choose N “randomly”?

Computing y. O(ln(p)2). Polynomial time in length of p.

El-Gamal operations

Encryption: z and d: O(ln(p)2) – polynomial time. E: O(ln(p)2) - polynomial time.

Decryption: x = yN mod p: O(ln(p)2) – polynomial time.X= x-1 mod p: O(ln(p) 2) – polynomial time. X*E mod p: O(ln(p)2) – polynomial time.

El-Gamal Security level

For discrete logarithms in this group:super polynomial time Lp(1/3,C), some constant C.

This is subexponential, and compares to the estimates for factoring classical RSA moduli

Hellman-Pohlig-Silver Algorithm [Stopped here: Sept. 25 (no signatures)]

Classical El-Gamal signatures

To sign message m: Choose, randomly, k with 0 < k < p-1. Compute t = gk mod p. Solve m = Nt+ks mod (p-1) for s Signature is (t,s)

NOTE: s = (m- Nt)/k mod (p-1) if gcd(k,p-1) = 1.

Security threats for signatures

1. Hellman Pohlig Silver algorithm.

2. Covert channels used to leak private keys.

3. Signature trapdoors by the key designers.

4. Careless choice of the primitive root.

a = x2 mod p, p prime and 0 < a<p:

Definition: a is a quadratic residue of p if a = x^2 mod p

has a solution. Else, a is a non-quadratic residue of p.

Definition: Legendre(a,p) = 1 if a is a quadratic residue of p, -1 otherwise.

Quadratic Reciprocity

Theorem: If Legendre(a,p) =1 then a = x^2 mod p has exactly two solutions. If p mod 4 =3, then exactly one of these two solutions is also a quadratic residue of p.

Theorem [Quadratic Reciprocity] For odd prime numbers p and q,Legendre(p,q)*Legendre(q,p) = (-1)(p-1)(q-1)/4

Solving a = x2 mod p

Theorem [Euler] If p mod 4 = 3 and Legendre(a,p) = 1, then b = a(p+1)/4 mod p solves a = x2 mod p, and b is a quadratic residue for p.

For p mod 4 =1, the Tonelli-Shanks algorithm can be used to compute solutions when they exist. This algorithm depends on finding first a quadratic nonresidue of p.

Theorem For each M there are infinitely many primes p such that M < least quadratic nonresidue of p < p-M

Theorem For each prime p, the least quadratic nonresidue of p is below 1 + p1/2.

Theorem [ERH] For each prime p there is a quadratic nonresidue of p of size O(ln(p)2).

Tonnelli-Shanks algorithm

1. Given a quadratic nonresidue, the algorithm finds a square root in time O(ln(p)4).

2. There is no known deterministic polynomial time algorithm for finding a quadratic nonresidue.

3. For each e>0 there is a polynomial time algorithm that finds a nonresidue with probability > 1-e:a. Randomly choose n<p [50% chance nonresidue]. b. Check in O(ln(p)3) time if n is a nonresidue.

This is a non-deterministic polynomial time algorithm. ERH indicates a deterministic polynomial time algorithm.

Elliptic Curves over Zp, p>3

Consider equations of form

y2 = x3 + ux2 +vx +w and u, v, w in Zp Examples over real line

With appropriate substitutions equivalent to:

E: y2 = x3 + a*x + b, and a, b in Zp .

If 4a3+27b2 mod p is nonzero we say: “E is an elliptic curve over Zp”.

E/Zp for 4a3+27b2 mod p nonzero

Solutions in Zp x Zp of E: y2 = x3 + ax + b, and a, b in Zp, together with extraneous point O is denoted E/Zp.

There is an Abelian group operation on E/Zp.

For A = [x1,y1] and B = [x2,y2] in E/Zp define:If A = B: L = (3x1

2 + a)/2y1

Else: L = (y2-y1)/(x2-x1)

Group operation

A+O = A = O+A

If x1 = x2 but y1 is not y2: A+B = O.

Else: A+B = [x3,y3] wherex3 = L2 – x1 – x2.y3 = y1 – L(x1 – x3).

Trace of Frobenius

t = || E/Zp | - (p+1)|Definition: If t = 1: E/Zp is anomalous.

If t mod p = 0: E/Zp is supersingular.

Theorem[Hasse] t < 2sqrt(p)+1.

Theorem:

1. p mod 4 = 3: E/Zp supersingular exactly if b = 0.

2. p mod 4 = 1: E/Zp supersingular exactly if a = 0.

The structure of E/Zp

Theorem E/Zp is isomorphic to Zd x Ze where d divides gcd(e,p-1). (d = 1 is possible).

Definition E/Zp[m] = {a in E/Zp:m*a = O}.

Theorem If m mod p > 0 then E/Zp[m] is isomorphic to Zm x Zm.

Computational Complexity and E/Zp

Group operation: O(ln(p)2) – polynomial time

Group order: O(ln(p)8) – polynomial time(Schoof’s algorithm)

Finding a point in E/Zp : Opr(ln(p)4) – probabilistic polynomial time

Examples

Discrete logarithms and E/Zp

Anomalous: O(ln(p)) – polynomial time.

Supersingular: Lk(1/3, c) wherec = (64/9)1/3) and k = pn, some “small” n.Subexponential but superpolynomial time.

General: O(eCln(p)) – exponential time.

Hellman-Pohlig-Silver algorithm also applies to E/Zp.

Koblitz embedding for E/Zp

Probabilistic parameter k: 30 < k < 51 Embedding F into Zp: range-values below p/k To embed m: Search x in [mk,(m+1)k) with

[x,y] in E/Zp for some y. This is image of m. Probability of failure about (1/2)k.

Recovering m from Koblitz point

m = floor(x/k).

.

El-Gamal on Elliptic Curve Groups

Public Key: E/Zp, g in E/Zp, y (= N*g)

(Specify p, a, b for E/Zp)

Private Key: N with 0 < N < |<g>|

EC El-Gamal Key Generation

Selection of prime number p Selection of parameters a and b Selection of point g: Opr(ln(p)4) Selection of private key N Computation of y: O(ln(N)ln(p)3)

EC-El Gamal Crypto-computations

Coding function: Opr(ln(p)4)

All other computations: O(ln(p)4) or better.

Example

EC-El Gamal vs RSA Security Level

General Elliptic curves E/Zp: O(eCln(p)).

General RSA parameter R: LR(1/3,c).

From Cln(p) = c(ln(R))1/3(ln(ln(R)))2/3:

Consider the graph of y3 = kxln(x)2