Cryptography

Lecture 7: RSAPrimality Testing

Piotr Faliszewski

Previous Class Number theory

Euler’s phi function

RSA Definition Correctness

Quiz Min = 14 Med = 21.75 Max = 30

The RSA Algorithm

Alice m – message to Bob m < n

otherwise cut m in blocks

Receives the public key (n, e) from Bob

Computesc = me (mod

n)

sends it to Bob

Bob p, q large primes

n = pq choose e s.t.

gcd(e, (pq) ) = 1 compute d s.t.

de = 1 (mod (n)) Bob sends his public

key (n,e) to Alice Bob decrypts the

messagem = cd

(mod n)

Security of RSA How to attack RSA?

we have public key (n,e) compute (n) and get d

Easier said than done!!

If we have n and n) then we can factor n

If we have e and dthen we can factor n

Adi Shamir

Attacks on RSA Various attacks on RSA

known digits attack low exponent attack short plaintext attack timing attack factoring

Len Adleman

Known Digits Attack Theorem

n = pq – has m digits

If we know the first or the last m/4 digits of either p or q, then we can efficiently factor n

Theorem(n,e) – RSA public keyn has m digits, and we know the last m/4 digits of d

We can find d in time linear in e log e

Conclusion: Need care about the choice of p and q!

Low Exponent Attack Theorem

p,q – RSA primes, q < p < 2q, n = pq 1 ≤ d, e < (n) de = 1 (mod (n)) If d < ⅓n1/4 , then d can be calculated quickly

Consequences cannot optimize decryption via small exponents how to choose good d’s?

Short Plaintext Attack Scenario

DES – symmetric cipher, used in the past Two banks exchange DES keys over RSA DES key m: 56 bits (m < 256 < 1017) RSA encryption

c = me (mod n) m small, but c will have many digits

Short Plaintext Attack c = me (mod n)

m < 256 < 1017

Eve prepares two lists: cx-e (mod n), 1 ≤ x ≤ 109

ye (mod n) 1 ≤ y ≤ 109

If there is a match on the lists then c = (xy)e (mod n) thus: m = xy (mod n)

If m is a multiple of two numbers <109 then this attack will succeed

Short Plaintext Attack Defense:

do not use short messages! pad with random bits Optimal Assymetric Encryption Padding

Optimal Assymetric Encryption Padding n – k bits k0, k1 – two numbers

s.t. k0 + k1 < k Message can have

k - k0 - k1 bits

r – random string of k0 bits

G: k0 k-k

0

H: k-k0 k

0

The method

x1 = m0k1 G(r)

x2 = r H(x1)

The message is x1x2

Bob decrypts and gets

m0k1 = x1 G(H(x1) x2)

Timing Attack Within RSA computation we do perform

exponentiaiton quick exponentiation procedure multiplications occur for each bit of the exponent

that is 1 these multiplications take „random” amounts of

time (variation)

Very hard in practice! Initiated a big discussion

Exponentiation Algorithm

Goal: yd (mod n) d = b1b2...bw

(in binary left-to-right) Algorithm

1. k = 1, s1 = 1

2. if bk = 1 then rk = sky (mod n)else rk = sk

3. sk+1 = rk2 (mod n)

4. if k = w, stopelse set k to k+1, goto 2

5. output rw

Factoring and Primality Testing

Factoring and Primality Testing

Factoring Input: n N Output: nontrivial factor of n

Primality testing Input n N Output:

the number is composite the number is probably

prime

Is there a difference? Yes! – primality testing

much easier!

You do not need to factor the number to see it is composite

Factoring and Primality Testing

Primality testing often need large primes

n = pq in RSA p, q – large primes

there are many primes (x) ≈ x/ln(x)

to choose p,q – choose a number randomly and test if it is a prime increase if composite

Factoring security of RSA =

hardness of factoring

quick factoring – allows us to break RSA

studying factoring – tells us how to choose p and q for RSA so pq is hard to factor

Fermat’s Test Fermat’s test:

n – input randomly choose a s.t.

1 < a < n-1 if an-1 = 1 then probably prime else certainly composite

Fermat’s Theorem

If:

p – a prime numbera – a number s.t. p does not divide a

Then:

ap-1 = 1 (mod p)

Fermat’s Test Fermat’s test is only

probabilistic usually gives a good

answer can give a wrong one!

A composite number n s.t.

an-1 = 1 (mod n)

is called pseudoprime for base a

Example2560 = 1 (mod 561)

561 = 3 * 11 * 17

But...3560 = 375 (mod 561)

Miller-Rabin Test Generalization of the

Fermat’s test

Principle if p is a prime then

x2 = 1 (mod p)

has only two solutions:

x = 1 and x = -1

Why does the principle hold?

Gist of the MR test find a number b such

that b2 = 1 (mod p) If b {-1,1} then

composite

Miller-Rabin Test

MR( int n ):let n-1 = 2kma random in {2, 3, ..., n-2 }

b0 = am (mod n)

if b0 = ±1 (mod n) then declare primefor j = 1 to k-1 do

bj = bj-12 (mod n)

if bj = 1 (mod n) then declare composite

if bj = -1 (mod n) thendeclare prime

declare composite

Why this works?

b0 = am (mod n)

b1 = a2m (mod n)

b2 = a4m (mod n) ... bj = a2jm (mod n) ... bk-1 = a(n-1)/2 (mod n)

Miller-Rabin Test: Examples n = 5*7*11 = 385

n -1 = 384 = 27*3

k = 7, m = 3

a = 9

b0 = 93 = 344 (mod 385)

b1 = 93*2 = 141 (mod 385)

b2 = 93*22 = 246 (mod 385)

b3 = 93*23 = 71 (mod 385)

b4 = 93*24 = 36 (mod 385)

b5 = 93*25 = 141 (mod 385)

n = 3*11*17 = 561n -1 = 560 = 24*35

k = 4, m = 35

a = 2

b0 = 235 = 263 (mod 561)

b1 = 235*2 = 166 (mod 561)

b2 = 235*22 = 67 (mod 561)

b3 = 235*23 = 1 (mod 561)

Proof of compositeness!

Miller-Rabin Test: Examples n = 401

n -1 = 400 = 24*25

k = 4, m = 25

a = 3

b0 = 325 = 268 (mod 401)

b1 = 325*2 = 45 (mod 401)

b2 = 325*22 = 20 (mod 401)

b3 = 325*23 = 400 (mod 401)= -1

(mod 401)

n = 401n -1 = 400 = 24*25

k = 4, m = 25

a = 2

b0 = 225 = 356 (mod 401)

b1 = 225*2 = 20 (mod 401)

b2 = 225*22 = 400 (mod 401)

Evidence of primality!

Miller-Rabin Test

if b0 = ±1 (mod n) all bi’s (i > 0) will be 1 can’t find nontrivial roots of 1

i {1, ..., k-1} if bi = 1 (mod n) then

bi-1 is neither 1 or -1 bi-1

2 = 1 (mod n) we found a nontrivial root

if bi = -1 (mod n) then bi+1 through bk are all 1 (mod

n) can’t find nontrivial roots of 1

Why this works?

b0 = am (mod n) b1 = a2m (mod n) b2 = a4m (mod n) ... bj = a2jm (mod n) ... bk-1 = a(n-1)/2 (mod n)

Miller-Rabin Test: Quality MR test is probabilistic Answer

composite – the number is certainly composite

prime – the number is prime with high probability

Errors MR(n) says prime but n

is composite Pr[error] · ¼ Repeat the test to

downgrade the prob. of error

Other Primality Tests Solovay-Strassen Test

similar in nature to MR uses so called Jacobi

symbol fast in practice probabilistic

Deterministic test Agrawal, Kayal, and

Saxena 2002 extremely slow

Tests that prove primality MR tests

compositeness! fairly slow needed in very few

cases