Mathematics of Cryptography

Report
Inside PK Cryptography:
Math and Implementation
Sriram Srinivasan (“Ram”)
[email protected]
Agenda


Introduction to PK Cryptography
Essential Number Theory






Fundamental Number Theorem
GCD, Euclid’s algorithm
Linear combinations
Modular Arithmetic
Euler’s Totient Function
Java implementation of RSA
Sriram Srinivasan
2/47
Security Issues



Authentication, Authorization, and
Encryption, Non-repudiation
Shared Secrets (e.g passwords, Enigma)
Something shared, something (else) secret

Concept by Ellis, Cocks and Williams


Popularly attributed to Diffie and Hellman
Algorithm by Rivest, Shamir and Adelman

Used everywhere: https, SSL, email, certificates.
Sriram Srinivasan
3/47
Public Key Cryptography

Consider a pair of magic pens.



You want to send a message to me




Write with one, use the other to decode.
Symmetric: either can be used to encode
You borrow one of my pens and write with it.
I decode it with my other pen.
Avoids problems of shared secrets
Same tools for authentication, encryption and
non-repudiation.
Sriram Srinivasan
4/47
Mathematics
Fundamental Theorem of
Arithmetic

All numbers are expressible as a unique
product of primes


10 = 2 * 5,
60 = 2 * 2 * 3 * 5
Proof in two parts


1. All numbers are expressible as products
of primes
2. There is only one such product sequence
per number
Sriram Srinivasan
6/47
Fundamental Theorem proof

First part of proof

All numbers are products of primes
Let S = {x | x is not expressible as a product of primes}
Let c = min{S}.
c cannot be prime
Let c = c1 . c2
c1, c2 < c  c1, c2  S (because c is min{S})
\ c1, c2 are products of primes  c is too
\ S is an empty set
Sriram Srinivasan
7/47
Fundamental Theorem proof

Second part of proof

The product of primes is unique
Let n = p1p2p3p4… = q1q2q3q4…
Cancel common primes. Now unique primes on both sides
Now, p1 | p1p2p3p4
 p1 | q1q2q3q4…
 p1 | one of q1, q2, q3, q4…
p1 = qi which is a contradiction
Sriram Srinivasan
8/47
GCD (Greatest Common Divisor)


gcd(a,b) = the greatest of the divisors
of a,b
Many ways to compute gcd

Extract common prime factors
Express a, b as products of primes
 Extract common prime factors
 gcd(18, 66) = gcd(2*3*3, 2*3*11) = 2*3 = 6
 Factoring is hard. Not practical


Euclid’s algorithm
Sriram Srinivasan
9/47
Euclid’s algorithm
a
1
b
r=a%b
b
2
3
r
r
r1
r1 = b % r
r % r1 = 0.
\ gcd (a,b) = r1
Sriram Srinivasan
10/47
Euclid’s algorithm proof

Proof that r1 divides a and b
r1 | r
b = r1 + r
a = qb + r
r1 | b
r1 | r
r1 | b
r1 | a
Sriram Srinivasan
11/47
Euclid’s algorithm proof

(contd)
Proof that r1 is the greatest divisor
Say, c | a and c | b
c | qb + r
c|r
c | q’b + r1
c | r1
Sriram Srinivasan
12/47
Linear Combination

ax + by = “linear combination” of a and b


12x + 20y = {…, -12,-8,-4,0,4,8,12, … }
The minimum positive linear combination
of a & b = gcd(a,b)

Proof in two steps:
1. If d = min(ax+by) and d > 0, then d | a, d | b
 2. d is the greatest divisor.

Sriram Srinivasan
13/47
GCD & Linear combination
(contd.)
Let S = {z = ax + by | z > 0 }
Let d = min{S} = ax1 + by1
Let a = qd + r. 0 <= r < d
r = a - qd = a - q(ax1 + by1)
r = a(1 - qx1) + (-qy1)b
If r > 0, r  S
But r < d, which is a contradiction, because d = min{S}
\r=0
d | a
Sriram Srinivasan
14/47
GCD & Linear combination

(contd.)
Second part of proof

Any other divisor is smaller than d
Let c | a, c | b, c > 0
a = cm, b = cn
d = ax1 + by1 = c(mx1 + ny1)
c | d
d is the gcd
Sriram Srinivasan
15/47
Summary 1




All numbers are expressible as unique
products of prime numbers
GCD calculated using Euclid’s algorithm
gcd(a,b) = 1  a & b are mutually prime
gcd(a,b) equals the minimum positive
ax+by linear combination
Sriram Srinivasan
16/47
Modular/Clock Arithmetic

1:00 and 13:00 hours are the same



1:00 and 25:00 hours are the same
1  13 (mod 12)
a  b (mod n)




n is the modulus
a is “congruent” to b, modulo n
a - b is divisible by n
a%n=b%n
Sriram Srinivasan
17/47
Modular Arithmetic


a  b (mod n), c  d (mod n)
Addition
a - b = jn
c - d = kn
a + c - (b + d) = (j + k) n


a + c  b + d (mod n)
Multiplication

ac  bd (mod n)
Sriram Srinivasan
18/47
Modular Arithmetic (contd.)

Power

a  b (mod n)  ak  bk (mod n)
Using induction,
If ak  bk (mod n),
a . ak  b . bk (mod n), by multiplication rule
\ ak+1  bk+1 (mod n)

Going n times around the clock

a + kn  b (mod n)
Sriram Srinivasan
19/47
Chinese Remainder Theorem

m  a (mod p), m  a (mod q)
 m  a (mod pq) (p,q are primes)
m-a = cp.
Now, m-a is expressible as p1. p2 .p3 . . .
If m - a is divisible by both p and q,
p and q must be one of p1 , p2 , p3
 m - a is divisible by pq
Sriram Srinivasan
20/47
GCD and modulus

If gcd(a,n) = 1, and a = b (mod n),
then gcd(b,n) = 1
a  b (mod n)  a = b + kn
gcd(a,n) = 1
ax1 + ny1 = 1, for some x1 and y1
(b + kn)x1 + ny1 = 1
bx1 + n(kx1 + y1) = bx1 + ny2 = 1
gcd(b,n) = 1
Sriram Srinivasan
21/47
Multiplicative Inverse

If a, b have no common factors, there
exists ai such that a.ai  1 (mod b)

ai is called the “multiplicative inverse”
gcd(a,b) = 1 = ax1+ by1, for some x1 and y1
ax1 = 1 – by1
ax1 = 1 + by2
(making y2 = -y1)
ax1 - 1 = by2
ax1  1 (mod b) (x1 is the multiplicative inverse)
Sriram Srinivasan
22/47
Summary 2

Modular arithmetic


Chinese Remainder Theorem


Addition, multiplication, power, inverse
If m  a (mod p) and m  a (mod q),
then m  a (mod pq)
Relationship between gcd and modular
arithmetic

gcd(a,b) = 1  aai  1 (mod b)
Sriram Srinivasan
23/47
Euler’s Totient function

f(n) = Totient(n)
= Count of integers n coprime to n



f(10) = 4 (1, 3, 7, 9 are coprime to 10)
f(7) = 6 (1, 2, 3, 4, 5, 6 coprime to 10)
f(p) = p - 1, if p is a prime
Sriram Srinivasan
24/47
Totient lemma #2: product

f(pq) = (p - 1)(q - 1) = f(p) . f(q)

if p and q are prime
Which numbers  pq share factors with pq?
1.p, 2.p, 3.p, … (q-1)p and
1.q, 2.q, 3.q, … (p-1)q and
pq
The rest are coprime to pq. Count them.
f(pq) = pq - (p - 1) - (q - 1) - 1 = (p - 1)(q - 1)
Sriram Srinivasan
25/47
Totient lemma #3: power

f(pk) = pk - pk-1 , if p is prime and k > 0
Only numbers that are a multiple of p have a
common factor with pk :
1.p, 2.p, 3.p, … pk-1 . p and
The rest don’t share any factors, so are coprime
\ f(pk) = pk - pk-1
Sriram Srinivasan
26/47
Totient lemma #4: product

f(mn) = f(m) . f(n)

if m and n are coprime ( gcd(m,n) = 1)
Organize into a matrix of m columns, n rows
1
2
3
…
r
…
m
m+1
m+2
m+3
m+r …
2m
2m+1
2m+2
2m+3
2m+r …
3m
(n-1)m+3
(n-1)m+r
nm
…
(n-1)m+1 (n-1)m+2
Sriram Srinivasan
27/47
Totient lemma #4

(contd.)
Step 1: Eliminate columns
If gcd(m,r) = 1, gcd(m,km+r) = 1
 All cells under that rth column have no common
factors with m
Others have a common factor with mn, so can be
eliminated
f(m) columns survive
Sriram Srinivasan
28/47
Totient lemma #4

(contd.)
Step 2: Examine cells in remaining
columns
No two cells in a column are congruent mod n
Because if im + r  jm + r (mod n), im + r - jm - r = kn
n |(i - j), which is not possible because i - j < n
Because there are n (non-congruent) cells in each
column, label them as 0, 1, 2, … n-1 in some order.
f(n) cells in each column coprime to n
f(n) f(m) cells left that are coprime to both m and n
Sriram Srinivasan
29/47
Totient lemma #5

If gcd(c,n) = 1 and x1,x2,x3 … xf(n) are
coprime to n, then cx1,cx2,… cxf(n) are
congruent to x1,x2,x3… in some order.

1, 3, 5, 7 are coprime to 8.

Multiply each with c=15, (also coprime to 8)

{15, 45, 75, 105}  {7, 5, 3, 1} (mod 8)
Sriram Srinivasan
30/47
Totient lemma #5
(contd.)
cxi is not  cxj (mod n). Because if cxi  cxj (mod n)
 c(xi - xj) = kn . But gcd(c,n) = 1
 n | (xi - xj), which is impossible because xi - xj < n
Remember the old identity:
gcd(a,n) =1 and a  b (mod n) gcd(b,n) = 1
Let cxi  b (mod n)
gcd(cxi, n) = 1  gcd(b,n) = 1
\ b must be one of xj
Sriram Srinivasan
31/47
Euler’s Theorem

If gcd(a,n) = 1, af(n)  1 (mod n)
Consider x1, x2, … xf(n) < n and coprime to n
Since a is also coprime to n, from previous result
ax1  xi (mod n), ax2  xj (mod n), … etc.
af(n) x1x2x3…xf(n)  x1x2x3…xf(n) (mod n)
af(n) x  x (mod n) where x = x1x2x3…xf(n)
n | x(af(n) - 1)
But n doesn’t divide x
n | (af(n) - 1)
af(n)  1 (mod n)
Sriram Srinivasan
32/47
Fermat’s little theorem

Special case of Euler’s theorem.

If gcd(a,p) = 1 and p is prime,
ap-1  1 (mod p)
Because f(p) = p - 1

We now have all the essential number
theory. Whew!
Sriram Srinivasan
33/47
RSA Algorithm

Bob generates public and private keys



public key : encrypting key e and modulus n
private key: decrypting key d and modulus n
Alice wants to send Bob a message m
m treated as a number
Alice encrypts m using Bob’s “public pen”
e
 encrypted ciphertext, c = m (mod n)



Bob decrypts using his own private key

To decrypt, compute cd (mod n). Result is m
Sriram Srinivasan
34/47
RSA Key Generation




Bob selects primes p, q computes n = pq
f(n) = f(p) f(q) = (p - 1) (q - 1)
Select e, such that gcd(e, f(n)) = 1
Compute the decrypting key, d, where




ed  1 (mod f(n))
Bob publishes public key info: e, n
Keeps private key: d, n
Important: m < n
Sriram Srinivasan
35/47
RSA Key Generation






Bob
selects
n = pq
p = 3,
q = 11 primes
 p, nq =computes
33
f(n) = f(p)
f(q) =- (p
(3 - 1)(11
1) -= 1)
20(q - 1)
Select
e = 7 e, such that gcd(e, f(n)) = 1
Compute
the20)
decrypting
d, where
7d = 1 (mod
 d = (1 key,
+ 20k)/7

edd=13(mod f(n))
Bob
publishes
Public
key = (7,public
33) key pair: e, n
Privateprivate
key = (3,
33)
Keeps
key:
d, n
Sriram Srinivasan
36/47
RSA algorithm

Treat eachletter
block
“RSA”
{18,or19,
1} as m (m < n)



n = 33, e = 7, d = 3
77
Encryption:
each
18
119
%%33
33 for {6,
{6
13m1}
13,
compute c=me (mod n)
6333%
113
%%33
33
33 for {18,
{18
19
Decryption:
each19,
c, 1}
compute cd (mod n)
Sriram Srinivasan
37/47
RSA proof

Prove c = me (mod n)  cd(mod n) = m
Review:
a  b (mod n)  ak  bk (mod n)
a<n
 a = a (mod n)
gcd(a,n) = 1
 af(n)  1 (mod n)
a (mod p)  a (mod q)  m = a (mod pq)
f(pq) = f(p)f(q)
ed  1 (mod f(n) )  ed = 1 + k f(n)
Sriram Srinivasan
38/47
RSA proof (contd.)
c = me (mod n)  c  me (mod n)
cd  med (mod n)
Consider, med (mod p) and med (mod q)
If p | m, med (mod p) = 0 = m (mod p)
If not,
med (mod p) m1+kf(n) (mod p)
m. mkf(p) f(q) (mod p)
m. (mf(p)) kf(q) (mod p)
m. (1) kf(q) (mod p) (by euler)
m (mod p)
Sriram Srinivasan
39/47
RSA proof (contd.)
So, in both cases, med m (mod p)
Similarly,
med m (mod q)
\ med m (mod pq)
(chinese remainder theorem)
m (mod n)
\ med (mod n) = m
Sriram Srinivasan
40/47
RSA Implementation

Creating a big random prime
SecureRandom r = new SecureRandom();
BigInteger p = new BigInteger(nbits, 100, r);

n = pq
n = p.multiply(q);

f(n) = (p - 1) (q - 1)
phi = p.subtract(BigInteger.ONE)
.multiply(q.subtract(BigInteger.ONE));
Sriram Srinivasan
41/47
RSA Implementation

Select e coprime to f(n)
e = new BigInteger("3");
while(phi.gcd(e).intValue() > 1)
e = e.add(new BigInteger("2"));

Select d, such that ed  1 (mod f(n))
d = e.modInverse(phi);
Sriram Srinivasan
42/47
RSA Implementation

Encrypt/decrypt
BigInteger encrypt (BigInteger message) {
return message.modPow(e, n);
}
BigInteger decrypt (BigInteger message) {
return message.modPow(d, n);
}
Sriram Srinivasan
43/47
Digital Signature



med (mod n) = mde (mod n)
Bob encrypts his name using private key
Alice, the recipient, decrypts it using
Bob’s public key
Sriram Srinivasan
44/47
RSA Deployment

If msg m > n, m chop it up in blocks < n

p and q are usually 512 bits, e = 65537.

Ensure p - 1 doesn’t have small prime
factors. Ensure d is large

Pad m with random bits

Never reuse n

Sign documents very carefully
Sriram Srinivasan
45/47
Examples of RSA Attacks

Exploiting algorithm parameter values


Exploiting implementation




Low e or d values
Measuring time and power consumption of
smart cards
Exploiting random errors in hardware
Exploiting error messages
Social Engineering: Blinding attack
Sriram Srinivasan
46/47
Ellis / Diffie-Hellman Key
Exchange

RSA is slow in practice



Encrypt AES’s keys using RSA
Alice and Bob agree publicly on a prime
p, and some integer, c < p. gcd(p,c) = 1
Alice chooses a privately, and Bob
chooses b. a, b < p
Sriram Srinivasan
47/47
Ellis / Diffie-Hellman Key
Exchange (contd)





Alice computes A=ca (mod p). Bob
computes B=cb (mod p)
They exchange these numbers.
Alice computes Ba. Bob computes Ab
Both of them compute cab (mod p)
Both use this number as a key for AES.
Sriram Srinivasan
48/47
References

“Cryptological Mathematics”, Robert Lewand

“Twenty Years of Attacks on the RSA
Cryptosystem”, Dan Boneh

http://crypto.stanford.edu/~dabo

pajhome.org.uk/crypt/index.html

“Concrete Mathematics”, Donald Knuth et al.

"The Code Book", Simon Singh
Sriram Srinivasan
49/47

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