Data Structures for Java Data Structures for Java
William H. Ford William H. Ford William R. Topp William R. Topp
Chapter 28 Chapter 28
Number Theory and Encryption Number Theory and Encryption
Bret Ford
Basic Number Theory Concepts Basic Number Theory Concepts
a a divides divides b if b = a * h for some b if b = a * h for some integer h.
integer h.
An integer p is a An integer p is a prime prime if p if p 2 and p has 2 and p has two only two divisors 1 and p.
two only two divisors 1 and p.
A A composite number composite number is the product of two is the product of two integers
integers 2 called factors. 2 called factors.
The The greatest common divisor greatest common divisor of a and b of a and b
(gcd(a,b))is the largest integer that divides (gcd(a,b))is the largest integer that divides
both a and b.
both a and b.
Basic Number Theory Concepts Basic Number Theory Concepts
(continued) (continued)
Integers a and b are relatively prime if Integers a and b are relatively prime if gcd(a,b) is 1.
gcd(a,b) is 1.
The Greek mathematician Euclid provided The Greek mathematician Euclid provided an elegant recursive algorithm for
an elegant recursive algorithm for
computing gcd(a,b). The algorithm, called computing gcd(a,b). The algorithm, called the the Euclidean Algorithm Euclidean Algorithm , computes , computes
gcd(a,b) using the identity gcd(a,b) using the identity
gcd(a,b) = gcd(b, a%b)
gcd(a,b) = gcd(b, a%b)
and the stopping condition
and the stopping condition
Basic Number Theory Concepts Basic Number Theory Concepts
(continued) (continued)
EuclidGCD(a,b)
Assume a and b are nonnegative integers if (b == 0)
gcd(a,b) = a; // stopping condition.
else
gcd(a,b) = g(b, a% b) // recursive step Examples
1. Let a = 54, b = 30
gcd(54,30) = gcd(30,54 % 30) = gcd(30,24) gcd(30,24) = gcd(24,30 % 24) = gcd(24,6) gcd(24,6) = gcd(6,24 % 6) = gcd(6,0) gcd(6,0) = 6 // stop: gcd(54,30) = 6 2. Let a = 45, b = 16
gcd(45,16) = gcd(16,45 % 16) = gcd(16,13) gcd(16,13) = gcd(13,16 % 13) = gcd(13,3) gcd(13,3) = gcd(3,13 % 3) = gcd(3,1)
Basic Number Theory Concepts Basic Number Theory Concepts
(continued) (continued)
An expression of the form b*x + c*y is a An expression of the form b*x + c*y is a linear combination
linear combination of b and c. of b and c.
An extension of the Euclidean algorithm An extension of the Euclidean algorithm uses the chain of recursive calls to
uses the chain of recursive calls to
represent gcd(a,b) as a linear combination represent gcd(a,b) as a linear combination
of a and b; that is, the algorithm of a and b; that is, the algorithm
determimes integers i and j such that determimes integers i and j such that
gcd(a,b) = a * i + b * j for some integers i and j
gcd(a,b) = a * i + b * j for some integers i and j
Basic Number Theory Concepts Basic Number Theory Concepts
(continued (continued
Examples
1. Let a = 54, b = 30. The extended Euclid's algorithm determines that gcd(54, 30) = 6 and finds integers i = -1 and j = 2 such that 6 is a linear combination of 54 and 30.
6 = gcd(54,30) = 54 * -1 + 30 * 2 // i = -1, j = 2 = -54 + 60
2. Let a = 45, b = 16. The extended Euclid's algorithm determines that gcd(45, 16) = 1 and finds integers i = 5 and j = -14 such that 1 is a linear combination of 45 and 16.
1 = gcd(45,16) = 45 * 5 + 16 * -14 // i = 5, j = -14 = 225 + -224
Modular Arithmetic Modular Arithmetic
A traditional view for the set of A traditional view for the set of
integers is a line that marks discrete values centered integers is a line that marks discrete values centered
about 0. The integers are an unbounded linear about 0. The integers are an unbounded linear
collection of numbers.
collection of numbers.
Taking the remainder after division by a positive Taking the remainder after division by a positive number n, maps the integers into a finite set of number n, maps the integers into a finite set of
integers in the range [0, n). The mapping uses the integers in the range [0, n). The mapping uses the
mod operator
mod operator % For an integer a, the mapping is a % For an integer a, the mapping is a -> a % n.
-> a % n.
Modular Arithmetic (continued) Modular Arithmetic (continued)
a = b (mod n) if and only if b - a = n * k a = b (mod n) if and only if b - a = n * k for some integer k. We say a is congruent for some integer k. We say a is congruent
to b modulo n.
to b modulo n.
Example: Assume n = 15
18 = 3 (mod 15) 18 - 3 = 15 = 15 * 1 // k = 1 50 = 5 (mod 15) 50 - 5 = 45 = 15 * 3 // k = 3
Modular Arithmetic (continued) Modular Arithmetic (continued)
Define Z(n) = {0,1,2. . . n-1} with Define Z(n) = {0,1,2. . . n-1} with
operators +, *, and ^ (exponent). Results operators +, *, and ^ (exponent). Results
are computed modulo n.
are computed modulo n.
Arithmetic Operations in Z(n)
Add (+): a + b = (a + b) (mod n) Multiply (*): a * b = (a*b) (mod n)
Exponent(^): (a)e = (ae) (mod n) // exponent is e
Add(+): 7 + 11 = 18 (mod 15) = 3 Multiply(*): 7 * 11 = 77 (mod 15) = 2 Exponent(^): 72 = 49 (mod 15) = 4
Modular Arithmetic (continued) Modular Arithmetic (continued)
If a is in Z(n) and gcd(a,n) = 1, then a has If a is in Z(n) and gcd(a,n) = 1, then a has a an inverse in Z(n); there exists an i in
a an inverse in Z(n); there exists an i in Z(n) such that a*i = 1 (mod n)
Z(n) such that a*i = 1 (mod n)
Let a be a number in Z(n), then a has a Let a be a number in Z(n), then a has a
multiplicative inverse if and only if a and n are multiplicative inverse if and only if a and n are
relatively prime (gcd(a,n) = 1).
relatively prime (gcd(a,n) = 1).
Examples: Assume n = 15
1. 1 = gcd(7, 15) = 7 * 13 + 15 * -6. Inverse of 7 is 13.
2. 1 = gcd(2, 15) = 2 * 8 + 15 * -1 Inverse of 2 is 8.
Euler's Totient Function Euler's Totient Function
Euler totient function Euler totient function (n) is the number (n) is the number of integers in Z(n) relatively prime to n.
of integers in Z(n) relatively prime to n.
Examples:
1. (5) = 4 (numbers 1, 2, 3, and 4 are relatively prime to 5) 2. (6) = 2 (numbers 1 and 5 are relatively prime to 6)
3. (7) = 6 (numbers 1, 2, 3, 4, 5, and 6 are relatively prime to 7)
4. (15) = 8 (numbers 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15)
Euler's Totient Function Euler's Totient Function
(continued) (continued)
If n is prime, If n is prime, (n) = n-1 (n) = n-1
If p and q are prime, If p and q are prime,
(pq) = (pq) = (p) * (p) * (q) = (p-1)(q-1). (q) = (p-1)(q-1).
Euler's Theorem: Euler's Theorem:
Let n be a positive integer and let a be an Let n be a positive integer and let a be an integer such that gcd(a,n) = 1. Then
integer such that gcd(a,n) = 1. Then
a a
(n(n))= 1 (mod n). = 1 (mod n).
Euler's Totient Function Euler's Totient Function
(concluded) (concluded)
(n) (n) a a gcd(a,n) = 1 gcd(a,n) = 1 a a
(n)(n)(mod n) (mod n)
(5) = 4 (5) = 4 3 3 gcd(3,5) = 1 gcd(3,5) = 1 3 3
44=81=1(mod 5) =81=1(mod 5)
(6) = 2 (6) = 2 5 5 gcd(5,6) = 1 gcd(5,6) = 1 5 5
22=25=1(mod 6) =25=1(mod 6)
(7) = 6 (7) = 6 2 2 gcd(2,7) = 1 gcd(2,7) = 1 2 2
66=64=1(mod 7) =64=1(mod 7)
(15) = 8 (15) = 8 4 4 gcd(4,15) = 1 gcd(4,15) = 1 4 4
88=65536=1(mod 15) =65536=1(mod 15)
Secure Message Passing Secure Message Passing
View the customer as a client process and View the customer as a client process and the retailer as a server process. To send a the retailer as a server process. To send a
message, the client encrypts the message, the client encrypts the
information into numeric data and information into numeric data and
transmits it across the Internet. The transmits it across the Internet. The
retailer decrypts the data back to the retailer decrypts the data back to the
original message. The techniques for original message. The techniques for
encryption and decryption are called encryption and decryption are called
cryptograpy
cryptograpy . .
Secure Message Passing Secure Message Passing
(continued) (continued)
RSA data encryption uses a public-key and RSA data encryption uses a public-key and a private-key to encrypt and decrypt a
a private-key to encrypt and decrypt a message.
message.
The server retains the secret key but sends the The server retains the secret key but sends the public key to the client who uses it to encrypt a public key to the client who uses it to encrypt a message. The term "public" is meaningful. The message. The term "public" is meaningful. The
server makes no attempt to hide the value from server makes no attempt to hide the value from
an eavesdropper when sending the key. The
an eavesdropper when sending the key. The
client uses the public key even though it may
client uses the public key even though it may
Secure Message Passing Secure Message Passing
(continued)
(continued)
Creating Keys for RSA Creating Keys for RSA
Encryption Encryption
Begin by selecting (at random) two Begin by selecting (at random) two
prime numbers p and q and form the product prime numbers p and q and form the product
n = p * q.
n = p * q.
Let t be the value of the Euler totient Let t be the value of the Euler totient function for a product of primes.
function for a product of primes.
t = t = (n) = (p - 1) * (q - 1) (n) = (p - 1) * (q - 1)
Select at random an encryption key e subject Select at random an encryption key e subject to the conditions e < t and
to the conditions e < t and
gcd(e, t) = 1. The public key is the pair (e,n).
gcd(e, t) = 1. The public key is the pair (e,n).
Creating Keys for RSA Creating Keys for RSA Encryption (concluded) Encryption (concluded)
Compute the decryption key d which is the Compute the decryption key d which is the inverse of e modulo t. Use the extended
inverse of e modulo t. Use the extended Euclidean algorithm to find d.
Euclidean algorithm to find d.
1 = gcd(e,t) = e * d + t * j 1 = gcd(e,t) = e * d + t * j
for some integers d and j for some integers d and j The private key is the pair (d,n).
The private key is the pair (d,n).
Using Keys for RSA Encryption Using Keys for RSA Encryption
Theorem (RSA) Theorem (RSA)
Let n = p * q where p and q are prime Let n = p * q where p and q are prime
numbers and let e and d be encryption and numbers and let e and d be encryption and
decryption keys; that is 1 = e * d (mod t) decryption keys; that is 1 = e * d (mod t)
where t = (p - 1) * (q - 1). Then for any where t = (p - 1) * (q - 1). Then for any
integer a in Z(n), integer a in Z(n),
a = a
a = a
e*de*d(mod n) (mod n)
Using Keys for RSA Encryption Using Keys for RSA Encryption
(continued) (continued)
Assume M in Z(n) is a numeric Assume M in Z(n) is a numeric
representation of a message and let representation of a message and let
C = M
C = M
ee(mod n). (mod n).
C C
dd(mod n) = (M (mod n) = (M
ee(mod n)) (mod n))
dd(mod n) = M (mod n) = M
e*de*d(mod n) = M.
(mod n) = M.
Using the RSA algorithm securely requires Using the RSA algorithm securely requires using large primes p and q for n = p*q. It using large primes p and q for n = p*q. It
relies on the fact that it is very, very hard to relies on the fact that it is very, very hard to
find the factors p, q of n.
find the factors p, q of n.
Using Big Integers Using Big Integers
The class BigInteger allows for the The class BigInteger allows for the
creation and manipulation of integers with creation and manipulation of integers with
an arbitrarily large number of digits. Aside an arbitrarily large number of digits. Aside
from addition, etc. there are methods that from addition, etc. there are methods that
implement modular arithmetic, evaluate implement modular arithmetic, evaluate
the GCD, and generate prime numbers.
the GCD, and generate prime numbers.
The static constant ONE defines the The static constant ONE defines the
BigInteger value 1. The class provides the BigInteger value 1. The class provides the
tools for RSA encryption.
tools for RSA encryption.
Using Big Integers (continued) Using Big Integers (continued)
// convert a String to a BigInteger representation String message;
BigInteger bigInt = new BigInteger(message.getBytes());
// convert a BigInteger to a String BigInteger bigInt;
String message = new String(bigInt.toByteArray())
BigInteger Class BigInteger Class
// The BigInteger constant one static BigInteger ONE
// returns a BigInteger whose value is (this + val) BigInteger add(BigInteger val)
// returns a BigInteger value which is the // inverse of this (mod n)
BigInteger modInverse(BigInteger n)
// returns a BigInteger value this^exp (mod n) BigInteger modPow(BigInteger exp, BigInteger n)
// returns a BigInteger whose value is (this * val) BigInteger multiply(BigInteger val)
// returns a BigInteger whose value is (this - val) BigInteger subtract(BigInteger val)
Program 28.1 Program 28.1
import java.math.BigInteger;
public class Program28_1 {
public static void main(String[] args) {
// define Z(n)
BigInteger p, q, n;
// used for keys BigInteger t, e, d;
// sent and received messages String clientMsg, serverMsg;
// BigInteger variables for data encryption
BigInteger strData, encryptedData, decryptedData;
// create BigInteger objects p = 5, q = 11
Program 28.1 (continued) Program 28.1 (continued)
// compute n = p * q n = p.multiply(q);
// use BigInteger operations to // compute t = (p-1)*(q-1)
t = p.subtract(BigInteger.ONE).multiply(
q.subtract(BigInteger.ONE));
// create BigInteger e = 3 which is
// relatively prime to t; that is, (e,t)=1 e = new BigInteger("3");
// modInverse() returns d, the inverse // of e (mod t); that is, e*d = 1 (mod t) d = e.modInverse(t);
Program 28.1 (continued) Program 28.1 (continued)
// convert the single character // string "1" to a BigInteger clientMsg = "1";
strData = new BigInteger(clientMsg.getBytes());
System.out.println("Client message: \"" + clientMsg + "\" Data value: " + strData);
// use modPow() to encrypt strData by // raising it to power e mod n
encryptedData = strData.modPow(e,n);
System.out.println("Encrypted data: " + encryptedData);
// decrypt the encrypted data by raising // it to power d mod n
decryptedData = encryptedData.modPow(d,n);
System.out.println("Decrypted data: " +
Program 28.1 (concluded) Program 28.1 (concluded)
// convert BigInteger back to a string
serverMsg = new String(decryptedData.toByteArray());
System.out.println("Server message: \"" + serverMsg + "\"");
} }
Run:
Client message: "1" Data value: 49 Encrypted data: 4
Decrypted data: 49 Server message: "1"
BigInteger Prime Numbers BigInteger Prime Numbers
The BigInteger class has methods that The BigInteger class has methods that produce random prime numbers of
produce random prime numbers of arbitrary size. The static method arbitrary size. The static method
probablePrime() returns a positive probablePrime() returns a positive
BigInteger object that is probably prime, BigInteger object that is probably prime,
with a specified bit length. The probability with a specified bit length. The probability
that the integer is prime is
that the integer is prime is ≥ ≥ 1-2 1-2
-100.-100.public static BigInteger probablePrime(int bitLength, Random rnd)
BigInteger Prime Numbers BigInteger Prime Numbers
(continued) (continued)
A brute-force method discovers a number A brute-force method discovers a number e such that gcd(e,t) = 1. It uses a
e such that gcd(e,t) = 1. It uses a constructor that creates randomly constructor that creates randomly
generated BigInteger prime numbers, generated BigInteger prime numbers,
which are uniformly distributed over the which are uniformly distributed over the
range 0 to 2
range 0 to 2
numBits 1numBits 1‑‑, inclusive. , inclusive.
BigInteger Prime Numbers BigInteger Prime Numbers
(concluded) (concluded)
// returns a number relatively prime to t
public static BigInteger randomModValue(BigInteger t) {
BigInteger k = null;
Random rnd = new Random();
// generate a sequence of random numbers and exit // the loop when the number is relatively prime to t do
{
// random number k is in the range 0 to 264 -1 k = new BigInteger(64,rnd);
} while(!t.gcd(k).equals(BigInteger.ONE));
return k;
RSA Client and Server RSA Client and Server
In the software supplement, you can find In the software supplement, you can find RSAServer and RSAClient programs that RSAServer and RSAClient programs that
use encryption for client/server message use encryption for client/server message
handling.
handling.
RSA Client and Server RSA Client and Server
(continued)
(continued)
Implementing Euclid's GCD Implementing Euclid's GCD
Algorithms Algorithms
// compute the greatest common divisor // of the nonnegative integers a and b // where both a and b cannot be 0
int gcd(int a, int b) {
if (b == 0)
return a; // a divides a and 0 else
return gcd(b, a%b); // recursive step }
Implementing Euclid's GCD Implementing Euclid's GCD
Algorithms (continued) Algorithms (continued)
// variables defined outside the scope of extGCD() static int i;
static int j;
...
// computes values i, j such that // gcd(a,b) = a * i + b * j
public static void extGCD(int a, int b) {
int x, y;
// stopping condition corresponds to // gcd(a,0) = a; the linear combination // is gcd(a,b) = a * 1 + b * 0; assign // i = 1 and j = 0
if (b == 0) {
Implementing Euclid's GCD Implementing Euclid's GCD
Algorithms (continued) Algorithms (continued)
else {
extGCD(b,a%b);
// gcd(b, a%b) = b*i + (a%b)*j;
// recompute i and j so gcd(a, b) = a*i + b*j // save i and j in x and y
x = i;
y = j;
// update i and j in terms of x, y, and a/b i = y;
j = x - (a/b) * y;
}
