Fermat’s Little Theorem

Applying Fermat’s Little Theorem
Applying Fermat’s Little Theorem

Fermat’s Little Theorem Claim 1: Assume gcd(k,n) = 1. If i·k j·k (mod n), then i j (mod n). Claim 2: Assume gcd(k,n) = 1. If i j (mod n), then i·k j·k (mod n) . In particular, when p is a prime & k not a multiple of p, then gcd(k,p)=1. If i j (mod p), then i·k j·k (mod p) Therefore, k mod p, 2k mod p, …, (p-1)k mod p are all different numbers. Claim1 was shown in earlier slides. So you need not prove claim 1. Claim 2 is just rephrasing Claim 1. So these two claims were shown. For example, when p=7 and k=3, 3 mod 7 = 3, 2·3 mod 7 = 6, 3·3 mod 7 = 2, 4·3 mod 7 = 5, 5·3 mod 7 = 1, 6·3 mod 7 = 4 Notice that in the above example every number from 1 to 6 appears exactly once.

Fermat’s Little Theorem In particular, when p is a prime & k not a multiple of p, then gcd(k,p)=1. If i j (mod p), then i·k j·k (mod p) Therefore, k mod p, 2k mod p, …, (p-1)k mod p are all different numbers. Each of ik mod p cannot be equal to 0, because p is a prime number Let ci = ik mod p. So 1 <= c1 <= p-1, 1 <= c2 <= p-1, …, 1< = cp-1 <= p-1 By the above we know that c1,c2,…,cp-2,cp-1 are all different. So for each i from 1 to p-1, there is exactly one cj such that cj = i. Therefore, we have (k mod p)·(2k mod p)·…·((p-1)k mod p) = c1·c2·…·cp-2·cp-1 = 1·2·3…·(p-2)·(p-1)

Fermat’s Little Theorem Theorem: If p is prime & k not a multiple of p 1 kp-1 (mod p) For example, when p=5, k=4, we have kp-1 mod p = 44 mod 5 = 1 “Proof” By the previous slide or direct calculation 4·3·2·1 [(4 mod 5) (2·4 mod 5) (3·4 mod 5) (4·4 mod 5)] (mod 5) [4 · (2·4) · (3·4) · (4·4)] (mod 5) [44 · (1·2·3·4)] (mod 5) Since gcd(1·2·3·4, 5)=1, we can cancel 1·2·3·4 on both sides. This implies 1 44 (mod 5)

Fermat’s Little Theorem Theorem: If p is prime & k not a multiple of p 1 kp-1 (mod p) Proof. 1·2···(p-1) (k mod p · 2k mod p·…·(p-1)k mod p) mod p (k·2k ··· (p-1)k) mod p (kp-1)·1·2 ··· (p-1) (mod p) So, by cancelling 1·2 ··· (p-1) on both sides applying Claim 1 (we can cancel them because gcd(1·2 ··· (p-1), p)=1), we have 1 kp-1 (mod p) By 2 slides before By the multiplication rule

Wilson’s Theorem (p-1)! -1 (mod p) Theorem: p is a prime if and only if (p-1)! -1 (mod p) First we consider the easy direction. If p is not a prime, assume p >= 5, (for p=4, 3! 2 (mod 4) ) Then p=qr for some 2 <= q < p and 2 <= r < p. If q ≠ r, then both q and r appear in (p-1)!, and so (p-1)! 0 (mod p). If q = r, then p = q2 > 2q (since we assume p > 5 and thus q > 2). then both q and 2q are in (p-1)!, and so again (p-1)! 0 (mod p).

Wilson’s Theorem (p-1)! -1 (mod p) Theorem: p is a prime if and only if (p-1)! -1 (mod p) To prove the more interesting direction, first we need a lemma. Lemma. If p is a prime number, x2 1 (mod p) if and only if x 1 (mod p) or x -1 (mod p) Proof. x2 1 (mod p) iff p | x2 – 1 iff p | (x – 1)(x + 1) iff p | (x – 1) or p | (x+1) iff x 1 (mod p) or x -1 (mod p) Lemma: p prime and p|a·b iff p|a or p|b.

Wilson’s Theorem (p-1)! -1 (mod p) Theorem: p is a prime if and only if (p-1)! -1 (mod p) Let’s get the proof idea by considering a concrete example. 10! 1·2·3·4·5·6·7·8·9·10 mod 11 1·10·(2·6)·(3·4)·(5·9)·(7·8) mod 11 1·-1·(1)·(1)·(1)·(1) mod 11 -1 mod 11 Besides 1 and 10, the remaining numbers are paired up into multiplicative inverse!

Wilson’s Theorem (p-1)! -1 (mod p) Theorem: p is a prime if and only if (p-1)! -1 (mod p) Proof. Since p is a prime, every number from 1 to p-1 has a multiplicative inverse. By the Lemma, every number 2 <= k <= p-2 has an inverse k’ with k≠k’. Since p is odd, the numbers from 2 to p-2 can be grouped into pairs (a1,b1),(a2,b2),…,(a(p-3)/2,b(p-3)/2) so that aibi 1 (mod p) Therefore, (p-1)! 1·(p-1)·2·3·····(p-3)·(p-2) (mod p) 1·(p-1)·(a1b1)·(a2b2)·····(a(p-3)/2b(p-3)/2) (mod p) 1·(-1)·(1)·(1)·····(1) (mod p) -1 (mod p)

Chinese Remainder Theorem Picture from http://img5.epochtimes.com/i6/801180520191974.jpg ……………………… ……………………… Dec 29

This Lecture In this lecture we will study the Chinese remainder theorem, which is a method to solve equations about remainders. One equation Ancient application Two equations and three equations Chinese Remainder theorem

Case Study ax b (mod n) How to solve the following equation? For example, consider the equation 2x 3 (mod 7) Suppose there is a solution x to the equation, then there is a solution x in the range from 0 to 6, because we can replace x by x mod 7. Then we can see that x=5 is a solution. Also, 5+7, 5+2·7, 5+3·7, …, 5-7, 5-2·7…, are solutions. Therefore the solutions are of the form 5+7k for some integer k.

One Equation ax b (mod n) How to solve the following equation? 2x 3 (mod 7) 5x 6 (mod 9) 4x -1 (mod 5) 4x 2 (mod 6) 10x 2 (mod 7) 3x 1 (mod 6) x = 5 + 7k for any integer k x = 3 + 9k for any integer k x = 1 + 5k for any integer k x = 2 + 3k for any integer k x = 3 + 7k for any integer k no solutions

One Equation: Relatively Prime ax b (mod n) Case 1: a and n are relatively prime Without loss of generality, we can assume that 0 < a < n. Because we can replace a by a mod n without changing the equation. e.g. 103x 6 (mod 9) is equivalent to 4x 6 (mod 9). Since a and n are relatively prime, there exists a multiplicative inverse a’ for a. Hence we can multiply a’ on both sides of the equation to obtain x a’b (mod n) Therefore, a solution always exists when a and n are relatively prime.

One Equation: Common Factor ax b (mod n) Case 2: a and n have a common factor c>=2. Case 2a: c divides b. ax b (mod n) ax = b + nk for some integer k a1cx = b1c + n1ck a1x = b1 + n1k a1x b1 (mod n1) we assume c|a and c|n and also c|b. In Case (2a) we can simplify the equation, and solve the new equation instead.

One Equation: Common Factor ax b (mod n) Case 2: a and n have a common factor c>=2. Case 2b: c does not divides b. ax b (mod n) ax = b + nk for some integer k a1cx = b + n1ck a1x = b/c + n1k This is a contradiction, since a1x and n1k are integers, but b/c is not an integer since c does not divide b. Therefore, in Case 2b, there is no solution.

One Equation ax b (mod n) Theorem. Let a, b, n be given integers. The above equation has a solution if and only if gcd(a,n) | b. Furthermore, if the condition gcd(a,n) | b is satisfied, then the solutions are of the form y mod (n/gcd(a,n)) for some integer y. Proof. First, divide b by gcd(a,n). If not divisible, then there is no solution by Case (2b). If divisible, then we simplify the solution as in Case (2a). Then we proceed as in Case (1) to compute the solution.

One Equation: Exercise 87x 3 (mod 15) Replace 87 by 87 mod 15 12x 3 (mod 15) Divide both sides by gcd(12,15) = 3 4x 1 (mod 5) Compute the multiplicative inverse of 4 modulo 5 x 4 + 5k 114x 5 (mod 22) Replace 114 by 114 mod 22 4x 5 (mod 22) Divide both sides by gcd(4,22) = 2 no solutions Because 2 does not divide 5. Important: to be familiar with the extended Euclidean algorithm to compute gcd and to compute multiplicative inverse.

Computing Multiplicative Inverse Example: n = 123, k=37 123 = 3·37 + 12 so 12 = n – 3k 37 = 3·12 + 1 so 1 = 37 – 3·12 = k – 3(n-3k) = 10k – 3n 12 = 12·1 + 0 done, gcd=1 So 1 = 10k-3n. Then we know that 10 is the multiplicative inverse of 37 under modulo 123.

This Lecture One equation Ancient application Two equations and three equations Chinese Remainder theorem

Form groups of 3 soldiers Ancient Application of Number Theory Starting from 1500 soldiers, after a war, about 400-500 soldiers died. Now we want to know how many soldiers are left. Form groups of 3 soldiers 韓信

Form groups of 5 soldiers Ancient Application of Number Theory ……………………… Form groups of 5 soldiers ……………………… There are 2 soliders left.

Form groups of 7 soldiers Ancient Application of Number Theory ……………………… Form groups of 7 soldiers There are 3 soliders left.

Ancient Application of Number Theory ……………………… We have 1073 soliders. There are 2 soliders left. How could he figure it out?!

+ The Mathematical Question x 2 (mod 3) x 3 (mod 5) x 2 (mod 7) How to solve this system of modular equations?

This Lecture One equation Ancient application Two equations and three equations Chinese Remainder theorem

Two Equations c1 x d1 (mod m1) c2 x d2 (mod m2) x a1 (mod n1) Find a solution to satisfy both equations simultaneously. c1 x d1 (mod m1) c2 x d2 (mod m2) First we can solve each equation to reduce to the following form. (Of course, if one equation has no solution, then there is no solution.) x a1 (mod n1) x a2 (mod n2) There may be no solutions simultaneously satisfying both equations. For example, consider x 1 (mod 3), x 2 (mod 3). x 1 (mod 6), x 2 (mod 4).

Two Equations x 2 (mod 3) x 4 (mod 7) Case 1: n1 and n2 are relatively prime. x 2 (mod 3) x 4 (mod 7) x = 2+3u and x = 4+7v for some integers u and v. 2+3u = 4+7v => 3u = 2+7v => 3u 2 (mod 7) 5 is the multiplicative inverse for 3 under modulo 7 Multiply 5 on both sides gives: 5·3u 5·2 (mod 7) => u 3 (mod 7) => u = 3 + 7w Therefore, x = 2+3u = 2+3(3+7w) = 11+21w So any x 11 (mod 21) is the solution. Where did we use the assumption that n1 and n2 are relatively prime?

Two Equations x 2 (mod 3) x 4 (mod 7) Assume n1 and n2 are relatively prime. x 2 (mod 3) x 4 (mod 7) The original idea is to construct such an x directly. Let x = 3·a + 7·b Note that when x is divided by 3, the remainder is decided by the second term. And when x is divided by 7, the remainder is decided by the first term. More precisely, x mod 7 = (3·a + 7·b) mod 7 = 3a mod 7 Similarly, x mod 3 = (3·a + 7·b) mod 3 = 7b mod 3 Therefore, to satisfy the equations, we just need to find a such that 3a mod 7 = 4 and b such that 7b mod 3 = 2.

Two Equations x 2 (mod 3) x 4 (mod 7) Assume n1 and n2 are relatively prime. x 2 (mod 3) x 4 (mod 7) The original idea is to construct such an x directly. Let x = 3·a + 7·b Therefore, to satisfy the equations, we just need to find a such that 3a mod 7 = 4 and b such that 7b mod 3 = 2. Since 3 and 7 are relatively prime, both the equations can be solved. The first equation is 3a 4 (mod 7), and the answer is a=6. Similarly, the second equation is 7b 2 (mod 3), and the answer is b=2. So one answer is x = 3a+7b = 3(6)+7(2) = 32.

Two Equations x 2 (mod 3) x 4 (mod 7) Assume n1 and n2 are relatively prime. x 2 (mod 3) x 4 (mod 7) The original idea is to construct such an x directly. Let x = 3·a + 7·b So one answer is x = 3a+7b = 3(6)+7(2) = 32. Are there other solutions? Note that 32 + 3·7·k is also a solution to satisfy both equations. Are there other solutions? The only solutions are of the form 32 + 21k for some integer k.

Three Equations x 2 (mod 3) x 3 (mod 5) x 2 (mod 7) Let x = 5·7·a + 3·7·b + 3·5·c So the first (second, third) term is responsible for the first (second, third) equation. More precisely, x mod 3 = (5·7·a + 3·7·b + 3·5·c) mod 3 = 5·7·a mod 3 x mod 5 = (5·7·a + 3·7·b + 3·5·c) mod 5 = 3·7·b mod 5 x mod 7 = (5·7·a + 3·7·b + 3·5·c) mod 7 = 3·5·c mod 7 Therefore, to satisfy the equations, we want to find a,b,c to satisfy the following: x mod 3 = 35a mod 3 = 2 x mod 5 = 21b mod 5 = 3 x mod 7 = 15c mod 7 = 2

Three Equations x 2 (mod 3) x 3 (mod 5) x 2 (mod 7) Let x = 5·7·a + 3·7·b + 3·5·c So the first (second, third) term is responsible for the first (second, third) equation. Now we just need to solve the following three equations separately. 35a 2 (mod 3), 21b 3 (mod 5), 15c 2 (mod 7). This is equal to 2a 2 (mod 3), b 3 (mod 5), c 2 (mod 7) Therefore, we can set a = 1, b = 3, c = 2. Then x = 35a+21b+15c = 35(1)+21(3)+15(2) = 128. Note that 128+3·5·7·k = 128+105k is also a solution. Since 韓信 knows that 1000 <= x <= 1100, he concludes that x = 1073. Wait, but how does he know that there is no other solution?

This Lecture One equation Ancient application Two equations and three equations Chinese Remainder theorem

Chinese Remainder Theorem Theorem: If n1,n2,…,nk are relatively prime and a1,a2,…,ak are integers, then x a1 (mod n1) x a2 (mod n2) x ak (mod nk) have a simultaneous solution x that is unique modulo n, where n = n1n2…nk. We will give a proof when k=3, but it can be extended easily to any k.

Proof of Chinese Remainder Theorem Let N1 = n2 n3 N2 = n1 n3 N3 = n1 n2 Since Ni and ni are relatively prime, this implies that there exist x1 x2 x3 N1x1 1 (mod n1) N2x2 1 (mod n2) N3x3 1 (mod n3) So, a1N1x1 a1 (mod n1), a2N2x2 a2 (mod n2), a3N3x3 a3 (mod n3) Let x = a1N1x1 + a2N2x2 + a3N3x3 Since n1|N2 and n1|N3, x a1 N1x1 (mod n1) Since N1x1 1 (mod n1), x a1 (mod n1) Similarly, x a2 (mod n2) x a3 (mod n3)

Uniqueness x a1 (mod n1) x a2 (mod n2) x ak (mod nk) Suppose there are two solutions, x and y, to the above system. Then x – y 0 (mod ni) for any i. This means that ni | x – y for any i. Since n1,n2,…,nk are relatively prime, this means that n1n2…nk | x – y. (why?) Therefore, x = y (mod n1n2…nk). So there is a unique solution in every n1n2…nk numbers.

General Systems x 3 (mod 10) x 8 (mod 15) x 5 (mod 84) What if n1,n2,…,nk are not relatively prime? x 3 (mod 2) 1 (mod 2) (a) x 3 (mod 10) x 8 (mod 15) x 5 (mod 84) x 3 (mod 5) (b) x 8 (mod 3) 2 (mod 3) (c) x 8 (mod 5) 3 (mod 5) (d) (e) x 5 (mod 4) 1 (mod 4) x 5 (mod 3) 2 (mod 3) (f) x 5 (mod 7) (g) So we reduce the problem to the relatively prime case. The answer is 173 (mod 420). (b) and (d) are the same. (c) and (f) are the same. (e) is stronger than (a).

Quick Summary First we talk about how to solve one equation ax b (mod n). The equation has solutions if and only if gcd(a,n) divides b. Then we talk about how to find simultaneous solutions to two equations a1x b1 (mod n1) and a2x b2 (mod n2). First use the technique in one equation to reduce to the form x c1 (mod n1) and x c2 (mod n2). By setting x = k1n1 + k2n2, then we just need to find k1 and k2 so that c2 k1 n1 (mod n2) and c1 k2 n2 (mod n1). These equations can be solved separately by using techniques for one equation. The same techniques apply for more than two equations. And the solution is unique mod n1 n2…nk if there are k equations. Finally, when n1 n2…nk are not relatively prime, we show how to reduce it back to the relatively prime case.

A Faster Method x 3 (mod 10) x 8 (mod 15) x 5 (mod 84) There is another method to solve the system of modular equations. x 3 (mod 10) x 8 (mod 15) x 5 (mod 84) From the third equation we know that x = 5+84u. Plug it into the second equation gives 5+84u 8 (mod 15) => 84u 3 (mod 15). Solving this would give us u 2 (mod 5) => u = 2 + 5v. Therefore x = 5+84u = 5+84(2+5v) = 173 + 420v. Plug it into the first equation gives 173+420v 3 (mod 10) => 420v = -170 (mod 10). This equation is always true. So we can conclude that x = 173+420v, or equivalently x 173 (mod 420). This method can also be used to prove the Chinese Remainder Theorem. It is much faster (no need to find factorization), requiring only k-1 Euclidean algorithm.

Cryptography Dec 29

This Lecture In this last lecture for number theory, we will see probably the most important application of number theory in computer science – the design of cryptosystem. Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem

Cryptography Cryptography is the study of methods for sending and receiving secret messages. Alice message Bob adversary Goal: Even though an adversary can listen to your conversation, the adversary can not learn what the message was.

Cryptography Goal: Even though an adversary can listen to your conversation, the adversary can not learn what the message was. Alice f(message) Bob encrypt the message decrypt the message adversary message -> f(message) f(message) -> message But the adversary has no clue how to obtain message from f(message) A difficult goal!

Key Goal: Even though an adversary can listen to your conversation, the adversary can not learn what the message was. Alice f(message, key) Bob encrypt the message using the key decrypt the message using the key adversary message -> f(message,key) f(message,key) -> message But the adversary can not decrypt f(message,key) without the key Use number theory!

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem

Turing’s Code (Version 1.0) The first step is to translate a message into a number “v i c t o r y” -> 22 09 03 20 15 18 25 Beforehand The sender and receiver agree on a secret key, which is a large number k. Encryption The sender encrypts the message m by computing: m* = m · k Decryption The receiver decrypts m by computing: m*/k = m · k/k = m

Turing’s Code (Version 1.0) Alice mk Bob m = message k = key encrypted message = mk mk = received message k = key decrypted message = mk/k=m adversary Why the adversary cannot figure out m? The adversary doesn’t have the key k, and so can only factor mk to figure out m, but factoring is a difficult task to do.

Turing’s Code (Version 1.0) Alice mk Bob m = message k = key encrypted message = mk mk = received message k = key decrypted message = mk/k=m adversary So why don’t we use this Turing’s code today? Major flaw: if you use the same key to send two messages m and m’, then from mk and m’k, we can use gcd(mk,m’k) to figure out k, and then decrypt every message.

Turing’s Code (Version 2.0) Beforehand The sender and receiver agree on a large prime p, which may be made public. (This will be the modulus for all our arithmetic.) They also agree on a secret key k in {1, 2, . . . , p − 1}. Encryption The message m can be any integer in the set {0, 1, 2, . . . , p − 1}. The sender encrypts the message m to produce m* by computing: m* = mk mod p Decryption Let k’ be the multiplicative inverse of k under modulo p. m* mk (mod p) m*k’ m (mod p) m*k’ = m

Turing’s Code (Version 2.0) Public information p Alice m* = mk mod p Bob m = message k = key encrypted message = mk mod p m* = received message k = key decrypted message = m*k’ =m adversary Why the adversary cannot figure out m? Many m and k can produce m* as output, just impossible to determine m without k.

Turing’s Code (Version 2.0) Public information p Alice m* = mk mod p Bob m = message k = key encrypted message = mk mod p m* = received message k = key decrypted message = m*k’ =m adversary So why don’t we use this Turing’s code today? If the adversary somehow knows m, then first compute m’ := multiplicative inverse of m m* mk (mod p) m*m’ k (mod p) So the adversary can figure out k. plain-text attack

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem

Private Key Cryptosystem Alice f(message, key) Bob encrypt the message using the key decrypt the message using the key adversary message -> f(message,key) f(message,key) -> message But the adversary can not decrypt f(message,key) without the key Two parties have to agree on a secret key, which may be difficult in practice. If we buy books from Amazon, we don’t need to exchange a secret code. Why is it secure?

Public Key Cryptosystem Public information: Key for Alice Public information: Key for Bob Alice f(message, Bob’s key) Bob encrypt the message using Bob’s key decrypt the message adversary message -> f(message,Bob’s key) f(message,Bob’s key) -> message But the adversary can not decrypt f(message, Bob’s key)! Only Bob can decrypt the message sent to him! There is no need to have a secret key between Alice and Bob. How is it possible???

RSA Cryptosystem RSA are the initials of three Computer Scientists, Ron Rivest, Adi Shamir and Len Adleman, who discovered their algorithm when they were working together at MIT in 1977.

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem Key generation, encryption, decryption Correctness Secure? Computational issues

Generating Public Key public key: e and n Alice Bob secret key: d How Bob creates his public keys? Secret key only known to Bob. Choose 2 large prime numbers p and q. Set n = pq and T = (p-1)(q-1) Choose e ≠1 so that gcd(e,T)=1 Calculate d so that de = 1 (mod T) Publish e and n as public keys Keep d as secret key > 150 digits

Encrypting Message Send y = xe mod n message x public key: e and n Alice Bob secret key: d How Alice sends a message to Bob? Look at Bob’s homepage for e and n. Send y = xe mod n Alice does not need to know Bob’s secret key to send the message.

Decrypting Message Send y = xe mod n message x public key: e and n Alice Bob secret key: d How Bob recover Alice’s message? Receive y = xe mod n Compute z = yd mod n Bob uses z is the original message that Alice sent.

RSA Cryptosystem Send y = xe mod n message x public key: e and n Alice Bob Encrypting message secret key: d Key generation Choose 2 large prime numbers p and q. Set n = pq and T = (p-1)(q-1) Choose e ≠1 so that gcd(e,T)=1 Calculate d so that de = 1 (mod T) Publish e and n as public keys Keep d as secret key Decrypting message Compute z = yd mod n

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem Key generation, encryption, decryption Correctness Secure? Computational issues

RSA Cryptosystem Send y = xe mod n message x public key: e and n Alice Bob secret key: d For the RSA cryptosytem to work, we need to show: 1) z = x 2) Without the secret key d, we can not compute the original message before the sun burns out. Compute z = yd mod n with additional assumptions…

Correctness Send y = xe mod n message x public key: e and n Alice Bob secret key: d 1) z = x Compute z = yd mod n Note that z = yd mod n = xed mod n. Therefore we need to prove x = xed mod n. p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) (a) x mod p = xed mod p (b) x mod q = xed mod q (c) x mod n = xed mod n Therefore, if Alice sends x < n, then Bob can recover correctly.

Correctness Send y = xe mod n = x·(xk(q-1))(p-1) mod p message x public key: e and n Send y = xe mod n Alice Bob secret key: d 1) z = x (a) x mod p = xed mod p Compute z = yd mod n = 1 + k(p-1)(q-1) Note that de = 1 + kT p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) Hence, xed mod p = x1+k(p-1)(q-1) mod p = x·xk(p-1)(q-1) mod p = x·(xk(q-1))(p-1) mod p

Correctness Send y = xe mod n = x·(xk(q-1))(p-1) mod p message x public key: e and n Send y = xe mod n Alice Bob secret key: d 1) z = x (a) x mod p = xed mod p Compute z = yd mod n Fermat’s little theorem: If p | a, then ap-1 1 mod p p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) Hence, xed mod p = x1+k(p-1)(q-1) mod p = x·xk(p-1)(q-1) mod p = x·(xk(q-1))(p-1) mod p = x mod p a

Correctness Send y = xe mod n = x·(xk(q-1))(p-1) mod p message x public key: e and n Send y = xe mod n Alice Bob secret key: d 1) z = x (a) x mod p = xed mod p Compute z = yd mod n This means p | xk(q-1), implying p | x, since p is prime p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) Hence, xed mod p = x1+k(p-1)(q-1) mod p = x·xk(p-1)(q-1) mod p = x·(xk(q-1))(p-1) mod p What if p | a? a Since p | x, we have xed mod p = x mod p = 0.

Correctness Send y = xe mod n message x public key: e and n Alice Bob secret key: d 1) z = x Compute z = yd mod n Note that z = yd mod n = xed mod n. Therefore we need to prove x = xed mod n. p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) (a) x mod p = xed mod p (b) x mod q = xed mod q (c) x mod n = xed mod n The same proof. (c) can be proved directly, also follows from Chinese Remainder theorem.

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem Key generation, encryption, decryption Correctness Secure? Computational issues

Why is this Secure? Send y = xe mod n message x public key: e and n Alice Bob adversary secret key: d 2) Without the secret key d, we can not compute the original message before the sun burns out. Compute z = yd mod n p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) Method 1: From y=xe mod n, don’t know how to compute x. Thus not possible to work backward. It is an example of an “one-way” function.

Why is this Secure? Send y = xe mod n message x public key: e and n Alice Bob adversary secret key: d 2) Without the secret key d, we can not compute the original message before the sun burns out. Compute z = yd mod n p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) Method 2: Factor n = pq. Compute secrete key d. Then decrypt everything! No one knows an efficient way to do factoring. The security is based on assumptions that some computational problems are hard.

This Lecture Introduction to cryptograph “Turing code” Public key cryptography RSA cryptosystem Key generation, encryption, decryption Correctness Secure? Computational issues

RSA Example Send y = xe mod n message x public key: e and n Alice Bob secret key: d Then Alice sends the encrypted message. Compute z = yd mod n x=33 y = 3323 mod 55 p=5 q=11 n = 55 T = 40 e = 7 d = 23 p, q prime n = pq T = (p-1)(q-1) e s.t. gcd(e,T)=1 de = 1 (mod T) y = 84298649517881922539738734663399137 mod 55 How to compute it efficiently? First Bob generated his keys.

Exponentiation To compute exponentiation mod n 1444 mod 713 This is much more efficient. This still takes too long when the exponent is large.

Repeated Squaring 1442 mod 713 = 59 1444 mod 713 = 629 1448 mod 713 = 1444·1444 mod 713 = 629·629 mod 713 = 639 14416 mod 713 = 1448·1448 mod 713 = 639·639 mod 713 = 485 14432 mod 713 = 14416·14416 mod 713 = 485·485 mod 713 = 648 Note that 50 = 32 + 16 + 2 14450 mod 713 = 14432 14416 1442 mod 713 = 648·485·59 mod 713 = 242

Generating Public Key Choose 2 large prime numbers p and q. Set n = pq and T = (p-1)(q-1) Choose e ≠1 so that gcd(e,T)=1 Calculate d so that de = 1 (mod T) Publish e and n as public keys Keep d as secret key How to choose large prime numbers efficiently? Given a large number, how to check whether it is prime efficiently?

Primality Testing Given a large integer n, determine quickly whether n is prime First test: for i = 1,…,√n, check if i divides n. We are talking about n with 150 digits. This simply takes too long (2150 steps, sun will burn out). We are looking for an exponential improvement (instead of n, we can only afford roughly log(n) steps), like we did in the extended GCD algorithm. Need some number theory!

Primality Testing (n-1)! -1 (mod n) Theorem: n is a prime if and only if (n-1)! -1 (mod n) It doesn’t seem to help, since we don’t know how to compute (n-1)! mod n quickly (in roughly log(n) steps).

Primality Testing 1 an-1 (mod n) Theorem: If n is prime & a not a multiple of n 1 an-1 (mod n) Contrapositive. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Example. Show that 1763 is composite (not a prime number). Let a=2, n=1763. 21762 (mod 1763) = 142 ≠ 1 Therefore, it is composite by (the contrapositive of) Fermat’s little theorem.

Primality Testing Contrapositive. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Example. Show that 1387 is composite (not a prime number). Let a=2, n=1387. can not tell whether n is prime or not. 21386 (mod 1387) = 1 Try a=3 31386 (mod 1387) = 1238 ≠ 1 this shows n is composite.

Primality Testing Contrapositive. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. “Fermat” test Given n, choose a < n Compute an-1 (mod n) If an-1 (mod n) ≠ 1 conclude that n is a composite number If an-1 (mod n) = 1 try another a Each test takes about log(n) steps. It depends on how many a that we need to try…

Primality Testing Contrapositive. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. “Fermat” test Given n, choose a < n Compute an-1 (mod n) If an-1 (mod n) ≠ 1 conclude that n is a composite number If an-1 (mod n) = 1 try another a Unfortunately, there exists n which is composite, but an-1 (mod n) = 1 for every a! These are called Carmichael numbers (e.g. 561, 1105, 1729, etc…)

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Lemma. If n is a prime number, x2 1 (mod n) if and only if x 1 (mod n) or x -1 (mod n) Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Example Note that it is (2693)2 For n=1387 and a=2, Fermat’s test fails because 21386 1 (mod 1387). However 2693 512 (mod 1387) 1 (mod 1387) By contrapositive 2, we can conclude that 1387 is a composite number.

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Strong primality test Let n-1 = 2kd Pick an a. Compute a2kd (mod n), a2k-1d (mod n), a2k-2d (mod n),…, ad (mod n) ≠1 Composite by contrapositive 1.

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Strong primality test Let n-1 = 2kd Pick an a. Compute a2kd (mod n), a2k-1d (mod n), a2k-2d (mod n),…, ad (mod n) =1 ≠1 & ≠-1 Composite by contrapositive 2.

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Strong primality test Let n-1 = 2kd Pick an a. Compute a2kd (mod n), a2k-1d (mod n), a2k-2d (mod n),…, ad (mod n) =1 =1 Continue to go backward and check

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Strong primality test Let n-1 = 2kd Pick an a. Compute a2kd (mod n), a2k-1d (mod n), a2k-2d (mod n),…, ad (mod n) =1 =1 =-1 End the test and say it is a “probable” prime.

Primality Testing Contrapositive 1. If 1 an-1 (mod n) and a is not a multiple of n, then n is not a prime number. Contrapositive 2. If x2 1 (mod n) but x 1 (mod n) and x -1 (mod n) then n is a composite number. Strong primality test Let n-1 = 2kd Pick an a. Compute a2kd (mod n), a2k-1d (mod n), a2k-2d (mod n),…, ad (mod n) =1 =1 =1 =1 =1 End the test and say it is a “probable” prime.

Primality Testing Strong primality test Given n, pick an a Let n’ = n-1 (so n’ is an even number) If an’ (mod n) ≠ 1, then stop and say “n is composite”. n’ := n’/2; While n’ is an integer do If an’ (mod n) = -1, then stop and say “n is a probable prime”. Stop and say “n is a probable prime”.

Primality Testing For a particular a, the strong primality test takes “about” log(n) steps. But again, there exists n which is composite but pass the test… Theorem: if n is composite, for more than half of a < n, the strong primality test will say n is composite! So, given a composite n, if we pick a random a, the strong primality test will be incorrect with probability <= 1/2. Thus, if we repeat the procedure for 10000 times, then the probability that the strong primality test is still incorrect is very small (e.g. much smaller than our computer will suddenly crash). This is the most efficient method used in practice!

Generating Public Key Choose 2 large prime numbers p and q. Set n = pq and T = (p-1)(q-1) Choose e ≠1 so that gcd(e,T)=1 Calculate d so that de = 1 (mod T) Publish e and n as public keys Keep d as secret key Similar idea Prime number theorem: From 1 to n, there are roughly n/log(n) prime numbers. How to choose large prime numbers efficiently? Pick a random large number, do the (randomized) strong primality tests, until we find a prime!

Remarks We have derived everything from basic principle. RSA cryptosystem is one of the most important achievements in compute science. (The researchers won the Turing award for their contribution.) Number theory is also very useful in coding theory (e.g. compression). Mathematics is very important in computer science.

More Remarks Theorem: if n is composite, for more than half of a < n, the strong primality test will say n is composite! The proof uses Chinese Remainder theorem and some elementary number theory. (Introduction to Algorithms, MIT press) Conjecture: It is enough to try a to up to roughly log(n). Theroem (Primes is in P, 2004) There is an efficient and deterministic primality test. Major Open Problem: Is there an efficient algorithm to compute the prime factorization?

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