# RSA Encryption. Tom Davis October 10, 2003

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1 RSA Encryption Tom Davis October 10, Public Key Cryptography One of the biggest problems in cryptography is the distribution of keys. Suppose you live in the United States and want to pass information secretly to your friend in Europe. If you truly want to keep the information secret, you need to agree on some sort of key that you and he can use to encode/decode messages. But you don t want to keep using the same key, or you will make it easier and easier for others to crack your cipher. But it s also a pain to get keys to your friend. If you mail them, they might be stolen. If you send them cryptographically, and someone has broken your code, that person will also have the next key. If you have to go to Europe regularly to hand-deliver the next key, that is also expensive. If you hire some courier to deliver the new key, you have to trust the courier, et cetera. 1.1 Trap-Door Ciphers But imagine the following situation. Suppose you have a special method of encoding and decoding that is one way in a sense. Imagine that the encoding is easy to do, but decoding is very difficult. Then anyone in the world can encode a message, but only one person can decode it. Such methods exist, and they are called one way ciphers or trap door ciphers. Here s how they work. For each cipher, there is a key for encoding and a different key for decoding. If you know the key for decoding, it is very easy to make the key for encoding, but it is almost impossible to do the opposite to start with the encoding key and work out the decoding key. So to communicate with your friend in Europe, each of you has a trap door cipher. You make up a decoding key D a and generate the corresponding encoding key E a. Your friend does exactly the same thing, but he makes up a decoding key D b and generates the corresponding encoding key E b. You tell him E a (but not D a ) and he tells you E b (but not D b ). Then you can send him messages by encoding using E b (which only he can decode) and vice-versa he encodes messages to you using E a (which only you can decode, since you re the only person with access to D a ). Now if you want to change to a new key, it is no big problem. Just make up new pairs and exchange the encoding keys. If the encoding keys are stolen, it s not a big deal. The person who steals them can only encode messages they can t decode them. In fact, the encoding keys (sometimes called public keys ) could just be published in 1

3 plying together two integers. Can you tell me what they are? This is a very difficult problem. A computer can factor that number fairly quickly, but (although there are some tricks) it basically does it by trying most of the possible combinations. For any size number, the computer has to check something that is of the order of the size of the square-root of the number to be factored. In this case, that square-root is roughly Now it doesn t take a computer long to try out possibilities, but what if the number to be factored is not ten digits, but rather 400 digits? The square-root of a number with 400 digits is a number with 200 digits. The lifetime of the universe is approximately seconds an 18 digit number. Assuming a computer could test one million factorizations per second, in the lifetime of the universe it could check possibilities. But for a 400 digit product, there are possibilties. This means the computer would have to run for times the life of the universe to factor the large number. It is, however, not too hard to check to see if a number is prime in other words to check to see that it cannot be factored. If it is not prime, it is difficult to factor, but if it is prime, it is not hard to show it is prime. So RSA encryption works like this. I will find two huge prime numbers, p and q that have 100 or maybe 200 digits each. I will keep those two numbers secret (they are my private key), and I will multiply them together to make a number N = pq. That number N is basically my public key. It is relatively easy for me to get N; I just need to multiply my two numbers. But if you know N, it is basically impossible for you to find p and q. To get them, you need to factor N, which seems to be an incredibly difficult problem. But exactly how is N used to encode a message, and how are p and q used to decode it? Below is presented a complete example, but I will use tiny prime numbers so it is easy to follow the arithmetic. In a real RSA encryption system, keep in mind that the prime numbers are huge. In the following example, suppose that person A wants to make a public key, and that person B wants to use that key to send A a message. In this example, we will suppose that the message A sends to B is just a number. We assume that A and B have agreed on a method to encode text as numbers. Here are the steps: 1. Person A selects two prime numbers. We will use p = 23 and q = 41 for this example, but keep in mind that the real numbers person A should use should be much larger. 2. Person A multiplies p and q together to get pq = (23)(41) = is the public key, which he tells to person B (and to the rest of the world, if he wishes). 3. Person A also chooses another number e which must be relatively prime to (p 1)(q 1). In this case, (p 1)(q 1) = (22)(40) = 880, so e = 7 is fine. e is also part of the public key, so B also is told the value of e. 3

4 4. Now B knows enough to encode a message to A. Suppose, for this example, that the message is the number M = B calculates the value of C = M e (mod N) = 35 7 (mod 943) = and (mod 943) = 545. The number 545 is the encoding that B sends to A. 7. Now A wants to decode 545. To do so, he needs to find a number d such that ed = 1(mod (p 1)(q 1)), or in this case, such that 7d = 1(mod 880). A solution is d = 503, since = 3521 = 4(880) + 1 = 1(mod 880). 8. To find the decoding, A must calculate C d (mod N) = (mod 943). This looks like it will be a horrible calculation, and at first it seems like it is, but notice that 503 = (this is just the binary expansion of 503). So this means that = = But since we only care about the result (mod 943), we can calculate all the partial results in that modulus, and by repeated squaring of 545, we can get all the exponents that are powers of 2. For example, (mod 943) = = (mod 943) = 923. Then square again: (mod 943) = (545 2 ) 2 (mod 943) = = (mod 943) = 400, and so on. We obtain the following table: (mod 943) = (mod 943) = (mod 943) = (mod 943) = (mod 943) = (mod 943) = (mod 943) = (mod 943) = (mod 943) = 324 So the result we want is: (mod 943) = (mod 943) = 35. Using this tedious (but simple for a computer) calculation, A can decode B s message and obtain the original message N = 35. 4

5 2.1 RSA Exercise OK, now to see if you understand the RSA decryption algorithm, suppose you are person A, and you have chosen as your two primes p = 97 and q = 173, and you have chosen e = 5. Thus you told B that N = (which is just pq) and you told him that e = 5. He encodes a message (a number) for you and tells you that the encoding is Can you figure out the original message? The answer appears at the end of this document. 3 How It Works RSA cryptography is based on the following theorems: Theorem 1 (Fermat s Little Theorem) If p is a prime number, and a is an integer such that (a, p) = 1, then a p 1 = 1(mod p). Proof: Consider the numbers (a 1), (a 2),...(a (p 1)), all modulo p. They are all different. If any of them were the same, say a m = a n(mod p), then a (m n) = 0(mod p) so m n must be a multiple of p. But since all m and n are less than p, m = n. Thus a 1, a 2,..., a (p 1) must be a rearrangement of 1, 2,...,(p 1). So modulo p, we have: p 1 p 1 p 1 i = a i = a p 1 i, so a p 1 = 1(mod p). Theorem 2 (Fermat s Theorem Extension) If (a, m) = 1 then a φ(m) = 1(mod m), where φ(m) is the number of integers less than m that are relatively prime to m. The number m is not necessarily prime. Proof: Same idea as above. Suppose φ(m) = n. Then suppose that the n numbers less than m that are relatively prime to m are: a 1, a 2, a 3,..., a n. Then a a 1, a a 2,..., a a n are also relatively prime to m, and must all be different, so they must just be a rearrangement of the a 1,..., a n in some order. Thus: n a i = modulo m, so a n = 1(mod m). n n a a i = a n a i, 5

6 Theorem 3 (Chinese Remainder Theorem) Let p and q be two numbers (not necessarily primes), but such that (p, q) = 1. Then if a = b(mod p) and a = b(mod q) we have a = b(mod pq). Proof: If a = b(mod p) then p divides (a b). Similarly, q divides (a b). But p and q are relatively prime, so pq divides (a b). Consequently, a = b(mod pq). (This is a special case with only two factors of what is usually called the Chinese remainder theorem but it is all we need here.) 3.1 Proof of the Main Result Based on the theorems above, here is why the RSA encryption scheme works. Let p and q be two different (large) prime numbers, let 0 M < pq be a secret message 1, let d be an integer (usually small) that is relatively prime to (p 1)(q 1), and let e be a number such that de = 1(mod (p 1)(q 1)). (We will see later how to generate this e given d.) The encoded message is C = M e (mod pq), so we need to show that the decoded message is given by M = C d (mod pq). Proof: Since de = 1(mod (p 1)(q 1)), de = 1 + k(p 1)(q 1) for some integer k. Thus: C d = M de = M 1+k(p 1)(q 1) = M (M (p 1)(q 1) ) k. If M is relatively prime to p, then M de = M (M p 1 ) k(q 1) = M (1) k(q 1) = M(mod p) (1) By the extension of Fermat s Theorem giving M p 1 = 1(mod p) followed by a multiplication of both sides by M. But if M is not relatively prime to p, then M is a multiple of p, so equation 1 still holds because both sides will be zero, modulo p. By exactly the same reasoning, M de = M M q 1 = M(mod q) (2) If we apply the Chinese remainder theorem to equations 1 and 2, we obtain the result we want: M de = M(mod pq). Finally, given the integer d, we will need to be able to find another integer e such that de = 1(mod (p 1)(q 1)). To do so we can use the extension of Fermat s theorem to get d φ((p 1)(q 1)) = 1(mod (p 1)(q 1)), so d φ((p 1)(q 1)) 1 (mod (p 1)(q 1)) is a suitable value for e. 3.2 Solution to the Exercise The secret message is If the message is long, break it up into a series of smaller messages such that each of them is smaller than pq and encode each of them separately. 6

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