TheEuler ϕ-functionis the mapϕ:N→Ndefined byϕ(n) = 1forn= 1, and, forn >1, ϕ(n) is the number of positive integersm with1≤m < nand gcd(m, n) = 1.
From Proposition3.4, we know that the order ofU(n), the group of units inZn, isϕ(n).
For example, |U(12)|=ϕ(12) = 4 since the numbers that are relatively prime to 12 are 1, 5, 7, and 11. For any primep,ϕ(p) =p−1. We state these results in the following theorem. Theorem 6.17. Let U(n) be the group of units in Zn. Then |U(n)|=ϕ(n).
6.4. EXERCISES 73 Theorem 6.18 Euler’s Theorem. Letaandnbe integers such thatn >0andgcd(a, n) = 1. Then aϕ(n)≡1 (modn).
Proof. By Theorem 6.17 the order of U(n) is ϕ(n). Consequently, aϕ(n) = 1 for all a∈U(n); oraϕ(n)−1is divisible by n. Therefore,aϕ(n)≡1 (mod n).
If we consider the special case of Euler’s Theorem in which n= p is prime and recall thatϕ(p) =p−1, we obtain the following result, due to Pierre de Fermat.
Theorem 6.19 Fermat’s Little Theorem. Letp be any prime number and suppose that p∤a(p does not divide a). Then
ap−1≡1 (modp). Furthermore, for any integer b, bp ≡b (modp).
Sage Sage can create all the subgroups of a group, so long as the group is not too large. It can also create the cosets of a subgroup.
Historical Note
Joseph-Louis Lagrange (1736–1813), born in Turin, Italy, was of French and Italian descent. His talent for mathematics became apparent at an early age. Leonhard Euler recognized Lagrange’s abilities when Lagrange, who was only 19, communicated to Euler some work that he had done in the calculus of variations. That year he was also named a professor at the Royal Artillery School in Turin. At the age of 23 he joined the Berlin Academy. Frederick the Great had written to Lagrange proclaiming that the “greatest king in Europe” should have the “greatest mathematician in Europe” at his court. For 20 years Lagrange held the position vacated by his mentor, Euler. His works include contributions to number theory, group theory, physics and mechanics, the calculus of variations, the theory of equations, and differential equations. Along with Laplace and Lavoisier, Lagrange was one of the people responsible for designing the metric system. During his life Lagrange profoundly influenced the development of mathematics, leaving much to the next generation of mathematicians in the form of examples and new problems to be solved.
6.4
Exercises
1. Suppose that G is a finite group with an element g of order 5 and an element h of order 7. Why must |G| ≥35?
2. Suppose that G is a finite group with 60 elements. What are the orders of possible subgroups ofG?
3. Prove or disprove: Every subgroup of the integers has finite index. 4. Prove or disprove: Every subgroup of the integers has finite order. 5. List the left and right cosets of the subgroups in each of the following.
(a) ⟨8⟩ inZ24 (b) ⟨3⟩ inU(8) (c) 3Zin Z (d) A4 inS4 (e) An inSn (f) D4 in S4 (g) T inC∗ (h) H ={(1),(123),(132)} inS4
6. Describe the left cosets ofSL2(R)inGL2(R). What is the index ofSL2(R)inGL2(R)?
7. Verify Euler’s Theorem forn= 15and a= 4.
8. Use Fermat’s Little Theorem to show that ifp= 4n+ 3is prime, there is no solution to the equationx2 ≡ −1 (modp).
9. Show that the integers have infinite index in the additive group of rational numbers. 10. Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.
11. Let H be a subgroup of a group G and suppose that g1, g2 ∈ G. Prove that the
following conditions are equivalent. (a) g1H=g2H
(b) Hg−11 =Hg−21 (c) g1H⊂g2H
(d) g2∈g1H
(e) g−11g2 ∈H
12. If ghg−1 ∈ H for all g ∈ G and h ∈H, show that right cosets are identical to left cosets. That is, show that gH =Hg for allg∈G.
13. What fails in the proof of Theorem 6.8ifϕ:LH → RH is defined byϕ(gH) =Hg?
14. Suppose that gn=e. Show that the order ofg dividesn.
15. Show that any two permutationsα, β ∈Snhave the same cycle structure if and only
if there exists a permutationγ such that β =γαγ−1. If β =γαγ−1 for someγ ∈Sn, then
α and β are conjugate.
16. If|G|= 2n, prove that the number of elements of order 2 is odd. Use this result to show thatGmust contain a subgroup of order 2.
17. Suppose that [G:H] = 2. Ifaand b are not inH, show thatab∈H. 18. If[G:H] = 2, prove thatgH =Hg.
19. LetH and K be subgroups of a groupG. Prove that gH∩gK is a coset of H∩K inG.
20. LetH and K be subgroups of a groupG. Define a relation∼on Gbya∼bif there exists anh∈H and a k∈K such that hak=b. Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of H={(1),(123),(132)}inA4.
21. Let Gbe a cyclic group of order n. Show that there are exactly ϕ(n) generators for G. 22. Letn=pe1 1 p e2 2 · · ·p ek
k , where p1, p2, . . . , pk are distinct primes. Prove that
ϕ(n) =n ( 1− 1 p1 ) ( 1− 1 p2 ) · · · ( 1− 1 pk ) . 23. Show that n=∑ d|n ϕ(d)
7
Introduction to Cryptography
Cryptography is the study of sending and receiving secret messages. The aim of cryptogra- phy is to send messages across a channel so that only the intended recipient of the message can read it. In addition, when a message is received, the recipient usually requires some assurance that the message is authentic; that is, that it has not been sent by someone who is trying to deceive the recipient. Modern cryptography is heavily dependent on abstract algebra and number theory.
The message to be sent is called theplaintext message. The disguised message is called the ciphertext. The plaintext and the ciphertext are both written in an alphabet, con- sisting of letters or characters. Characters can include not only the familiar alphabetic characters A, . . ., Z and a,. . ., z but also digits, punctuation marks, and blanks. A cryp- tosystem, orcipher, has two parts: encryption, the process of transforming a plaintext message to a ciphertext message, and decryption, the reverse transformation of changing a ciphertext message into a plaintext message.
There are many different families of cryptosystems, each distinguished by a particular encryption algorithm. Cryptosystems in a specified cryptographic family are distinguished from one another by a parameter to the encryption function called a key. A classical cryptosystem has a single key, which must be kept secret, known only to the sender and the receiver of the message. If person A wishes to send secret messages to two different people B and C, and does not wish to have B understand C’s messages or vice versa, A must use two separate keys, so one cryptosystem is used for exchanging messages with B, and another is used for exchanging messages with C.
Systems that use two separate keys, one for encoding and another for decoding, are called public key cryptosystems. Since knowledge of the encoding key does not allow anyone to guess at the decoding key, the encoding key can be made public. A public key cryptosystem allows Aand B to send messages toC using the same encoding key. Anyone is capable of encoding a message to be sent toC, but onlyC knows how to decode such a message.
7.1
Private Key Cryptography
In single orprivate key cryptosystems the same key is used for both encrypting and decrypting messages. To encrypt a plaintext message, we apply to the message some func- tion which is kept secret, say f. This function will yield an encrypted message. Given the encrypted form of the message, we can recover the original message by applying the inverse transformation f−1. The transformationf must be relatively easy to compute, as mustf−1; however,f must be extremely difficult to guess from available examples of coded messages.
Example 7.1. One of the first and most famous private key cryptosystems was the shift code used by Julius Caesar. We first digitize the alphabet by letting A= 00,B= 01, . . . ,Z= 25. The encoding function will be
f(p) =p+ 3mod26;
that is,A7→D, B7→E, . . . , Z7→C. The decoding function is then f−1(p) =p−3mod26 =p+ 23mod26.
Suppose we receive the encoded message DOJHEUD. To decode this message, we first digitize it:
3,14,9,7,4,20,3. Next we apply the inverse transformation to get
0,11,6,4,1,17,0,
or ALGEBRA. Notice here that there is nothing special about either of the numbers 3 or
26. We could have used a larger alphabet or a different shift.
Cryptanalysisis concerned with deciphering a received or intercepted message. Meth- ods from probability and statistics are great aids in deciphering an intercepted message; for example, the frequency analysis of the characters appearing in the intercepted message often makes its decryption possible.
Example 7.2. Suppose we receive a message that we know was encrypted by using a shift transformation on single letters of the26-letter alphabet. To find out exactly what the shift transformation was, we must compute b in the equation f(p) =p+bmod26. We can do this using frequency analysis. The letter E = 04 is the most commonly occurring letter in the English language. Suppose that S = 18 is the most commonly occurring letter in the ciphertext. Then we have good reason to suspect that 18 = 4 +bmod26, or b = 14. Therefore, the most likely encrypting function is
f(p) =p+ 14mod26. The corresponding decrypting function is
f−1(p) =p+ 12mod26. It is now easy to determine whether or not our guess is correct.
Simple shift codes are examples ofmonoalphabetic cryptosystems. In these ciphers a character in the enciphered message represents exactly one character in the original message. Such cryptosystems are not very sophisticated and are quite easy to break. In fact, in a simple shift as described in Example 7.1, there are only26 possible keys. It would be quite easy to try them all rather than to use frequency analysis.
Let us investigate a slightly more sophisticated cryptosystem. Suppose that the encoding function is given by
f(p) =ap+bmod26.
We first need to find out when a decoding function f−1 exists. Such a decoding function exists when we can solve the equation
7.2. PUBLIC KEY CRYPTOGRAPHY 77