Discrete Mathematics/Modular arithmetic

Introduction
We have already considered moduli and modular arithmetic back in ../Number theory/, however in this section we will take a more in depth view of modular arithmetic.

For revision, you should review the material in number theory if you choose.

Simultaneous equations
When we speak of simultaneous equations with relation to modular arithmetic, we are talking about simultaneous solutions to sets of equations in the form
 * x &equiv; a1 (mod m1)
 * x &equiv; ak (mod mk)
 * x &equiv; ak (mod mk)
 * x &equiv; ak (mod mk)

There are two principal methods we will consider, successive substitution and the Chinese remainder theorem.

Successive substitution
The method of successive substitution is that where we use the definition of the modulus to rewrite these simultaneous equations, and then successively make substitutions.

It will probably be best to motivate the idea with an example.

Example: Solve 3x &equiv; 10 (mod 19), and x &equiv; 19 (mod 21) using successive substitution.

First:


 * 3x &equiv; 10 (mod 19)

Find the inverse of 3 in Z19; 3-1=-6, then
 * x &equiv; -60 (mod 19)
 * x &equiv; 16 (mod 19)
 * x = 16 + 19j &exist; j&isin;Z (*)

Substitute in the second equation
 * (16+19j) &equiv; 19 (mod 21)
 * 19j &equiv; 3 (mod 21)

Find the inverse of 19 in Z21; 19-1=10


 * j = 30 (mod 21)


 * j = 9 (mod 21)

Writing in the equivalent form
 * j = 9 + 21k &exist; k&isin;Z

Substituting back j in (*)
 * x = 16 + 19(9+21k)
 * x = 187+399k

Writing back in the first form
 * x &equiv; 187 (mod 399)

which is our solution.

Chinese remainder theorem
The Chinese remainder theorem is a method for solving simultaneous linear congruences when the moduli are coprime.

Given the equations
 * x &equiv; a1 (mod m1)
 * x &equiv; ak (mod mk)
 * x &equiv; ak (mod mk)
 * x &equiv; ak (mod mk)

multiply the moduli together, i.e. N=m1m2...mk, then write n1=N/m1, ..., nk=N/mk.

We then set yi be the inverse of ni mod mi for all i, so yini=1 mod mi.

Our solution will be
 * x &equiv; a1y1n1+...+akyknk (mod N)

To see why this works consider what values x mod mk takes. The term akyknk mod mk becomes equal to ak as yknk=1 mod mk, and all the terms ajyjnj mod mk become equal to zero as when $$j \neq k$$ mk is a factor of nj.

The Chinese Remainder Theorem is of immense practical use, as if we wish to solve an equation mod M for some large M, we can instead solve it mod p for every prime factor of M and use CRT to obtain a solution mod M.

Powers and roots
This section deals with looking powers of numbers modulo some modulus. We look at efficient ways of calculating
 * ab (mod m)

If we tried to calculate this normally - by calculating ab and then taking the modulus - it would take an exorbitant amount of time. However some of the theory behind modular arithmetic allows us a few shortcuts.

We will look at some of these and the theory involved with them.

Fermat's (little) Theorem
Fermat's theorem allows us to see where ab (mod m) is 1. This has an application in disproving primality.

It states
 * If p is prime, and gcd(a,p)=1, then, in Zp
 * ap-1=1.

So, for example, 1310=1 in Z11.

Primitive elements
If in Zn, can we write some elements as powers of an element? This is conceivably possible.

Let's look at Z3.
 * 20=1
 * 21=2
 * 22=1

The elements {1,2} constitute in fact :Z3*.

Generally, we have
 * If p prime, then there is an element g&isin;Zp* such that every element of Zp* is a power of g.

Orders
We can express this idea in a different way, using the concept of the order. We denote the order of a &isin; Zn* by the smallest integer k written On(a) such that
 * ak=1 in Zn.

For example, On(-1)=2 for all n except 2, since
 * (-1)2=1

except when n = 2, since in that field -1 = 1 and thus has order 1.

Note if gcd(a,n)&ne;1, that is, a &notin; Zn*, the order is not defined.

Properties of orders
The orders obey some properties, the first of which was originally proven by Lagrange:

If p prime, gcd(a,p)=1,
 * Op(a) divides p-1
 * a is primitive iff Op(a)=p-1

Orders and finding primitive elements
Given these facts above, we can find primitive elements in Zp for p > 2 fairly easily.

Using the above facts, we only need to check a(p-1)/pi=xi in Zp for all i, where the pi are the prime factors of p-1. If any of the xi are 1, a is not a primitive element, if none are, it is.

Example: Find a primitive element of Z11.

Try 2. p-1 = 10 = 2. 5 Check:
 * 210/2=25=10
 * 210/5=22=4

Neither is 1, so we can say that 2 is a primitive element in Z11.

Problem set
Given the above, answer the following. (Answers follow to even-numbered questions)
 * 1) Is 4 primitive in Z13?
 * 2) Is 5 primitive in Z23?
 * 3) Find a primitive element of Z5.
 * 4) Find a primitive element of Z19.

Answers

 * 2. Yes: In Z23, (23-1)=2*11, and 522/11=2, 522/2=22 and then 522=1. No lesser base gives this.
 * 4. 2. Check: (19-1) has distinct prime factors 2 and 3. In Z19, 218/2&ne;1 and 218/3&ne;1 but 218=1 so 2 is primitive.

Euler's totient function
Euler's totient function is a special function that allows us to generalize Fermat's little theorem above.

It is defined as
 * &phi;(n) = |Zn*|
 * =|{a&isin;Z|1 &le; a &le; n and gcd(a,n) = 1}|
 * that is the number of elements that have inverses in Zn

Some results
We have the following results leading on from previous definitions. In other symbols: $$\sum_{d\ divisor\ of\ n} \phi(d) = n$$.
 * 1) &phi;(p) = p - 1
 * 2) &phi;(pk) = pk-pk-1
 * 3) &phi;(mn)=&phi;(m)&phi;(n) for gcd(m,n)=1
 * 4) For any integer n, the sum of the totient values of each of its divisors equals n.

Proof of 2.: There are pk elements in Zpk. The non-invertible elements in Zpk are the multiples of p and there are pk-1 of them: p, 2p, 3p, ..., (pk-1-1)p, pk. Removing the non-invertible elements from the invertible ones leaves pk-pk-1 left. ∎

Corollary to 1, 2 and 3: If n has distinct prime factors (i.e. not counting powers) pi for i=1,...,r we have
 * $$\phi(n)=n\prod_{i=1}^r (1-\frac{1}{p_i})$$

For example:
 * 16=24, so &phi;(16)=(16)(1-1/2)=16/2=8
 * &phi;(11)=(11)(1-1/11)=(11)(10/11)=10
 * (confirm from before 11 prime so &phi;(11)=11-1=10).

Proof of 3.: We can prove this equality using a special case of the Chinese Remainder Theorem, where the CRT is now just a system of 2 congruences, namely:
 * x == a1 (mod m)
 * x == a2 (mod n)

(remember that the CRT is applicable here because m and n are assumed coprime in the equality).

Note that a1 can take on m values (from 0 to m-1), and a2 can take on n values (from 0 to n-1). Also note that, for each and everyone of the m*n (a1, a2) tuples, there is a unique solution x that is strictly smaller than m*n. Moreover, for each x strictly smaller than m*n, there is a unique tuple (a1, a2) verifying the congruence system (these two assertions are a component of the Chinese Remainder Theorem: a solution to the congruence system is unique modulo m*n).

With this bijective uniqueness property in mind, the proof is simple. Go through each x, from 0 to m*n-1, and show that if x is a totient of m*n (i.e., gcd (x,m*n) = 1), then a1 is a totient of m and a2 is a totient of n. Furthermore, you must also show that if a1 and a2 are totients of m and n respectively, then it follows that x must be a totient of m*n.

If gcd (x,m*n) = 1, then according to Bezout's identity, there exist X and Y integers such that x*X + m*n*Y = 1. Furthermore, we have:


 * x = a1 + k*m
 * x = a2 + q*n

Therefore, a1*X + m*(k + n*Y) = 1, ''' should this be a1*X + m*(k*X + n*Y) = 1 ?? ''' so gcd (a1,m) = 1, and therefore a1 is a totient of m. Proceed similarly to prove that a2 is a totient of n.

Proving the other direction is very similar in that it requires some simple replacement algebra.

So what have we shown? In the above we have shown that for every totient x of m*n, there is a unique tuple of totients of m on the one hand and n on the other hand. Furthermore, that for each tuple of totients of m on the one hand and n on the other hand, there is a unique totient of m*n. Therefore, phi(m*n) = phi(m)*phi(n).

Proof of 4.: Let Q(g) be the set of all integers between 1 and n inclusive, such that gcd(x,n) = g. Q(g) is nonempty if and only if g divides n. If g doesn't divide n, then good luck finding an x such that g is the greatest common DIVISOR of x and n. Secondly, if x belongs to Q(g) for a given g, then it can't belong to another Q(...), since, if n is fixed, then gcd(x,n) is unique, by definition of the GREATEST common divisor. Thirdly, for all x between 1 and n inclusive, there exists a g such that gcd (x,n) = g (in the "worst" case, it's 1). Put together, these three properties imply that the union of all the Q(g) sets (for each g a divisor of n), which are pairwise mutually exclusive, is the set {1,2,3,...,n}. And therefore, the sum of the cardinalities of each Q(g) equals n.

Now we show that |Q(g)| = &phi;(n/g).

One direction: Let x be an arbitrary member of Q(g) for some g. Therefore, we have that gcd (x,n) = g => gcd (x/g, n/g) = 1 => x/g belongs to the set of numbers coprime to n/g (whose cardinality of course is &phi;(n/g)). For diff\ erent x's, the two values x1/g and x2/g are distinct. So for each x in Q(g), there is a correspondingly unique x/g in the set of numbers coprime to n/g.

Other direction: Let x be an arbitrary member of the set of numbers coprime to n/g. This implies gcd (x,n/g) = 1 => gcd (xg,n) = g => xg belongs to Q(g). For different x's, the two values x1g and x2g are distinct. So for each x in the set of numbers coprime to n/g, there is a correspondingly unique xg in Q(g).

Therefore, |Q(g)| = &phi;(n/g).

Euler's theorem
We can now generalize Fermat's theorem to extend past just Zn.

Euler's theorem says:
 * If a &isin; Zn*, in Zn*,
 * a&phi;(n)=1
 * equivalently if gcd(a,n)=1,
 * a&phi;(n)&equiv;1 (mod n)

Example: Find 3216 in Z14. We need to calculate firstly &phi;(14)=&phi;(7)&phi;(2)=(7-1)(2-1)=6. Then write the exponent as: 216 = 6 &times; 36 So: 3216=(36)36

But Euler's theorem tells us 36=1 in Z14 (i.e., mod 14) since 3&phi;(14)=1 in Z14 as above. So we have: 3216=136=1.

Calculating large powers efficiently
When Euler's or Fermat's theorem fails us in the calculation of a high power, there is a way to decompose an exponent down so calculation is still easy.

Let us work through an example as motivation.

Example. 528 in Z4.

First write 28 in base 2 = (11100)2 = 24+23+22 = 16 + 8 + 4

Now 528 = 516+8+4 = 516 58 54 Now rewrite these powers of 2 as repeated exponents:
 * (((52)2)2)2 &times; ((52)2)2 &times; (52)2

When you calculate each exponent, reduce mod 4 each time.

Problem set
Given the above, calculate the following powers. (Answers follow to even-numbered questions)
 * 1) 312 (mod 13)
 * 2) 242 (mod 43)
 * 3) 6168 (mod 30)
 * 4) 2252 (mod 19)
 * 5) 261 (mod 22)
 * 6) 813 (mod 5)
 * 7) 1110 (mod 11) (Tricky!)

Answers

 * 2. Since gcd(2,43)=1 and the exponent is one less than the modulus, use Fermat's theorem - the answer is 1
 * 4. Observe that &phi;(19)=18 and 18|252. 252/18=14. Decompose the exponent then as 218&times;14=(218)14=1.
 * 6. Use fast exponentiation by squaring: the answer is 3