Description
1 Prime Factorization
A prime number is an integer greater than one and is divisible by one and itself only. The sequence
of prime numbers starts with 2, 3, 5, 7, 11, 13, 17, 19, . . .. There are infinitely many primes. The
current largest known prime is 282589933 − 1.
An integer greater than one that is not prime is composite. For example, 4 is composite because it
is also divisible by 2 in addition to 1 and itself. By definition, 1 is neither a prime number nor a
composite number.
The fundamental theorem of arithmetic states that every integer greater than one can be uniquely
factored into a product of one or more primes. For example, 25480 = 23
· 5 · 7
2
· 13. More formally,
any integer n ≥ 2 can be written in the form
n = p
α1
1
· p
α2
2
· . . . · p
αk
k
,
where p1 < p2 < · · · < pk are the prime factors, and the exponents α1, α2, . . . , αk > 0 are their
respective multiplicities. For example, 2 has multiplicity 3 in the factorization of 25480.
1.1 Factorization and Encryption
Prime factorization is widely believed to be a very difficult computational problem. No efficient
prime factorization algorithm for large numbers is known. More specifically, there is no known
algorithm that can factor all integers in polynomial time, i.e., factor d-digit numbers in time O(d
k
)
for some constant k.
In 2009, several researchers successfully concluded two years of effort to factor the following 232-
digit number (named RSA-768) via the use of hundreds of machines:
123018668453011775513049495838496272077285356959533479219732245215172640050
726365751874520219978646938995647494277406384592519255732630345373154826850
791702612214291346167042921431160222124047927473779408066535141959745985690
2143413
=
334780716989568987860441698482126908177047949837137685689124313889828837938
78002287614711652531743087737814467999489
×
367460436667995904282446337996279526322791581643430876426760322838157396665
11279233373417143396810270092798736308917
The presumed difficulty of prime factorization lies at the foundation of cryptosystems such as RSA
that are used for secure data transmission. In such a system, a user publishes a large integer, their
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public key, while holding onto another large integer, their private key. Messages sent to the receiver
are encrypted using the public key, but are presumed impossible to decrypt without the knowledge
of the private key, because doing so would require prime factorization of very large numbers —
even larger than RSA-768.
1.2 Direct Search Factorization
In this project we will use a brute force strategy named the direct search factorization. It is based
on the observation that a number n, if composite, must have a prime factor less than or equal to
√
n. Clearly, the number cannot have more than one prime factor greater than √
n.
This method initializes m ← n and systematically tests divisors d from 2 up to b
√
nc.
1
In your
implementation, test the condition d · d ≤ n instead of d ≤
√
n for efficiency. If d divides m (i.e.,
is a factor of m), then set m ← m/d, and repeat this check with the divisor d until it no longer
divides m. Then increment d to continue the testing until d · d > m.
For example, to factorize 25480, we start by trying to divide the number by 2. Since 2 divides
25480, we record it and divide 25480 by 2 to obtain 12740. We find that the number 12740 can be
divided by 2 two more times. The original number thus contains the prime factor 2 with multiplicity
3. Now we work on 25480 / 8 = 3185. The next number to test is 3, which does not divide 3185;
neither does 4. The number 5 divides 3185 only one time, yielding the quotient 637. Moving on, 6
does not divide 637. We find that 7 divides 637 twice to yield 13. The algorithm terminates since
8 · 8 > 13.
You can cut down the number of test candidates by half by ignoring all the even numbers greater
than 2 (because they are apparently not prime).
2 List Structure
The class PrimeFactorization houses a doubly-linked list structure to store the prime factors of
a number. Every node on the list stores a distinct prime factor and its multiplicity, which are
together packaged into an object of the PrimeFactor class.
A node is implemented by the private class Node inside PrimeFactorization. The class Node also
has two links, previous and next, to reference the preceding and succeeding nodes.
For example, 25480 has the prime factor 2 with multiplicity 3, so the PrimeFactorization list for
25480 will have a Node whose data is a PrimeFactor object whose fields prime and multiplicity
take values 2 and 3, respectively.
In the linked list inside the class PrimeFactorization, the nodes are connected by the next link
in the increasing order of prime factor. The list also has two dummy nodes: head and tail. The
factorization of 25480 is represented by the list below with six nodes, four of which represent its
prime factors 2, 5, 7, 13 with multiplicities 3, 1, 2, 1, respectively.
1The symbols b c denote the floor operator, which gives the largest integer less than or equal to the operand. For
example, bπc = 3 and b5c = 5.
2
The class PrimeFactorization also stores the factored number in the instance variable value of
the long type. If this number exceeds the maximum 263 −1 representable by the type long, value
is set to be -1 (defined to be a constant OVERFLOW in the template code). For this reason, we make a
distinction between value and the object’s represented integer. These two integers are equal when
value != -1.
This class has four constructors:
• public PrimeFactorization() is a default constructor that constructs an empty list to
represent the number 1.
• public PrimeFactorization(long n) throws IllegalArgumentException performs the
direct search factorization described in Section 1.2 on the integer parameter n. An exception
will be thrown if n < 1.
• public PrimeFactorization(PrimeFactorization pf) is a copy constructor which assumes
that the argument object pf stores a correct prime factorization.
• public PrimeFactorization(PrimeFactor[] pfList) takes an array of PrimeFactor objects. This is especially useful for the case of an overflowing number.
3 Arithmetic Operations
The list structure of this PrimeFactorization object needs to be updated when its represented
integer is multiplied with or divided by another integer, either of type long or represented by
another PrimeFactorization object. After every arithmetic operation, the nodes in the linked list
must remain in the increasing order of the prime factors they store.
3.1 Multiplication
There are three methods to carry out multiplication, each of which must update the linked list of
the PrimeFactorization object that the method belongs to.
• public void multiply(long n) throws IllegalArgumentException can be implemented
via a traversal of the linked list referenced by head using an iterator (which is an object of
the private class PrimeFactorizationIterator) as follows.
Factor n using the direct search described in Section 1.2. For each prime factor found (with
a multiplicity), move the iterator forward from its current position to the first node storing
an equal or larger prime. In the first case, increment the multiplicity field of that node.
In the second case, create a new node for this factor and add it to the linked list before the
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larger prime. If the cursor reaches the end of the list, simply add the new node as the last
node.
• public void multiply(PrimeFactorization pf) updates the linked list by traversing it
and the linked list of the argument object pf simultaneously. Essentially, it merges a copy of
the second list into the first list by doing the following on the first list:
– Copy over the nodes from the second list corresponding to prime factors that do not
appear in the first list.
– For each prime factor shared with the second list, increase the multiplicity field of
the PrimeFactor object stored at the corresponding node in the first list.
The implementation requires the use of two iterators, one for each list, and works in a way
similar to merging two sorted arrays into one.
For example, suppose a PrimeFactorization object for 25480 (with the linked list shown in
Section 2) calls multiply() on another object storing the factorization of 405, which is 34
· 5.
After the call, the first linked list will become:
A new node (drawn in red) has been added, and the PrimeFactor object stored at an existing
node has its multiplicity field updated from 1 to 2 (shown in red).
• public static PrimeFactorization multiply
(PrimeFactorization pf1, PrimeFactorization pf2) is a static method that takes two
PrimeFactorization objects and returns a new object storing the product of the two numbers
represented by the first two objects.
The first two multiply() methods also call Math.multiplyExact() to carry out the multiplication
of this.value (if not -1) with n or pf.value. An ArithmeticException will be thrown if an
overflow happens. In this case, set the value field to -1. Otherwise, store the product in value.
Note that this.value == - 1 only means that the number represented by the linked list cannot
be represented in Java as a long integer. We can still perform arithmetic operations on the large
number by this PrimeFactorization object.
3.2 Division
The PrimeFactorization class provides three methods to perform division — by an integer, by
the integer encapsulated in another PrimeFactorization object, and of the first such object by a
second object. The linked list is updated to store the result.
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First, the public boolean dividedBy(long n) immediately returns false if this.value != -1
and this.value < n. Otherwise, it constructs a PrimeFactorization object for n, and then calls
the second dividedBy() method with the constructed object as the argument.
Second, public boolean dividedBy(PrimeFactorization pf) immediately returns false if
this.value != -1 && pf.value != -1 && this.value < pf.value, or if
this.value != -1 && pf.value == -1.
If the dividend and the divisor have equal values, it removes all the nodes storing prime factors by
calling the private method clearList(). In this case, it sets value to 1 and returns true.
Otherwise, it makes a copy of the linked list of this object and does the following.
Traverse the list copy and the list of the object pf using two separate iterators, say, iterCopy and
iterPf, respectively. Advance iterCopy.cursor until
iterCopy.cursor.pFactor.prime >= iterPf.cursor.pFactor.prime
or
(!iterCopy.cursor.hasNext() \&\& iterPf.hasNext()).
In the latter case, return false immediately. In the former case, there are three possible situations:
• iterCopy.cursor.pFactor.prime > iterPf.cursor.pFactor.prime. Then the number is
not divisible by the number represented by pf, so return false immediately.
• iterCopy.cursor.pFactor.prime == iterPf.cursor.pFactor.prime but
iterCopy.cursor.pFactor.multiplicity < iterPf.cursor.pFactor.multiplicity.
Again, the number is not divisible by the number represented by pf, so return false immediately.
• Otherwise, the following two conditions must hold:
iterCopy.cursor.pFactor.prime = iterPf.cursor.pFactor.prime
iterCopy.cursor.pFactor.multiplicity >= iterPf.cursor.pFactor.multiplicity
In this case, reset iterCopy.cursor.pFacor.multiplicity to the difference between itself
and iterPf.cursor.pFactor.multiplicity. If the difference is zero, the node needs to be
removed. Advance both iterators.
Repeat until either false is returned or iterPf.cursor == iterPf.tail eventually holds. If the
latter happens, the represented integer of this object is divisible by that of pf. Set the copy list to
be the linked list of this object by updating its head and tail. Also, update the object’s size and
value fields accordingly. Return true after the updates.
Example: Suppose the earlier PrimeFactorization object representing 25480 calls divideBy(98).
A PrimeFactorization object for 98 = 2 · 7
2
is first constructed with a linked list containing four
nodes: head, a node whose data is a PrimeFactor object with prime value 2 and multiplicity
value 1, a node whose data is a PrimeFactor object with prime value 7 and multiplicity value
2, and tail.
Then, this temporary object is provided as the argument for a call to the second divideBy()
method. The new linked list for the original object is shown below with the multiplicity of 2
decreased to 2 (colored red) and the node for the factor 7 deleted.
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In the case that this object represents a number that overflowed before the division, which is indicated by this.value == -1, the quotient to replace this number in the object might not overflow.
When that happens, the value field should store the quotient instead of -1. Checking can be done
at the end of divideBy() using a method called updateValue(). Initialize a variable to have value
1. Keep multiplying it with the power of every prime factor (determined by its multiplicity) stored
in the new linked list representing the quotient. Use Math.multiplyExact() to carry out the
multiplications, which should stop as soon as the method throws an ArithmeticException.
Finally, public static PrimeFactorization dividedBy(PrimeFactorization pf1,
PrimeFactorization pf2) is a static method that takes two PrimeFactorization objects and
returns a new object storing the quotient of the first corresponding number divided by the second
number, or null if the first number is not divisible by the second one.
4 Greatest Common Divisor and Least Common Multiple
The class PrimeFactorization provides methods to compute the greatest common divisor of two
positive integers and return the result as a PrimeFactorization object. The linked list in a
returned PrimeFactorization object must have its nodes store prime factors in the increasing
order.
4.1 The Euclidean Algorithm
A common divisor d of two positive integers a and b is a positive integer that divides both a and
b. The greatest common divisor (GCD) of a and b is the biggest of all their common divisors.
For example, 12 and 42 have four common divisors 1, 2, 3, 6. Their GCD is 6.
The GCD of two numbers can be efficiently computed using the Euclidean algorithm, named for
the Greek mathematician Euclid, who recorded it in his treatise Elements circa 300 BCE.
Suppose we are to find the GCD of 184 and 69. We divide the bigger number by the smaller one,
obtaining
184 = 2 · 69 + 46 .
The remainder 46 is less than the divisor 69, and any common divisor of 184 and 69 must also
divide
46 = 184 − 2 × 69 .
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Therefore, the GCD of 184 and 69 is also the GCD of 69 and 46. Next, we divide 69 by 46,
yielding
69 = 1 · 46 + 23 .
Thus, the problem further boils down to finding the GCD of 46 and 23. One more round of division
results in
46 = 2 · 23 + 0 .
Since the remainder becomes 0, 23 divides 46. We have thus obtained the GCD:
GCD(184, 69) = GCD(69, 46) = GCD(46, 23) = 23 .
In general, suppose we want to find the GCD of two numbers a and b, with a > b. Let r0 = a and
r1 = b.
The Euclidean algorithm carries out the following sequence of divisions:
r0 = q1 · r1 + r2 (0 < r2 < r1)
r1 = q2 · r2 + r3 (0 < r3 < r2)
r2 = q3 · r3 + r4 (0 < r4 < r3)
.
.
.
rk−1 = qk · rk + 0 .
The dividend and divisor in a division were respectively the divisor and remainder in the previous
division. Since the remainder is always less than the divisor, it follows that
r1 > r2 > · · · > rk > rk+1 = 0 .
The algorithm will terminate because the remainder decreases by at least one after each division
step. It holds that
GCD(r0, r1) = GCD(r1, r2) = · · · = GCD(rk−1, rk) = rk .
The following static method implements the Euclidean algorithm:
public static long Euclidean(long m, long n) throws IllegalArgumentException
The Euclidean() method must be used by the gcd() method below to compute the GCD of
this.value and another number n when this.value != -1.
public PrimeFactorization gcd(long n) throws IllegalArgumentException
5 Computing GCD from Prime Factorizations
If the second number is encapsulated in a PrimeFactorization object, the GCD computation is
carried out as follows.
• Traverse the linked lists of the two objects to find the common prime factors of the two
numbers.
• For every common prime factor p, create a new node to store p and min(α1, α2) , the minimum
of the multiplicities α1 and α2 of p stored in the two lists.
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• Link all the new nodes together to construct a PrimeFactorization object to represent the
GCD.
The work can be done during one round of simultaneous traversals of the linked lists of both
PrimeFactorization objects. The above procedure is implemented by the following method:
public PrimeFactorization gcd(PrimeFactorization pf)
Even when one or both of the numbers cause overflows, their GCD may not. Thus, if this.value
== -1 or pf.value == -1, call updateValue() to check if the GCD overflows, and set the value
field of the generated PrimeFactorization object properly.
When this.value != -1 and the second number for a GCD computation is provided as an integer,
an alternative to calling the first gcd() method (described in Section 4.1) would be to construct
a PrimeFactorization object pf over the second number, and call the second gcd() method
above. However, it is more efficient to directly use the first gcd() method, even though it returns
a PrimeFactorization object constructed through factoring the GCD. This factorization is less
expensive than factoring n, which has to be done to construct the object pf to call the second
gcd() method with. The reason is that n could be significantly larger than the GCD.
The third GCD method is static and takes two PrimeFactorization objects.
public static PrimeFactorization gcd(PrimeFactorization pf1,
PrimeFactorization pf2)
6 Iterators
The iterator class PrimeFactorizationIterator implements ListIterator
to the Javadocs of these methods, especially add() and remove().
7 Submission
Write your classes in the edu.iastate.cs228.hw3 package. Turn in the zip file, not your class
files. Please follow the submission guidelines posted on Canvas. You are not required to submit
any JUnit test cases. Nevertheless, you are encouraged to write JUnit tests for your code. Since
these tests will not be submitted, feel free to share them with other students. Include the Javadoc
tag @author in every class source file you have made changes to. Your zip file should be named
Firstname_Lastname_HW3.zip
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