The new integer factorization algorithm based on fermat’s factorization algorithm and euler’s theorem

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 2, April 2020, pp. 1469~1476
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i2.pp1469-1476  1469
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
The new integer factorization algorithm based on fermat’s
factorization algorithm and euler’s theorem
Kritsanapong Somsuk
Department of Computer and Communication Engineering, Faculty of Technology,
Udon Thani Rajabhat University, Udon Thani, 41000, Thailand
Article Info ABSTRACT
Article history:
Received Apr 7, 2019
Revised Oct 7, 2019
Accepted Oct 20, 2019
Although, Integer Factorization is one of the hard problems to break RSA,
many factoring techniques are still developed. Fermat’s Factorization
Algorithm (FFA) which has very high performance when prime factors are
close to each other is a type of integer factorization algorithms. In fact, there
are two ways to implement FFA. The first is called FFA-1, it is a process to
find the integer from square root computing. Because this operation takes
high computation cost, it consumes high computation time to find the result.
The other method is called FFA-2 which is the different technique to find
prime factors. Although the computation loops are quite large, there is no
square root computing included into the computation. In this paper,
the new efficient factorization algorithm is introduced. Euler’s theorem is
chosen to apply with FFA to find the addition result between two prime
factors. The advantage of the proposed method is that almost of square root
operations are left out from the computation while loops are not increased,
they are equal to the first method. Therefore, if the proposed method is
compared with the FFA-1, it implies that the computation time is decreased,
because there is no the square root operation and the loops are same.
On the other hand, the loops of the proposed method are less than the second
method. Therefore, time is also reduced. Furthermore, the proposed method
can be also selected to apply with many methods which are modified from
FFA to decrease more cost.
Keywords:
Computation loops
Euler’s theorem
Fermat’s factorization
Integer factorization
Prime factors
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Kritsanapong Somsuk,
Department of Computer and Communication Engineering,
Faculty of Technology, Udon Thani Rajabhat University,
Udon Thani, 41000, Thailand
Email: kritsanapong@udru.ac.th
1. INTRODUCTION
Nowadays, the significant information is always exchanged via the communication channel
connected to computer system such as internet. Generally, this channel is the insecure channel. That means
attackers can access data easily by using various techniques. With this reason, security for the information
becomes very important. At present, many security algorithms were introduced to protect the secret data
sending over insecure channel. Cryptography is one of techniques to defend data from attackers by using
encryption and decryption processes. In addition, there are two types about cryptography. The first is
symmetric key cryptography using the same key which is called secret key for encryption and decryption
processes. The second is asymmetric key cryptography (or public key cryptography) [1] using a pair of keys
for encryption and decryption. In addition, one key which is always distributed to keep in the key center is
called public key. On the other hand, the other key which is always kept secretly by owner’s key is called
private key. RSA [2] is the most well-known public key cryptography used for both of digital signature and
data encryption. This algorithm is one-way function. That means it is very easy to compute the production of

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 1469 - 1476
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two large prime numbers. Nevertheless, it becomes very difficult to factor the modulus. Furthermore,
RSA has been improved continuously to avoid attacking from intruders [3].
In fact, if attackers can recover two large prime factors of the modulus, then RSA is broken.
Although, Integer Factorization is one of the hard problems to break RSA, many integer factorization
algorithms, such as [4-10] are still developing to find the weakness of RSA which is not still disclosed.
In general, the efficiency of each algorithm is based on characteristic of prime factors. For example, if one of
two prime factors is very small, Trial Division Algorithm (TDA) [11, 12] is the best algorithm for this
situation. However, TDA becomes inefficient algorithm whenever both of prime factors are very large.
The other example is that if all prime factors of p–1 or q–1 are very small, where p and q are represented as
two large prime factors of the modulus, Pollard’s p-1 [13] should be chosen to recover both of them.
Fermat’s Factorization Algorithm (FFA) [14, 15] discovered by Pierre De Fermat in 1600 is one of integer
factorization algorithms. FFA can factor modulus very fast when the difference between two large prime
factors is small. In addition, the other form of modulus which is equal to the difference between two perfect
square numbers is considered instead of the form of the production between two prime numbers.
In fact, many improvement of FFA algorithms [16-21] were presented to reduce loops. In 2017, Specific
Fermat's Factorization Algorithm Considered from X (SFFA-X) [22] was proposed to reduce the time
complexity. The last m digits of modulus are considered to leave many unexpected values out from
the computation. Furthermore, SFFA-X can be increased performance by changing the value of m which
must be larger. However, assuming only traditional FFA is considered, there are two techniques to implement
FFA. Both of them have the different advantage and disadvantage. The advantage of the first technique
which is called FFA-1 is small loops when it is compared with the other technique which is called FFA-2.
However, every loops have to compute square root operation which is not included to FFA-2.
The aim of this paper is to propose the new integer factorization algorithm. Euler’s theorem is applied with
FFA to get the new idea behind the proposed method. In fact, the proposed method is from the combination
of the advantage from both of FFA-1 and FFA-2.
2. RELATED WORKS
2.1. Fermet’s factorization algorithm: FFA
FFA is the one of the methods to find two large prime factors. It suits to solve modulus which both
of prime factors are close to each other. Assume p, q are odd prime numbers and n is the modulus that n=p*q.
By using Fermat’s technique, n must be rewritten in the other form as following:
n=x2
–y2
(1)
where, x=
p + q
2
and y=
p - q
2
FFA is distinguished as two methods which have different advantage and disadvantage as following:
2.2. FFA-1
Assigning, the initial value of x is n 
  , from (1), y is calculated from the following equation:
y= 2
-x n (2)
Generally, if y is an integer, then p=x+y and q=x–y are two prime factors of n. On the other hand, x must be
increased by 1 to find the new value of y whenever it is not the integer. In fact, the process must be repeated
until the integer of y is found. Assigning l1 is the number of loops for FFA-1, then
l1=
p + q
2
- n 
  (3)
Nowadays, many improvements of FFA-1 were proposed to reduce l1 such as [16-18]. However,
the disadvantage of FFA-1 and it’s improving algorithms are about computing square root of integer.

Int J Elec & Comp Eng ISSN: 2088-8708 
The new integer factorization algorithm based on Fermat’s Factorization … (Kritsanapong Somsuk)
1471
2.3. FFA-2
FFA-2 is different from FFA-1, because there is no square root computing in the main process.
From (1), the equation can be rewritten as following:
4n=u2
–v2
(4)
where, u=p+q and v=p–q. Assuming the initial value of u and v are 2 n 
  and 0, respectively, then r is
computed by following equation:
r=u2
–v2
–4n (5)
From (5), if r is equal to 0, p and q are recovered from p=
u + v
2
and q=
u + v
2
. On the othe hand,
if r is not equal to 0, the process is divided as two cases:
- Case 1: r>0, v will be increased by 2 in order to reduce r until r which is equal to 0 is found.
- Case 2: r<0, u will be increased by 2 in order to enlarge r until r which is equal to 0 is found.
Assigning l2 is represented as the number of loops computation for FFA-2, l2 can be computed from,
l2=lu+lv (6)
where, lu is loops of u and lv is loops of v,
Considering lu: the initial and target of u are 2 n 
  and p+q. Moreover, it is always increased by 2
then it implied that lu is equal to l1. Considering lv: the target result of v is p-q and the increment is 2, then
lv=
p - q
2
(7)
As FFA-1, there are many algorithms improved from FFA-2 such as Estimated Prime Factor (EPF) for
estimating the new initial values [15] and Specific Fermat's Factorization Algorithm Considered from
X (SFFA-X) [22] to leave unrelated integers from the computation. Although, integer square root is not
computed, time to find prime factors is still high because of large loops. Table 1 is shown advantage and
disadvantage between FFA-1 and FFA-2.
Table 1. Advantage and disadvantage between FFA-1 and FFA-2
Factorization Algorithm Square root Computing Loops
FFA-1 Every Time Small (Compared with FFA-2)
FFA-2 None Large
2.4. Euler’s theorem
Euler’s Theorem [23] is the theorem modified from Fermat’s little Theorem. Assigning a

 and
gcd(a, n)=1,  (n)=(p–1)*(q–1)=n–(p+q)+1 and a is relative prime to  (n), then
(n)
a 1

 mod n (8)
In fact, this theorem is one of the main cores for implementing the proposed method mentioned in
the next section.
3. THE PROPOSED METHOD
In this paper, the new algorithm based on FFA is proposed. The main core is to replace square root
computing with modular multiplication. In dept, modular multiplication takes low cost in comparison to
square root operation. Moreover, the loops are same as FFA-1. In addition, the method is from
the combination between euler’s theorem and FFA. In fact, (8) is rewritten as following:
an–(p +q) +1  1 mod n (9)

 ISSN: 2088-8708
1472
Algorithm: The Proposed Method
Input: n
Output: p, q
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
u←2 n 
 
Select c, where gcd(c, n)=1
a←c-1
mod n
s←c2
mod n
t←an–u+1
mod n
IF t=1 then
x←
u
2
y← 2
-x n
Else
y←0.1
End IF
While y is not an integer do
t←t*s
IF t>n then
t←t mod n
End IF
u←u+2
IF t=1 then
x←
u
2
y← 2
-x n
End IF
End While
p←x+y
q←x–y
Because lu=
p + q
2
- n 
  (x is increased by 1) or p+q-2 n 
  (u is increased by 2), therefore,
if u increased by 2 is considered, then p+q=lu+2 n 
  .
Assigning b, c

 , b=
n 2 n 1
a
  
 
mod n and c=a-1
mod n then,
b*clu
=(
n 2 n 1
a
  
 
)*(a-1
)lu
mod n
=
n 2 n 1-lu
a
  
 
mod n
=
n (2 n lu) 1
a
   
 
=an–(p +q)+1
mod n
In fact, the process begins with b=
n 2 n 1
a
  
 
mod n and b is recomputed by using modular
multiplication with c2
mod n whenever the result which is equal to 1 is not found as follow:
b=b*c2
mod n (10)
From (10), the result is certainly equal to 1 whenever the process is repeated lu times.
Assigning x= u
l
2
, then y can be recovered by using (2). However, it is possible that there are some
integers, assigned as li where i=0, 1, 2, ∙∙∙, u-1 and li<lu, that
n (2 n li) 1
a
   
 
mod n is equal to 1. Nevertheless,
y is not certainly an integer. Therefore, the process must be repeated until the b=1 and integer of y is found.

1473
In fact, square root operation will be processed only when b is equal to 1. Because, lu is represented as loops
of the proposed method and the increasing step is 2, it implies that loops between the method and FFA-1 are
same. In addition, c2
mod n should be calculated as the constant at first, because it is often operated in (10).
Furthermore, it should be assigned as small integer to decrease the cost. Therefore, c will be assigned before
finding a: a=c-1
mod n.
Example 1: Factoring n=340213 by using the proposed method
Sol. Each step from the algorithm is at following:
Step 1-2: u=1168, c=2
Step 3: a=2-1
mod 340213=170107
Step 4: s=22
mod 340213=4
Step 5: t=170107339046
mod 340213=186054
Step 6-11: Because t  1, y=0.1
Step 12–19 (Loops)
Loop 1:
Step 13-16: t=186054*4=744216
Because t>340213 then 744216 mod 340213=63790
Step 17: u=1168+2=1170
Loop 2:
Step 13: t=63790*4 mod 340213=255160
Step 17: u=1170+2=1172
Loop 3:
Step 13-16: t=255160*4=1020640
Because t>340213 then 1020640 mod 340213=1
Step 17: u=1172+2=1174
Step 18–21: Because t=1, then
Step 19-20: x=
1174
2
=587 and y= 2
587 340213 =66
Because t=1 and y is an integer (end of loop), then
Step 23-24: p=587+66=653, q=587-66=521
Therefore, there are only three loops to implement n=340213 by using the proposed method.
Considering FFA-1: l1=
653 + 521
2
-584=3
Considering FFA-2: lv=
653 - 521
2
=66, and l2=3+66=69
Therefore, it implies that loops of FFA-2 are larger than the proposed method. However,
FFA-1 takes time complexity higher than the proposed method, because modular multiplication takes low
time complexity in comparison to square root operation [24]. Because n=340213 is too small, all mentioned
algorithms can solve the problem very quick. The information in Table 2 is shown the main process and
loops of FFA-1, FFA-2 and the proposed method to factor n = 340213. Assuming n=788582867650121563
which is from the production between p=1066200463 and q=739619701 (p and q have the same bits’ length),
Table 3 is shown the comparison during three algorithms to find p and q of n=788582867650121563.
Table 2. The comparison during three algorithms to solve problem with n = 340213
Factorization Algorithm Loops Main Process
FFA-1 3 Square root
FFA-2 69 Multiplication
The Proposed Method 3 Multiplication
Table 3. The comparison during three algorithms to solve problem with n=788582867650121563
Factorization Algorithm Loops Time (Second)
FFA-1 902910079 19.04
FFA-2 1066200460 25.75
The Proposed Method 902910079 2.82

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From Table 3, although the loops between FFA-1 and the proposed method are same, the proposed
method takes time lower than FFA-1. In addition, FFA-2 consumes the highest computation cost.
In the example, the method is faster than FFA-1 and FFA-2 about 75% and 89%, respectively.
Next, assuming n=1047329636821139813 which is from the production between p=1971074143 and
q=531349691 (p and q have the different bits’ length). Table 4 is shown the comparison during three
algorithms to find p and q of n=1047329636821139813.
Table 4. The comparison during three algorithms to solve problem with n=1047329636821139813
Factorization Algorithm Loops Time (Second)
FFA-1 227820673 290.02
FFA-2 947682899 142.53
The Proposed Method 227820673 41.17
In Table 4, the proposed method still takes the lowest computation time. However, FFA-2 becomes
faster than FFA-1. The reason is that, although loops of FFA-2 are the highest, FFA-1 is slower than FFA-2,
because square root computing becomes inefficient operation when loops are large (bits length between p and
q are different). In the example, the method is faster than FFA-1 and FFA-2 about 75% and 72%.
4. COMPLEXITY ANALYSIS
4.1. The proposed method Vs. FFA-1
The multiplication of the main process from the proposed method is different from square root
operation. The value of s which is the multiplication of t is selected at first. Then, the cost for the production
between t and s is equal to the s-1 times of the addition between t and itself. For example, the compared
process of proposed method in the example 1 is t+t+t+t = 4t. For each iteration, it implies that the complexity
of the main process for the proposed method is close to the addition operation that is  (m) when m is
represented as binary based on t. Therefore, it is less than square root cost which is (M(m)), for Newton’s
method.
4.2. The proposed method Vs. FFA-2
In fact, the multiplication with the small multiple value is the main process for FFA-2. Therefore,
the cost complexity for each iteration is equal to the proposed method. However, loops of FFA-2 are very
larger than the proposed method. Therefore, the proposed method’s cost is less than the compared algorithm.
5. APPLIED PROPOSED METHOD WITH IMPROVED ALGORITHMS
In addition, the proposed method can be applied with most of algorithms that modified from both of
FFA-1 and FFA-2. However, two algorithms are selected as the examples to combine with the proposed
method. First, EPF [15] improved from FFA-2 is the method to estimate the new initial values (u and v) for
unbalanced modulus. In fact, the new initial value of u is also applied with the proposed method.
Example 2: Factoring n=1783647329 (p=84449 and q=21121)
Sol. Assign ui is the new initial value, the loop's equation is changed as: i
p + q - u
2
Because, n 
  =42234, then from (3) the loops are 10551. However, ui is computed as 97430, then the new
loops can be re-estimated as 4070 that is less than traditional proposed method.
Second method is Multi Forms of Modulus for Fermat Factorization Algorithm (Mn-FFA) [25].
The key of this method is to find the pattern of x which is distinguished as 16 cases from the considering
forms of n mod 4, n mod 6 and n mod 20 together. In fact, this method can be applied the proposed method
by converting pattern of x as pattern of u.
Example 3: Factoring n=571187 (p=941 and q=607)
Sol. Because, 571187 mod 4=3, 571187 mod 6=5 and 571187 mod 20=7, then from Table 2 in [15], u must
be divided by 12 and the last digit is 2 or 8.
Because 2 n 
  =1512 that is divided by 12 and the last digit is 2, ui is equal to 1512.
Assigning c=2, then a=285594 and t=an–u+1
mod n=411191

1475
Usually u=u+2=1514 must be computed for the proposed method. However, for applying Mn-FFA
with the proposed method, u can be computed as follow: u=u+36=1548, because it is the minimum value
which is larger than 1512 and still in the condition. Therefore, the multiple value is not c2
but it is c36
as
following: c36
mod n=539953 and t=t*c36
mod n=1, then x=774 and y=167. Because y is an integer, then p
and q can be recovered. In fact, there are only 2 loops for the implementation which are less than loops of
tradition proposed method which are equal to 18.
6. CONCLUSION
In this paper, the new factorization algorithm from the applied method between Fermat’s
factorization algorithm and euler’s theorem is proposed. The key is that the almost square root operations are
left from the computation while the loops are not increased. In fact, multiplication operation with small
multiple value is the main process of the proposed method. Then, it implies that the complexity for each
iteration is only  (m), where m is binary based of computed number. Moreover, the proposed method can
be also chosen to apply with the other algorithms modified from FFA to decrease more costs. In addition,
this paper shows the proposed method which is applied with EPF and Mn-FFA.
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[24] Computational complexity of mathematical operations, https://en.wikipedia.org/wiki/Computational_
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BIOGRAPHY OF AUTHOR
Kritsanapong Somsuk is an assistant professor of the department of Computer and Communication
Engineering in Faculty of Technology, Udon Thani Rajabhat University, Udon Thani, Thailand.
He received his M.Eng. (Computer Engineering) from department of Computer Engineering in
Faculty of Engineering, Khonkaen University, M.Sc. (Computer Science) from department of
Computer Science in Faculty of Science, Khonkaen University and his Ph.D. (Computer
Engineering) from department of Computer Engineering in Faculty of Engineering, Khonkaen
University. His research interests include computer security, cryptography and integer factorization
algorithms.

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