EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5659
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Septupling Point 7P Arithmetic Formula for
López-Dahab Coordinate and Affine Over Binary

Elliptic Curve Cryptosystem

Waleed K. Abdulraheem

Department of Information System and Network, The World Islamic Sciences and
Education University, Amman, Jordan

Abstract. The Elliptic Curve Cryptography (ECC) is one of the most prominent Asymmetric-
based cryptosystems as it affords a higher level of security with small keys. According to National
Institute of Standards and Technology (NIST), ECC gains the smallest secure key over the binary
curve. In literature, the best field over binary curves is Lopez-Dahab (LD) and Affine coordinates,
and it considered fit for lightweight cryptography in resource-constrained devices such as Internet
of Things (IoTs). ECC consists of three operational levels; scalar multiplication, point arithmetic
and field arithmetic. This research focuses on point arithmetic precomputation useful in scalar
multiplication and then for field arithmetic. There is no existing formula for Septupling point
7P over binary curves in LD and Affine coordinates. A new precomputed Septupling point 7P is
introduced in this paper using LD and non-supersingular affine coordinate over the binary field
E(F2m). This paper uses the form 7P = 2(3P ) + P , consisting of Doubling point, Tripling point
and the point addition. Also, the form 2(3P ) means the Sixtupling point is also proposed. Results
show that the 7P is characterized by a cost of 19 S + 28 M , while the cost of Sixtupling point
6P is 14 S + 19 M . The point is mathematically proved as valid. The proposed point can be
implemented for different scalar multiplication such as w−NAF for w = 4, and multi-based scalar
such as {2, 3, 7}-based scalar multiplication method.

2020 Mathematics Subject Classifications: 94A60, 11T71, 14G50, 68P25

Key Words and Phrases: Elliptic Curve Cryptography, Information Security, Binary curves,
Affine coordinates, Point Septupling Arithmetic, Lightweight Cryptography

1. Introduction

Recently, Information security has become one of the most important research topics
affecting, organizations, countries [1]. Cryptography is considered the most fundamental
method used to ensure the information and cyber security [2]. As a mechanism for achiev-
ing security , cryptography is vital to protect our data, information, and communication
from threats, intruders, and attackers [3]. Cryptography codes the information to ensure

DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5659

Email address: waleed.amin@wise.edu.jo (W. K. Abdulraheem)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 2 of 13

the main three security services which are confidentiality, integrity and authenticity, [4],
[5], and [6] as shown in Fig. 1. Confidentiality ensures the data is secret, integrity ensures
that the data is not changed, while authentication verifies the message origin [7], [8] and
[9].

Figure 1: C.I.A Triad [7]

Cryptography can be divided into three types; symmetric, Asymmetric, and hashing
algorithms [10] and [11]. According to national institute of standards and technology
NIST, Symmetric key cryptography or private key refers to cryptographic algorithms that
uses the same key for encryption and decryption operation, while Asymmetric or public
cryptography refers to algorithms that uses two different keys to exchange information,
one to digital sign or encrypt information and the other to decrypt it or to verify the
digital signature [12] and [13] as in Fig. 2.

Although symmetric cryptosystems are very effective with respect to consumed time
and resources, they still suffer from inherent problems in key distribution through the
unsecure channels [14] and [15]. Asymmetric cryptosystem mechanisms facilitate key dis-
tribution over unsecure channels which assures both confidentiality and non-repudiation,
however, the performance expectations are lower compared to symmetric cryptosystem
[16] and [17].



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 3 of 13

Figure 2: Symmetric and Asymmetric cryptosystems [18]

Elliptic curve cryptography ECC is a public key cryptosystem which has recently
gained higher acceptance and popularity within the research community. This is because it
achieves a significantly higher security level using short keys compared to other asymmetric
cryptosystems [19] and [20]. As in Fig. 3, the ECC scalar multiplication is composed of
three computational levels; scalar arithmetic, point arithmetic and finally field arithmetic.
Scalar multiplication is to find the result of the operation Q = kP such that k ∈ N ,
P and Q are two points on the curve. On the other hand, point arithmetic is adding two
points on the curve P +Q or P + P (for point doubling). Field arithmetic are related to
coordinates where the point operations are performed [21].

Figure 3: Operations in ECC divided by levels.[21]

Precomputing the points will help faster computation at level 2. Instead of repeating
the addition and doubling many times, precomputation will increase the overall perfor-
mance of the scalar and ECC protocols. This include point Tripling 3P , point Quintupling
5P , and point Septupling 7P in addition to other precomputed points [22] and [23].

Point Septupling 7P is to find (x7, y7) by calculating the 7-fold of the point P = (x, y).
According to [24] who proposed the first Septupling formula of the point introduced by
[25], the Septupling is calculated over the affine binary field to be used for multi-base
number (MTB) as a scalar recoding. Its cost is 3i+7s+18m, where i, s, and m is denote
the cost of inversion, squaring, and multiplication respectively. Later, [26] proposed a new
7P calculation over binary field in terms of affine coordinates, the cost of the proposed



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 4 of 13

algorithm is 2i + 7s + 14m. In [27], the authors studied the literature on multiplying an
odd scalar to the point P over prime field. They found that Sakai’s method which uses
the formula [α + 4d + 1 + 4d + 1 β] to calculate the cost gives the best performance
with an overall cost of 67 operations. In [28], the authors proposed a new Septupling
point 7P formula in Jacobian coordinates and used it in the multi-base non-adjacent form
(mbNAF) and Window-based (wmbNAF) methods. The cost over Jacobian is 14m + 15s.
Similarly in [24], they proposed a new point Septupling 7P formula for triple-based chain
(TBC) representation of the scalar using {2, 3, 7}-basis over prime field. Similarly, they
used Co Z operation method and calculate the overall cost of the triple-based together.

2. Motivation

According to NIST report [29] page 88, using binary field in ECC is outperforms the
prime field since it yields the same level of security with smaller key size (163 vs. 192). In
addition, using Lopez-Dahan LD coordinate over the binary curve yields the best perfor-
mance for ECC [30], [22], [23] and [31]. Over the binary curves, most known formulas for
adding points together are incomplete [32]. Also, in literature, the precomputed formula
of the point Septupling 7P is not available [31] and [33]. Thus, the point Septupling can
be efficiently implemented by scalar multiplication such as w−NAF , double or multi-base
numbering system. In view of the aforementioned, the contributions of this are as follows:

• A new precomputed Septupling point 7P arithmetic formula is proposed for LD
coordinate over the binary curve using the form 7P = 6P +P using mixed addition.

• A new precomputed Sixtupling point 6P arithmetic formula is proposed for LD
coordinate over the binary curve using the form 6P = 3(2P ) using mixed addition
which is to be deployed in the Septupling precomputed formula.

The remaining sections of this paper are organized as follows: Section 2 introduces the
literature and related work. Section 3 presents the proposed formulas, and finally Section
4 concludes the paper and highlights suggestions for future work.

3. Literature Review

Elliptic curve cryptography ECC utilizes the elliptic curve over a finite field Fq, denoted
by E(Fq) which contains the points (x, y) ∈ Fq x Fq that is in affine coordinates and satisfy
Weierstrass as provided in Equation (1).

y2 + a1xy + a3y = x3 + a2x
2 + a4x+ a5 (1)

Where ai ∈ Fq

The point at infinity O is a special point that form abelian group with the finite field
E(Fq), where O is a neutral element in the group operation [7].



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 5 of 13

Over ECC, different forms for ai will represent different curves. Some curves are
suitable for use in the prime field such as short Weierstrass curve which uses the form in
Equation (2), Barreto-Naehrig curve which uses the form in Equation (3) below, as well as
other curves. Although there are several applicable formulas within the family of binary
curves, the most suitable curve for the binary field is Koblitz curve as shown in Equation
(4), where a ∈ {0, 1}.

y2 = x3 + ax+ b (2)

y2 = x3 + b (3)

y2 + xy = x3 + ax2 + 1 (4)

Anomalous binary curve (ABC) or Koblitz curve is recommended by NIST standard
[29] for the curve K−163 where a = 1. It allows for fast and efficient scalar multiplication
while efficiently using both hardware and software. Particularly, it is attractive because of
its lightweight implementation [34]. Over LD projective coordinates, the equation of the
projective point (X : Y : Z) where Z ̸= 0 has its Equation as shown in (5) [35] .

Y 2 +XY Z = X3 + aX2Z2 + bZ6 (5)

The first proposed point addition over LD by [36] costs 6S+14M +8A as provided in
the formula in Equation (6) below.

(X0, Y0, Z0) + (X1, Y1, Z1) = (X2, Y2, Z2)

Where :

A0 = Y1.Z0

A1 = Y0.Z
2
1

B0 = X1.Z0

B1 = X0.Z1

C = A0 +A1

D = B0 +B1

E = Z0.Z1

F = D.E

Z2 = F 2

G = D2.(F + aE2)

H = C.F



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 6 of 13

X2 = C2 +H +G

I = D2.A0 +X2

J = D2.A0 +X2

Y2 = H.I + Z2.J (6)

To reduce the cost, the formula in (6) can be improved when Z = 1 for its special case
as shown in the formula provided in Equation (7).

(X0, Y0, Z0) + (X1, Y1, 1) = (X2, Y2, Z2)

Such that :

A = Y1.Z
2
0 + Y0

B = X1.Z0 +X0

C = Z0.B

D = B2.(C + aZ2)

Z2 = C2

E = A.C

X2 = A2 +D + E

F = X2 +X1.Z2

G = X2 + Y1.Z2

Y2 = E.F + Z2.G (7)

The cost was reduced later by [37] who introduced a new formula with cost 4S +
13M + 9A. Afterwards, [38] proposed a new formula which reduces the cost of the point
addition using mix addition techniques in the projective as shown in the formula provided
in Equation (8) below.

(X1, Y1, 1) + (X2, Y2, Z2) = (X3, Y3, Z3)

Where :

U = Z2
2 Y1

S = Z2 X1 +X2

T = Z2 S

Z3 = T 2

V = Z3 X1

X3 = U2 + T (U + S2 + a2T )



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 7 of 13

Y3(V +X3)(TU + Z3) + Z2
3 (Y1 +X1) (8)

The point doubling 2P implies adding the point P to itself such that 2P = P +P . The
first proposed 2P over LD coordinate as the projected form is as shown in the following
Formula provided in Equation (9) below with the cost 4S + 5M + 5A [36].

2(X1.Y1, Z1) = (X3, Y3, Z3)

Where

A = Z2
1

B = bA2

C = X2
1

Z3 = AC

X3 = C2 +B

Y3 = (Y 2
1 + aZ3 +B) X3 + Z3 B (9)

Later, [39] improved the cost of the point doubling to 4S + 5M + 5A. The proposed
projective form is as shown in in the formula provided in Equation (10) below:

2(X1, Y1, Z1) = (X3, Y3, Z3)

Where

S = X2
1

U = S + Y1

T = X1Z1

Z3 = T 2

T = UT

X3 = U2 + T + a2Z3

Y3 = (Z3 + T ) X3 + S2Z3 (10)

Using these formulas, different point compositions have been implemented such as point
Tripling 3P using the mixed addition, and using the projective form for LD coordinate over
binary coordinates. The best cost point Tripling uses the form 3P = 2P +P proposed by
[22], with a cost of 7S+12M where the addition were neglected. A new two Quintupling
points 5P was also proposed by [23] using two forms; 5P = 3P + 2P and the form
5P = 2(2P ) + P with cost of 13S + 23M and 12S + 17M respectively outperforming the
former with respect to the second form.



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 8 of 13

4. Proposed Algorithm

This paper introduces a new formula for point Septupling 7P for the projective coor-
dinates using affine LD over short Weiestrass curve. To achieve a higher performance, the
mixed formula is used for addition in the affine. The proposed point Septupling used the
form 7P = 2(3P ) + P .with details shown in Algorithm 1.

Algorithm 1: A new Septupling point of the form 7P = 2(3P ) +P using LD coordi-
nate in affine and mixed addition over binary curve.

Let the point P = (X1, Y1, Z1) be a point on the Weiestrass curve as Y 2 + X Y =
X3 + aX2 + b using LD coordinate, let the points 2P , 3P, and 6P is (X2, Y2, Z2),
(X3, Y3, Z3) and (X6, Y6, Z6) respectively. Then the 7P formula is formulated as shown
below:

First, the Doubling point 2P is found using the Formula in (10) as shown in (11) below:

2(X1, Y1, Z1) = (X2, Y2, Z2)

A2 ←− X2
1

B2 ←− A2 + Y1

C2 ←− X1 Z1

D2 ←− B2 C2

Z2 ←− C2
2

X2 ←− B2 +D2 + a Z2

Y2 ←− (Z2 +D2) X2 +A2
2 Z2

Cost of Doubling point is : 4S + 5M (11)

Then, finding the Tripling point 3P using (8) yields the formula in (12) below:

(X3, Y3, Z3) = (X1, Y1, Z1) + (X2, Y2, Z2) such that Z1 = 1

A3 ←− Y2 + Y1 Z2
2

B3 ←− X2 +X1 Z2

C3 ←− B3 Z2

Z3 ←− C2
3

D3 ←− X1 Z3

X3 ←− A2
3 + C3(A3 +B2

3 + a C3)

Y3 ←− (D3 +X3)(A3 C3 + Z3) + (Y1 +X1) Z
2
3

Cost of Doubling point is : 5S + 9M (12)



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 9 of 13

Both the formulas in (11) and (12) can be used to find the Sixtupling point 6P as
shown in (13) below:

6P = 2(3P ) = (X6, Y6, Z6) = 2 ∗ (X3, Y3, Z3)

A6 ←− X2
3

B6 ←− A6 + Y3

C6 ←− X3 Z3

D6 ←− B6 C6

Z6 ←− C2
6

X6 ←− B2
6 +D6 + a Z6

Y6 ←− (Z6 +D6) X6 +A2
6 Z6

Cost of Doubling point is : 4S + 5M (13)

Finally, the Septupling point 7P is found using the form 7P = 6P + P using the
formulas in (10) and (13) as shown in (14) below:

(X7, Y7, Z7) = (X1, Y1, Z1) + (X6, Y6, Z6) such that Z1 = 1

A7 ←− Y6 + Y1 Z2
6

B7 ←− X6 +X1 Z6

C7 ←− B7 Z6

Z7 ←− C2
7

D7 ←− X1 Z7

X7 ←− A2
7 + C7(A7 +B2

7 + C7 a)

Y7 ←− (D7 +X7)(A7 C7 + Z7) + (Y1 +X1) Z
2
7

Cost of Doubling point is : 5S + 9M (14)

The cost of the point is obtained by adding the number of squaring to number of
multiplications. So, the Septupling point cost is the total of cost of Formulas (11), (12),
(13), and (14); which 19 S + 28 M , while the Sixtupling point’s cost is the total cost of
(11), (12) and (13); which 14 S + 19 M .

To prove the Septupling point, a non-supersingular affine coordinate is used over the
binary field E(F2m) for the equation y2+xy = x3+ax2+b which includes the set of points
P (x, y) such that x, y ∈ F2m . Proof of the used method follows from [38], [22], [40], [23]
and [31] and proposed as in Lemma (1) below.

Lemma 1: The proposed Septupling point of form 7P = 2(3P ) + P has a valid
formula.

Proof:



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 10 of 13

To prove the formulas in Algorithm 1, the Affine coordinate is used. In Affine coordi-
nates, for adding the two points (X7, Y7, Z7) = (X1, Y1, Z1) + (X6, Y6, Z6) the following
equations should be satisfied:

λ =
(y6 + y1)

(x6 + x1)

x7 = λ2 + λ+ x6 + x1 + a

y7 = λ(x1 + x7) + x7 + y1 (15)

Using Formulas (11), (12), (13), and (14), the forms X7
Z7

= x7 and Y7

Z2
7
= y7 needs to be

proven. The process is:

X7

Z7
=

A2
7 + C7(A7 +B2

7 + aC7)

C2
7

=
A2

7

C2
7

+
A7

C7
+

B2
7

C7
+ a

=
(Y6 + Y1Z

2
6 )

2

(B7Z6)2
+

Y6 + Y1Z
2
6

B7Z6
+ a

=
(Y6 + Y1Z

2
6 )

2

(X6 +X1Z6)2(Z6)2
+

Y6 + Y1Z
2
6

(X6 +X1Z6)Z6
+

X6 +X1Z6

Z6
+ a (16)

Let Z6 = 1

=
(Y6 + Y1)

2

(X6 +X1)2
+

Y6 + Y1
X6 +X1

+X6 +X1 + a

= (
Y6 + Y1
X6 +X1

)2 +
Y6 + Y1
X6 +X1

+X6 +X1 + a

= (
Y6 + Y1/Z

2
1

X6 +X1/Z1
)2 +

Y6 + Y1/Z
2
1

X6 +X1/Z1
+X6 +

X1

Z1
+ a

= (
y6 + y1
x6 + x1

)2 +
y6 + y1
x6 + x1

+ x6 + x1 + a

x7 = λ2 + λ+ x6 + x1 + a (As x7 form in Formula (15))

Y7
Z2
7

=
(D7 +X7)(A7C7 + Z7) + (Y1 +X1)Z

2
7

C4
7

=
A7C7(X1Z7 +D7) + (Y1 +X1)Z

2
7

C4
7

=
(Y6 + Y1Z

2
6 )(X1Z7 +X7)

C3
7

+
X7

C2
7

+ Y1

=
(Y1 + Y6/Z

2
6 )

(X1 +X6/Z6)
(X1 +X7Z7) +

X7

Z7
+ Y1

=
(y1 + y6)

(x1 + x6)
(x1 + x7) + x7 + y1

y5 = λ(x1 + x7) + x7 + y1 (Proven as in Formula (15))



W. K. Abdulraheem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5659 11 of 13

5. Conclusion and Future Work

This paper introduces the formula of the Septupling point 7P , using LD coordinates
and a non-Supersingular Affine over the binary field E(F2m) for the equation y2 + xy =
x3 + ax2 + b which includes the set of points P (x, y) such that x, y ∈ F2m . The point
is using the form 7P = 2(3P ) + P where the point P is in the Affine and the Sixtupling
point 2(3P ) = 6P form is in the projective coordinates. The cost of the Septupling point
is 19 S + 28 M , while the cost of Sixtupling point 6P is 14 S + 19 M . The point is not
comparable to any Septupling point since it is the first formula for 7P over LD coordinate
and Affine. Since it is precomputed and precalculated, the point can be used with scalar
multiplication operation in ECC operation methods such as w − NAF where w = 4 or
multi-base scalar multiplication such as {2 3, 7}-base methods. This work can also be
improved using other forms if available, and higher point calculations can be precomputed
such as Nineupling 9P or much higher degrees.

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