Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

856 
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Elliptic Curves over 𝒑-adic field 𝑸𝒑 and its 𝒑-adic point addition 

 

P. Anuradha Kameswari1 and T. Sai Tejaswini2 

1,2Department of Mathematics, Andhra University, Visakhapatnam - 530003, Andhra Pradesh, 

India., panuradhakameswari@yahoo.in,  tejutanniru89@gmail.com. 

 

Article History: 

Received: 28-10-2024 

Revised:12-11-2024 

Accepted:19-12-2024 

Abstract: Public-key cryptosystems with the Elliptic curve 𝐸(𝐹𝑝) over finite field 𝐹𝑝 

are an alternative to RSA with finite fields. In the context of improving the efficiency 

of cryptosystems with elliptic curves, the study of elliptic curves 𝐸(𝑄𝑝) over p-adic 

number field 𝑄𝑝 was consequential. In this paper we first obtain all the points in the 

elliptic curve 𝐸(𝑄𝑝) over p-adic number field 𝑄𝑝 as lifts of the points in 𝐸(𝐹𝑝) to 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and then 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) to 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝3) and so on and then 

to evaluate the arithmetic on 𝐸(𝑄𝑝), we first proceed to describe the implementation 

of the arithmetic of points on 𝐸(𝑄𝑝) to points on 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and then give the 

algorithms for the computations. 

Keywords: Elliptic curves, 𝑝-adic field, Arithmetic of 𝑝-adic numbers, Elliptic curve 

over a field 𝜅. 

1. Introduction 

Elliptic curve cryptography is one of majorly using cryptosystems in the current century which 

provides higher security and greater efficiency. It provides considerable security with smaller key size 

compared to any other public key cryptosystems. In the context of improving the efficiency of 

cryptosystems with elliptic curves, the study of cryptosystems with elliptic curves 𝐸(𝑄𝑝) over 𝑝-adic 

field 𝑄𝑝 was consequential and the arithmetic of points in 𝐸(𝑄𝑝) is implemented to points on 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) over 𝑝 - adic number field 𝑄𝑝 . In this paper, we obtain all the points in the elliptic 

curve 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) over p-adic number field 𝑄𝑝 as lifts of the points in 𝐸(𝐹𝑝) then describe the 

implementation of the arithmetic of points on 𝐸(𝑄𝑝) to points on 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and then give the 

algorithms for the computations. For this, In section 2, we describe the 𝑝-adic field 𝑄𝑝 and arithmetic 

operations on 𝑝-adic numbers and In section 3, we describe elliptic curve over 𝑝-adic field 𝑄𝑝 and 

obtain the points in elliptic curve 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) over 𝑄𝑝 by considering the lift of points in 𝐸(𝐹𝑝) 

to 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2). In section 4, we implement the arithmetic of points on 𝐸(𝑄𝑝) to points on elliptic 

curve 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) over 𝑝-adic field 𝑄𝑝 . 

1.1 Elliptic Curves over a field 𝜿 

Definition 1.1 (Elliptic curve equation over field 𝜅 with 𝐶ℎ𝑎𝑟 𝜅 ≠ 2,3).  For any field 𝜅 with 

characteristic 𝜅 ≠ 2,3 the elliptic curve 𝐸 over 𝜅 is denoted by 𝐸(𝜅) and is given as 

𝐸(𝜅) = {(𝛼, 𝛽) ∈ 𝜅 × 𝜅/𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵} ∪ {𝒪} 

Where {𝒪} is the point at infinity and 𝐴,𝐵 ∈ 𝜅 such that the discriminant 𝛥 = −(4𝐴3 + 27𝐵2) ≠ 0. 

The equation 𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 is called Weierstrass equation. 



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Definition 1.2 (Elliptic curve equation over field 𝜅 with 𝐶ℎ𝑎𝑟 𝜅 = 3).  For any field 𝜅 with 

characteristic 𝜅 = 3 the elliptic curve 𝐸 over 𝜅 is denoted by 𝐸(𝜅) and is given as 

                      𝐸(𝜅) = {(𝛼, 𝛽) ∈ 𝜅 × 𝜅/𝛽2 = 𝛼3 + 𝐴𝛼2 + 𝐵𝛼 + 𝐶} ∪ {𝒪}  

Where {𝒪} is the point at infinity and 𝐴,𝐵, 𝐶 ∈ 𝜅 such that the discriminant 𝛥 = −4𝐴3𝐶 + 𝐴2𝐵2 +

18𝐴𝐵𝐶 − 4𝐵3 − 27𝐶2 ≠ 0. 

Definition 1.3 (Elliptic curve equation over field 𝜅 with 𝐶ℎ𝑎𝑟 𝜅 = 2).  For any field 𝜅 with 

characteristic 𝜅 = 2 the elliptic curve 𝐸 over 𝜅 is denoted by 𝐸(𝜅) and is given as  

                              𝐸(𝜅) = {(𝛼, 𝛽) ∈ 𝜅 × 𝜅/𝛽2 + 𝛼𝛽 = 𝛼3 + 𝑎2′𝛼
2 + 𝑎6′} ∪ {𝒪}  

𝑜𝑟 

𝐸(𝜅) = {(𝛼, 𝛽) ∈ 𝜅 × 𝜅/𝛽2 + 𝑎3′𝛽 = 𝛼
3 + 𝑎2′𝛼

2 + 𝑎6′} ∪ {𝒪} 

Where {𝒪} is the point at infinity and 𝑎2′, 𝑎3′, 𝑎4′, 𝑎6′ ∈ 𝜅 such that 𝑎3′ ≠ 0 and 𝑎6′ ≠ 0. 

Remark 1.  The discriminant 𝛥 ≠ 0 assures that the roots of the cubic equation 

𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 

are distinct as 𝛥 = ((𝑒1 − 𝑒2)(𝑒2 − 𝑒3)(𝑒3 − 𝑒1))
2
 for 𝑒1, 𝑒2, 𝑒3 are the cube roots and basing on this 

point the arithmetic on elliptic curve 𝐸(𝜅) is established. 

Example 1.1.  Find the points on an elliptic curve 𝐸: 𝑦2 = 𝑥3 + 𝑥 + 1 over 𝐹5. 

In finding points on 𝐸, we first consider all the possible values of 𝑥 which are 0,1,2,3,4 and then find 

𝑦 which is a square of 𝑥3 + 𝑥 + 1(𝑚𝑜𝑑5) and the following table represents the points in 𝐸(𝐹5). 

𝒙 𝒙𝟑 + 𝒙𝟐 + 𝟏 𝒚 Points on 𝑬(𝑭𝟓) 

0 1 1,4 (0,4), (0,1) 

1 3 - - 

2 1 1,4 (2,1), (2,4) 

3 1 1,4 (3,1), (3,4) 

𝒪 𝒪 𝒪 𝒪 

 

The points in 𝐸(𝐹5) are {(0,1), (0,4), (2,1), (2,4), (3,1), (3,4)(4,2), (4,3)} ∪ {𝒪}. 

1.2 Arithmetic on elliptic curve over a field 𝜿 

The hidden beauty of ECC lies in adding two points on elliptic curve in such a way that it is completely 

different from any other point additions that are generally used. The addition law on elliptic curves is 

explained geometrically below. Let us suppose an elliptic curve over a field of characteristic 𝜅 ≠ 2,3 

then the curve equation over the field 𝜅 is given as 

𝐸(𝜅): 𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 



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Let 𝑃1 =(𝛼1, 𝛽1) and 𝑃2 =(𝛼2, 𝛽2) be two points on the given elliptic curve E. Draw a line 𝐿 through 

𝑃1 and 𝑃2, then 𝐿 intersects 𝐸 in a third point 𝑃3′ as 𝛥 ≠ 0. Reflect 𝑃3′ across 𝑋-axis to obtain 𝑃3 . 

Now we define the sum of 𝑃1 and 𝑃2 as 𝑃3 and is denoted as 𝑃1 + 𝑃2 = 𝑃3 given as 𝑃3 = (𝛼3, 𝛽3) as 

shown in fig. 1. 

The set of points on an elliptic curve over a field 𝜅 forms a group which is also abelian with respect to 

point addition defined above.  

 

Figure 1: Geometric Interpretation of Point Addition on curve of the form β2 = α3 +Aα + B 

The point addition and doubling formula for points on an elliptic curve on a field 𝜅 with 𝐶ℎ𝑎𝑟𝜅 = 2 

and 𝐶ℎ𝑎𝑟𝜅 = 3 and 𝐶ℎ𝑎𝑟𝜅 ≠ 2,3 under different conditions at points 𝑃1 and 𝑃2 is given in the table 

below. 

Field  

κ 
           Elliptic curve 

             Slope 𝒎                                            𝑷𝟏 + 𝑷𝟐 = 𝑷𝟑 

 P1 ≠ 

P2 
   𝑃1 = 𝑃2 

 P1 ≠ P2 

∝1≠∝2 

P1≠ 

P2 

∝1

=∝2 

𝑃1  = 𝑃2  

Char  

κ  ≠ 

2,3  

𝛽2 = ∝3+𝐴 ∝ +𝐵 
𝛽2−𝛽1

∝2−∝1
   

3 ∝1
2+ 𝐴

2𝛽1
 

∝3=𝑚
2─ ∝1 ─ ∝2 

𝛽3 = 𝑚(∝1−∝3) ─ 𝛽1 
𝒪 

∝3= 𝑚
2─2 ∝1 

𝛽3 = 𝑚(∝1−∝3) 

─ 𝛽1 

Char  

κ = 3 

𝛽2 = ∝3+𝐴 ∝2+𝐵

∝ +𝐶 

𝛽2−𝛽1

∝2−∝1
   
3 ∝1

2+2𝐴 ∝ +𝐵

2𝛽1
 
∝3= 𝑚

2─𝐴─ ∝1 ─ ∝2 

𝛽3 = 𝑚(∝1−∝3) ─ 𝛽1 
𝒪 

∝3= 𝑚
2─𝐴─2 ∝1 

𝛽3 = 𝑚(∝1−∝3) 

─ 𝛽1 

Char  

κ  = 2 

𝛽2+∝ 𝛽 = ∝3+𝑎2 ∝
2

+ 𝑎6 

               or 

𝛽2 + 𝑎′3𝛽

= ∝3+ 𝑎′2 ∝
2+𝑎′6 

𝛽2−𝛽1

∝2−∝1
  

 

𝛽2−𝛽1

∝2−∝1
  

∝1
2+𝛽1
∝1

 

 

∝1
2+ 𝑎4
𝑎3

 

∝3= 𝑚
2 +𝑚 +∝1

+∝2

+ 𝑎2 

𝛽3 = 𝑚(∝1+∝3) 

+∝3+  𝛽1 

𝒪 

 

 

𝒪 

∝3=
∝1
4+𝑎6

∝1
2  

𝛽3=∝ +𝛽 

∝3=
∝1
4+𝑎4

2

𝑎3
2  

𝑦3=𝑎3 + 𝛽 



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∝3=𝑚
2 +∝1+∝2 

𝛽3 = 𝑚(∝1+∝3) 

+∝3+ 𝑎3 

Table 1: Table for Arithmetic of Points in E(κ) 

2. 𝒑-adic numbers 

Definition 2.1 (p-adic valuation).  The 𝑝-adic valuation 𝑣𝑝(𝛼) is given as for any 𝛼 ∈ 𝑄∗ and 𝑥 =

𝑝𝜌 .
𝑚

𝑛
 with 𝑚,𝑛, 𝜌 ∈ 𝑍, 𝑝 is a prime, and 𝑝 ∤ 𝑚𝑛. Also, 𝑣𝑝(0) = ∞. 

Remark 2.  The p-adic valuation satisfies the following properties: 

For any 𝑎, 𝑏 ∈ 𝑄 

1. 𝑣𝑝(𝑎𝑏) = 𝑣𝑝(𝑎) + 𝑣𝑝(𝑏) 

2. 𝑣𝑝(𝑎 + 𝑏) ≥ 𝑚𝑖𝑛{𝑣𝑝(𝑎), 𝑣𝑝(𝑏)} with 𝑣𝑝(𝑎) ≠ 𝑣𝑝(𝑏) 

3. 𝑣𝑝(𝑎) = ∞ if and only if 𝑥 = 0 

Definition 2.2 (𝑝-adic norm).  Let 𝑝 be a prime and 𝛼 ∈ 𝑄 then 𝑝-adic norm is given as 

             |𝛼|𝑝 = {𝑝
−𝑣𝑝(𝛼)   if  𝛼 ≠ 0
0        if    𝛼 = 0

 𝑝-adic norm |𝑥|𝑝 is non-archimedean norm of 𝑥 on 𝑄. 

Definition 2.3 (𝑝-adic numbers).  For any fixed prime 𝑝. The completion of 𝑄 with respect to 𝑝-adic 

norm | |𝑝 is denoted as 𝑄𝑝 which is called the field of 𝑝-adic numbers. 

we have 𝑄𝑝 as a field of characteristic 0. 

Proposition 1.  If 𝛼 ∈ 𝑄𝑝 then there exists a unique sequence of integers 𝛼𝑖’s with 0 ≤ 𝑥𝑖 ≤ 𝑝 − 1 

and 𝑥𝑖 = 0 for 𝑖 sufficiently negative such that 𝛼 = ∑ 𝛼𝑖
∞
𝑖=−∞ 𝑝𝑖 

Note 1.  The partial sums of the series 𝛼 = ∑ 𝛼𝑖
∞
𝑖=−∞ 𝑝𝑖 form a Cauchy sequence and 𝑥 is the limit of 

this sequence. 

Remark 3.  Every 𝑥 ∈ 𝑄𝑝 has unique representation depending on its 𝑝-adic valuation |𝑥|𝑝 i.e., either 

|𝑥|𝑝 ≥ 1 or |𝑥|𝑝 ≤ 1. 

If 𝑥 ∈ 𝑄𝑝 with |𝑥|𝑝 ≤ 1, we can represent 𝑥 as a sequence given as 

𝑥 = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. . . +𝑥𝑘−1𝑝

𝑘−1 = ∑𝑥𝑛

∞

𝑛=0

𝑝𝑛 

where 𝑥𝑖 ∈ 0,1,2, . . . 𝑝 − 1. 

If 𝑥 ∈ 𝑄𝑝 with |𝑥|𝑝 ≥ 1, we can represent 𝑥 as a sequence given as 

𝑥 =. . . +𝑥−2𝑝
−2 + 𝑥−1𝑝

−1 + 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. . . +𝑥𝑘−1𝑝

𝑘−1 = ∑ 𝑥𝑛

∞

𝑛=−∞

𝑝𝑛 

where 𝑥𝑖 ∈ 0,1,2, . . . 𝑝 − 1. 



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Remark 4.  For every 𝑥 ∈ 𝑄𝑝 with 𝑥 = ∑ 𝑥𝑛
∞
𝑛=−𝑚 𝑝𝑛, the canonical expression of 𝑎 is given as 

𝑥 =. . . 𝑥𝑛 . . . 𝑥2𝑥1𝑥0. 𝑥−1𝑥−2𝑥−3. . . 𝑥−𝑚 

Definition 2.4 (𝑝-adic integers 𝑍𝑝).  A 𝑝-adic number 𝑥 ∈ 𝑄𝑝 is said to be 𝑝-adic integer if its 

canonical expression contains only non-negative powers of 𝑝. Set of 𝑝-adic integers is denoted by 𝑍𝑝 

which is a sub-ring of 𝑄𝑝. 

𝑍𝑝 = {𝑥 ∈ 𝑄𝑝 𝑤𝑖𝑡ℎ |𝑥| ≤ 1} 

                                                               = {𝑥 ∈ 𝑄𝑝 𝑤𝑖𝑡ℎ 𝑣𝑝(𝑥) ≥ 0} 

One can also perform arithmetic operations like addition, subtraction, multiplication and division of 

any two 𝑝-adic numbers. 

2.1 Arithmetic operations on 𝒑-adic numbers. 

Every number 𝛼 ∈ 𝑄𝑝 for 𝑝 being a prime has its 𝑝-adic expansion given as 

𝛼 = . . . +𝛼−2𝑝
−2 + 𝛼−1𝑝

−1 + 𝛼0 + 𝛼1𝑝 + 𝛼2𝑝
2 + 𝛼3𝑝

3+. .. 

while some are finite expansions including only positive powers of 𝑝 in its expansion. 

Example 2.1.1.  1. 320 = 5 + 3 × 7 + 6 × 72 = 635 in 𝑄7 

2. 108 = 3 + 1 × 7 + 2 × 72 = 213 in 𝑄7 

3. 
1

2
= 3+ 2 × 5 + 2 × 52+. . . = ...2223 in 𝑄5 

Also, negative of 𝑥 ∈ 𝑄𝑝 is given as −𝑥 = 𝑥 × (−1) with its 𝑝-adic expansion given as 

−1 = (𝑝 − 1) + (𝑝 − 1) × 𝑝 + (𝑝 − 1) × 𝑝2+. .. 

i.e., −1 = 4 + 4.5 + 4.52+. .. in 𝑄5 

Arithmetical operations in 𝑄𝑝 extend ordinary arithmetic operations on Natural numbers 𝑁. 𝑝-adic 

addition and multiplication are performed from right to left with a carry. Also, 𝑝-adic division will 

be performed from right to left but different from long division as in 𝑁. 

Example 2.1.2.  Addition, Multiplication and Division in 7-adic field 𝑄7 are given below for 320 
and 108 expanded in 𝑄7 in above examples. 

320 + 108 = 1151 given as 
5 + 3 × 7 + 6 × 72 

+ 
3 + 1 × 7 + 2 × 72 

−− −− −− −− −−− −− −− −− 
1 + 5 × 7 + 1 × 72 + 1 × 73 

−− −− −− −− −−− −− −− −− 
320 − 108 = 422 given as 

5 + 3 × 7 + 6 × 72 
− 

3 + 1 × 7 + 2 × 72 
−− −− −− −− −−− −− −− −− 



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2 + 2 × 7 + 4 × 72 
−− −− −− −− −−− −− −− −− 

Also, 320 × 108 = 202521 given as 
5 + 3 × 7 + 6 × 72 

×             3+ 1 × 7 + 2 × 72 
−− −− −− −− −− −−− −− −− −− −− 

1 + 4 × 7 + 5 × 72 + 2 × 73 
       5 × 7 + 3 × 72 + 6 × 73 

                                                         3 × 72 + 0 × 73 + 6 × 74 + 1 × 75 
− −− −− −−− −− −− −− −− −− −−− −− − 

                            1 + 2 × 7 + 5 × 72 + 2 × 73 + 0 × 74 + 2 × 75 
− −− −− −−− −− −− −− −− −− −−− −− − 

Also, 
302

108
= ....244 

3 + 1 × 7 + 2 × 72  )  5 + 3 × 7 + 6 × 72  (  4 + 4 × 7 + 2 × 72 
                  5 + 5 × 7 + 1 × 72 + 1 × 73 

−− −− −− −− −− −− −−− −− −− −− −− −− 
                                                  5 × 7 + 4 × 72 + 6 × 73 + 6 × 74 +⋯ 

                                        5 × 7 + 5 × 72 + 1 × 73 + 1 × 74 
  − − − −− −− −− −− −− −−− −− −− −− −− −− − 
                                                              6 × 72 + 4 × 73 + 6 × 74 +⋯ 

                                                     6 × 72 + 2 × 73 + 4 × 74 
−− −− −− −− −− −− −− −−− −− −− −− −− −− −−− −− −  

                                                                                              2 × 73 + 0 × 74 + 6 × 75 +⋯  
−− −− −− −− −− −−− −− −− −− −− −− −−− −− −− −− −− 

3. Elliptic curves 𝑬(𝑸𝒑) over 𝒑-adic field 𝑸𝒑 and points in 𝑬(𝑸𝒑)(𝒎𝒐𝒅 𝒑𝒏) for 𝒏 > 𝟏 

In the context of studying point addition on 𝐸(𝑄𝑝), we have to study the relation between the points 

in 𝐸(𝑄𝑝) and the points in 𝐸(𝐹𝑝). We first proceed to study the reduction of points on 𝐸(𝑄𝑝) to 𝑚𝑜𝑑𝑝. 

for any element �̃� ∈ 𝑍𝑝, there is a natural reduction �̃� → 𝑥 from 𝑍𝑝 → 𝐹𝑝. But, such reduction cannot 

be extended from 𝑄𝑝 to 𝐹𝑝 as reduction is not injection whereas any ring homomorphism from 𝑄𝑝 

would always be an injection. Therefore, an elliptic curve over 𝑄𝑝 in general cannot be reduced to 𝐹𝑝 

. But, note as for any 𝑥 ∈ 𝑍𝑝, as 𝑥 can be reduced to an element in 𝐹𝑝 naturally, to reduce an elliptic 

curve over 𝑄𝑝, we need to consider 𝐸(𝑄𝑝) to be defined over 𝑍𝑝 

For an elliptic curve defined over 𝑍𝑝, the point �̃� = (�̃�, �̃�) ∈ 𝐸(𝑄𝑝) defined over 𝑍𝑝 is such that either 

(�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝 or (�̃�, �̃�) ∉ 𝑍𝑝 × 𝑍𝑝 i.e., (�̃�, �̃�) ∈ 𝑍𝑝 × 𝑄𝑝 or (�̃�, �̃�) ∈ 𝑄𝑝 × 𝑍𝑝 or (�̃�, �̃�) ∈

𝑄𝑝 × 𝑄𝑝. Note for (�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝, we have �̃� = 𝛼0 + 𝛼1𝑝 + 𝛼2𝑝
2+. .. and �̃� = 𝛽0 + 𝛽1𝑝 +

𝛽2𝑝
2+. .. , Note there is a natural reduction to 𝐹𝑝(𝑚𝑜𝑑𝑝) i.e., (�̃�, �̃�) → (𝛼0, 𝛽0) from 𝐸(𝑄𝑝) → 𝐸(𝐹𝑝). 

Note for (�̃�, �̃�) ∉ 𝑍𝑝 × 𝑍𝑝, we consider the corresponding projective co-ordinates of 

�̃� = [𝐴:𝐵: 𝛤] ∋ 𝐴, 𝐵, 𝛤 ∈ 𝑍𝑝 

and obtain natural reduction �̃� → 𝑃 from 𝐸(𝑄𝑝) → 𝐸(𝐹𝑝). 



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Hence, for an elliptic curve 𝐸(𝑄𝑝) defined over 𝑍𝑝, its points can always be reduced to points in 𝐸(𝐹𝑝), 

the elliptic curve over a finite field 𝐹𝑝. However if our elliptic curve 𝐸(𝑄𝑝) is not defined over 𝑍𝑝, 

there is an elliptic curve 𝐸′(𝑄𝑝) such that 𝐸′ ≅ 𝐸 where 𝐸′ is the elliptic curve defined over 𝑍𝑝 i.e., 

the coefficients of curve 𝐸(𝑄𝑝) are in 𝑍𝑝 under the mapping (𝛼, 𝛽) → (𝑣−2𝛼, 𝑣−3𝛽). 

Hence for further study, without loss of generality, we consider 𝐸(𝑄𝑝) is always defined over 𝑍𝑝. 

Definition 3.1. (Lift of a point).  The set of all lifts of points from 𝐹𝑝 to 𝑍𝑝 is denoted as 𝐿(𝐹𝑝). For 

any (𝑥0, 𝑦0) ∈ 𝐹𝑝 × 𝐹𝑝 , the point (𝛼, 𝛽) = (�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝 is called a lift of a point (𝑥0, 𝑦0) from 𝐹𝑝 

to 𝑍𝑝 and is defined as 

𝐿(𝐹𝑝) = {(�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝/�̃� = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. . . 𝑎𝑛𝑑�̃� = 𝑦0 + 𝑦1𝑝 + 𝑦2𝑝

2+. .. with (𝑥0, 𝑦0) ∈

𝐹𝑝 × 𝐹𝑝} 

Definition 3.2. The set of all lifts of points on elliptic curve over finite field 𝐹𝑝 is given as for any 

(𝛼, 𝛽) = (𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝) the point (�̃�, �̃�) ∈ 𝐸(𝑄𝑝) is called a lift of a point (𝑥0, 𝑦0) from 𝐸(𝐹𝑝) to 

𝐸(𝑄𝑝) and is defined as 

 

𝐿 (𝐸(𝐹𝑝)) = {(�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝/�̃� = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2 +⋯ and �̃� = 𝑦0 + 𝑦1𝑝 + 𝑦2𝑝

2 +⋯ with 

(𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝) and �̃�2 = �̃�3 + 𝐴�̃� + 𝐵} 

Now in the following theorems, we describe the points in 𝐸(𝑄𝑝) defined over 𝑍𝑝 as as lift of each 

point (𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝). 

Theorem 3.1.  Let 𝐸 be an elliptic curve over 𝑄𝑝 defined over 𝑍𝑝 then each point in 𝐸(𝐹𝑝) is a reduction 

of a point in 𝐸(𝑄𝑝). 

Proof. By definition, 

𝐸(𝑄𝑝) = {(𝛼, 𝛽) ∈ 𝑄𝑝 × 𝑄𝑝/𝛽
2 = 𝛼3 + 𝐴𝛼 + 𝐵} ∪ {�̃�} 

𝑎𝑛𝑑 

𝐸(𝐹𝑝) = {(𝛼, 𝛽) ∈ 𝐹𝑝 × 𝐹𝑝/𝛽
2 = 𝛼3 + 𝐴𝛼 + 𝐵} ∪ {�̃�} 

Let 𝑃 = (𝑥0, 𝑦0) be a point in 𝐸(𝐹𝑝) such that 𝑃 ≠ 𝒪, then for 𝑃 = (𝑥0, 𝑦0) we have 𝑦0
2 = 𝑥0

3 +

𝐴𝑥0 + 𝐵. 

Now to find �̃� lift of 𝑃 such that �̃� ∈ 𝐸(𝑄𝑝). 

Let �̃� = (�̃�, �̃�) ∈ 𝑍𝑝 × 𝑍𝑝 be a lift of 𝑃 = (𝑥0, 𝑦0), then we have �̃�, �̃� are 𝑝-adic integers having the 

form 

�̃� = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. .. 

�̃� = 𝑦0 + 𝑦1𝑝 + 𝑦2𝑝
2+. .. 



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with 𝑥0, 𝑦0, 𝑥1, 𝑦1, 𝑥2, 𝑦2, . . . ∈ 𝐹𝑝. Now if �̃� = (�̃�, �̃�) ∈ 𝐸(𝑄𝑝) then �̃�2 = �̃�3 + 𝐴�̃� + 𝐵, i.e., for �̃� =

(𝑥0 + 𝑝𝑥1 + 𝑝
2𝑥2+. . . , 𝑦0 + 𝑝𝑦1 + 𝑝

2𝑦2+. . . ) we have 

(𝑦0 + 𝑦1𝑝 + 𝑦2𝑝
2+. . . )2 = (𝑥0 + 𝑥1𝑝 + 𝑥2𝑝

2+. . . )3 + 𝐴(𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. . . ) + 𝐵 

𝑦0
2 + 𝑦1

2𝑝2 + 2𝑦0𝑦1𝑝 + 𝑝
2𝑡 = 𝑥0

3 + 3𝑥0
2𝑥1𝑝 + 3𝑥0𝑥1

2𝑝2 + 𝑝3𝑥1
3 + 𝐴𝑥0 + 𝐴𝑝𝑥1 + 𝐵 + 𝑝

2𝑘 

let 𝑃1̃ be the lift of 𝑃 modulo 𝑝2 given as 𝑃1̃ = (�̃�1, �̃�1) with 

�̃�1 = 𝑥0 + 𝑥1𝑝 

�̃�1 = 𝑦0 + 𝑦1𝑝 

note �̃�1 ∈ 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) 

𝑦0
2 + 𝑦1

2𝑝2 + 2𝑦0𝑦1𝑝 = 𝑥0
3 + 3𝑥0

2𝑥1𝑝 + 3𝑥0𝑥1
2𝑝2 + 𝑝3𝑥1

3 + 𝐴𝑥0 + 𝐴𝑝𝑥1 + 𝐵 

𝑦0
2 + 2𝑝𝑦0𝑦1 ≡ 𝑥0

3 + 3𝑥0
2𝑥1𝑝 + 𝐴𝑥0 + 𝐴𝑝𝑥1 + 𝐵(𝑚𝑜𝑑 𝑝2) 

Now as (𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝) note 𝑦0
2 = 𝑥0

3 + 𝐴𝑥0 + 𝐵, substituting in above equation, we have 

2𝑝𝑦0𝑦1 ≡ 3𝑥0
2𝑥1𝑝 + 𝐴𝑝𝑥1(𝑚𝑜𝑑 𝑝2) 

As 𝑥0, 𝑦0 are known, we can obtain 𝑦1 in terms of 𝑥1 by assigning values for 𝑥1 in 𝐹𝑝 , Therefore 

𝑦1 ≡ (2𝑝𝑦0)
−1(3𝑥0

2𝑥1𝑝 + 𝐴𝑝𝑥1)(𝑚𝑜𝑑 𝑝2) 

𝑃1̃ = (�̃�1, �̃�1) = (𝑥0 + 𝑥1𝑝, 𝑦0 + 𝑦1𝑝) = (𝑥0 + 𝑝𝑥1, 𝑦0 + 𝑝(2𝑝𝑦0)
−1(3𝑥0

2𝑥1𝑝 + 𝐴𝑝𝑥1))(𝑚𝑜𝑑 𝑝2) 

Note �̃�1
2 = �̃�1

3 + 𝐴�̃�1 + 𝐵(𝑚𝑜𝑑 𝑝2) 

Consider (�̃�1)
2 = (𝑦0 + 𝑝(2𝑝𝑦0)

−1(3𝑥0
2𝑥1𝑝 + 𝐴𝑝𝑥1))

2
≡  (�̃�)2(𝑚𝑜𝑑 𝑝2) 

But note we have 

�̃�1
3 + 𝐴�̃�1 + 𝐵 =  (𝑥0 + 𝑥1𝑝)

3 + 𝐴(𝑥0 + 𝑥1𝑝) + 𝐵 

                                                           =  𝑥0
3 + 𝑥1

3𝑝3 + 3𝑝2𝑥0𝑥1
2 + 3𝑝𝑥0

2𝑥1 + 𝐴𝑥0 + 𝐴𝑝𝑥1 + 𝐵 

           ≡  𝑥0
3 + 𝐴𝑥0 + 𝐵(𝑚𝑜𝑑 𝑝2) 

≡  𝑦0
2(𝑚𝑜𝑑 𝑝2)       

≡  (�̃�)2(𝑚𝑜𝑑 𝑝2)    

Therefore, �̃�1
2 = �̃�1

3 + 𝐴�̃�1 + 𝐵(𝑚𝑜𝑑 𝑝2). 

Hence, (�̃�1, �̃�1) satisfies the given curve 𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵(𝑚𝑜𝑑 𝑝2), Now repeating the above 

argument for 𝑚𝑜𝑑 𝑝3 and using (�̃�1, �̃�1) ∈ 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2). let 𝑃2̃ be the lift of 𝑃 modulo 𝑝3 given 

as 𝑃2̃ = (�̃�2, �̃�2) with 

�̃�2 = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2 

�̃�2 = 𝑦0 + 𝑦1𝑝 + 𝑦2𝑝
2 

note �̃�2 ∈ 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝3) 

As 𝑥0, 𝑥1, 𝑦0 , 𝑦1 are known, we can obtain 𝑦2 in terms of 𝑥2 by assigning values for 𝑥2 in 𝐹𝑝 . Also 



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we can see that (�̃�2, �̃�2) satisfies the given curve 𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 modulo 𝑝3. Proceeding so on, 

let 𝑃�̃� be the lift of 𝑃 modulo 𝑝𝑖+1 given as 𝑃�̃� = (�̃�𝑖, �̃�𝑖) with 

�̃�𝑖 = 𝑥0 + 𝑥1𝑝 + 𝑥2𝑝
2+. . . +𝑥𝑖𝑝

𝑖 

�̃�𝑖 = 𝑦0 + 𝑦1𝑝 + 𝑦2𝑝
2+. . . +𝑦𝑖𝑝

𝑖 

note �̃�2 ∈ 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝𝑖+1) 

As 𝑥0, 𝑥1, . . . 𝑥𝑖−1, 𝑦0, 𝑦1, . . . 𝑦𝑖−1 are known, we can obtain 𝑦𝑖 in terms of 𝑥𝑖 by assigning values for 

𝑥𝑖 in 𝐹𝑝 . Also we can see that (�̃�𝑖, �̃�𝑖) satisfies the given curve 𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 modulo 𝑝𝑖+1. 

Hence, (�̃�, �̃�) = (𝑥0 + 𝑝𝑥1 + 𝑝
2𝑥2+. . . , 𝑦0 + 𝑝𝑦1 + 𝑝

2𝑦2+. . . ) satisfies the curve 𝛽2 = 𝛼3 + 𝐴𝛼 +

𝐵. i.e., the lift �̃� of 𝑃 is in 𝐸(𝑄𝑝). 

Therefore, for 𝑃 ≠ 𝒪 ∈ 𝐸(𝐹𝑝) there lies a lift �̃� in 𝑍𝑝 × 𝑍𝑝 such that �̃� ∈ 𝐸(𝑄𝑝), with reduction of �̃� 

is 𝑃 itself. 

Hence, for all 𝑃 ∈ 𝐸(𝐹𝑝) such that 𝑃 ≠ 𝒪, 𝑃 is the reduction of some �̃� ∈ 𝐸(𝑄𝑝) with �̃� in 𝑍𝑝 × 𝑍𝑝. 

Now for 𝑃 = 𝒪, the lift of point at infinity 𝒪 can be obtained by considering the corresponding 

elliptic curve involving 𝑍-coordinate is as 𝑌2𝑍 = 𝑋3 + 𝐴𝑋𝑍2 + 𝐵𝑍3. Now considering the point at 

infinity 𝒪 = [0:1: 0] which is an equivalence class of points (0, 𝑘, 0). We have for 𝒪 as (0, 𝑘, 0), the 

lift of 𝑃 = 𝒪 = (0, 𝑘, 0) is given as 

�̃� = [�̃�, �̃�, 𝑍] = (0 + 𝑋1𝑝 + 𝑋2𝑝
2+. . . , 𝑘 + 𝑌1𝑝 + 𝑌2𝑝

2+. . . ,0 + 𝑍1𝑝 + 𝑍2𝑝
2+. . . ) 

such that �̃� satisfies the curve 𝑌2𝑍 = 𝑋3 + 𝐴𝑋𝑍2 + 𝐵𝑍3. Hence, we have 

(𝑘 + 𝑌1𝑝 + 𝑌2𝑝
2+. . . )2(0 + 𝑍1𝑝 + 𝑍2𝑝

2+. . . ) =                                                                       

(0 + 𝑋1𝑝 + 𝑋2𝑝
2+. . . )3 + 𝐴(0 + 𝑋1𝑝 + 𝑋2𝑝

2+. . . )(0 + 𝑍1𝑝 + 𝑍2𝑝
2+. . . )2 + 𝐵(0 + 𝑍1𝑝 + 𝑍2𝑝

2+. . . )3
 

On reducing 𝑚𝑜𝑑𝑝2, we have 

(𝑘 + 𝑌1𝑝)
2(0 + 𝑍1𝑝) = (0 + 𝑋1𝑝)

3 + 𝐴(0 + 𝑋1𝑝)(0 + 𝑍1𝑝)
2 + 𝐵(0 + 𝑍1𝑝)

3(𝑚𝑜𝑑 𝑝2) 

(𝑘2 + 𝑌1
2𝑝2 + 2𝑘𝑌1𝑝)𝑍1𝑝 ≡ 𝑋1

3 + 𝐴𝑋1𝑍1𝑝
2 + 𝐵𝑍1𝑝

3(𝑚𝑜𝑑 𝑝2) 

𝑘2𝑍1𝑝 ≡ 0(𝑚𝑜𝑑 𝑝2) 

𝑍1𝑝 ≡ 0(𝑚𝑜𝑑 𝑝2) 

𝑍1 ≡ 0(𝑚𝑜𝑑 𝑝) 

Also, note 𝑍1 ≡ 0(𝑚𝑜𝑑 𝑝) ⇒ 𝑋1 ≡ 0(𝑚𝑜𝑑 𝑝) 

Therefore, substituting in �̃� we have for �̃�(𝑚𝑜𝑑 𝑝2) = �̃�1 say ⇒ �̃�1 = (0, 𝑘 + 𝑝𝑌1, 0)(𝑚𝑜𝑑 𝑝2) 

Proceeding as above for 𝑚𝑜𝑑 𝑝3, we have 

�̃�2 = (0, 𝑘 + 𝑌1𝑝 + 𝑌2𝑝
2, 0)(𝑚𝑜𝑑 𝑝3) 

On continuing so on, we have 𝑍𝑛 ≡ 0(𝑚𝑜𝑑 𝑝) for 𝑛 = 0,1,2, . .. Hence, 

�̃� = (0, 𝑘 + 𝑌1𝑝 + 𝑌2𝑝
2+. . . ,0) 

  = (0,1 + 𝑌1𝑝 + 𝑌2𝑝
2+. . . ,0) 



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Therefore, 

�̃� = [�̃�, �̃�, �̃�] = (0,1 + 𝑌1𝑝 + 𝑌2𝑝
2+. . . ,0) 

for all 𝑌𝑖 ∈ 𝐹𝑝 are the lifts of point at infinity 𝒪 ∈ 𝐸(𝐹𝑝) and is denoted as �̃�. 

Therefore, for 𝑃 = 𝒪 ∈ 𝐸(𝐹𝑝), there exists the lift �̃� ∈ 𝐸(𝑄𝑝) such that 𝒪 is the reduction of a point 

�̃� ∈ 𝐸(𝑄𝑝). 

Therefore, each point in 𝐸(𝐹𝑝) is a reduction of a point in 𝐸(𝑄𝑝).  

Theorem 3.2.  Let 𝐸 be an elliptic curve over 𝑄𝑝 defined over 𝑍𝑝 then each point in 𝐸(𝑄𝑝) defined 

over 𝑍𝑝 is a lift of a point in 𝐸(𝐹𝑝). 

Proof. By definition 

 

𝐸(𝑄𝑝) = {(𝛼, 𝛽) ∈ 𝑄𝑝 × 𝑄𝑝/𝛽
2 = 𝛼3 + 𝐴𝛼 + 𝐵} ∪ {�̃�} 

𝑎𝑛𝑑 

𝐸(𝐹𝑝) = {(𝛼, 𝛽) ∈ 𝐹𝑝 × 𝐹𝑝/𝛽
2 = 𝛼3 + 𝐴𝛼 + 𝐵} ∪ {�̃�} 

Now, note for any point �̃� = (�̃�, �̃�) ∈ 𝐸(𝑄𝑝) then we have two cases. 

(i) �̃� ∈ 𝑍𝑝 × 𝑍𝑝 (ii) �̃� ∉ 𝑍𝑝 × 𝑍𝑝. 

case (i): for �̃� ∈ 𝑍𝑝 × 𝑍𝑝, we have 

�̃� = (𝑥0 + 𝑝𝑥1 + 𝑝
2𝑥2+. . . , 𝑦0 + 𝑝𝑦1 + 𝑝

2𝑦2+. . . ) 

Reduction of �̃� is �̃�(𝑚𝑜𝑑 𝑝) = (𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝) Hence, �̃� is a lift of (𝑥0, 𝑦0) ∈ 𝐸(𝐹𝑝). 

case(ii): if �̃� ∉ 𝑍𝑝 × 𝑍𝑝 then we have either both �̃�, �̃� ∉ 𝑍𝑝 or exactly one of �̃�, �̃� is in 𝑍𝑝. 

Now for �̃� = (�̃�, �̃�) with both �̃�, �̃� ∉ 𝑍𝑝, �̃� is the lift of point at infinity 𝒪 ∈ 𝐸(𝐹𝑝), follows from [8]. 

Now for �̃� ∉ 𝑍𝑝 × 𝑍𝑝 with exactly one of �̃�, �̃� is in 𝑍𝑝 then we have(�̃�, �̃�) such that either �̃� ∈

𝑍𝑝, �̃� ∉ 𝑍𝑝 or �̃� ∉ 𝑍𝑝, �̃� ∈ 𝑍𝑝. 

then, if �̃� is not reduction of point at infinity 𝒪 ∈ 𝐸(𝐹𝑝) then �̃� reduction is of the form [𝑋, 𝑌, 𝑍] with 

𝑍 ≠ 0 ⇒ (𝑋, 𝑌, 𝑍) = (𝑥, 𝑦, 1) 

⇒ �̃� is the lift of (𝛼, 𝛽) ∈ 𝐸(𝐹𝑝), then note (�̃�, �̃�) is a lift of (𝛼, 𝛽) ∈ 𝐸(𝐹𝑝). But any lift of (𝛼, 𝛽) ∈

𝐸(𝐹𝑝) are in 𝑍𝑝 × 𝑍𝑝, which is a contradiction. 

Hence, there are no �̃� ∈ 𝐸(𝑄𝑝) such that �̃� = (�̃�, �̃�) ∉ 𝑍𝑝 × 𝑍𝑝 such that exactly one of �̃�, �̃� ∉ 𝑍𝑝. 

Hence, for all �̃� ∈ 𝐸(𝑄𝑝), �̃� is a lift of a point 𝑃 in 𝐸(𝐹𝑝). 

Remark 5.  We can obtain all the points in 𝐸(𝑄𝑝) by starting with points in 𝐸(𝐹𝑝) and lifting each 

point in 𝐸(𝐹𝑝) to 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) as described in theorem 3.1 and lifting of each point in 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) to 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝3) and so on. The lifting of points from 𝐸(𝐹𝑝) to obtain points in 

𝐸(𝑄𝑝) is depicted in the following figure 2. 



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Figure 2: Flowchart for points in 𝐸(𝑄𝑝) 

The points in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) which are obtained by lifting of points in 𝐸(𝐹𝑝) are given in the 

following example for 𝑝 = 5. 

Example 3.1.  Consider the elliptic curve 𝑦2 = 𝑥3 + 𝑥 + 1 , then we have 

𝐸(𝐹5) = {(0,1), (0,4), (2,1), (2,4), (3,1), (3,4)(4,2), (4,3)} ∪ {𝒪} 

A lift of any point in 𝐸(𝐹5) to a point in 𝑍5 × 𝑍5(𝑚𝑜𝑑5
2) is given as in the following process: 

       Let 𝑃 = (2,1) ∈ 𝐸(𝐹5). Lift 𝑃(2,1) to �̃� = (2 + 𝑝𝑥1, 1 + 𝑝𝑦1)(𝑚𝑜𝑑5
2), 0 ≤ 𝑥1, 𝑦1 < 5  

then �̃� ∈ 𝐸(𝑄5)(𝑚𝑜𝑑5
2) then as 𝑃 satisfies 𝑦2 = 𝑥3 + 𝑥 + 1 , note 

(1 + 𝑝𝑦1)
2 = (2 + 𝑝𝑥1)

3 + 2 + 𝑝𝑥1 + 1(𝑚𝑜𝑑5
2) 

Now expressing 𝑦1 in terms of 𝑥1, we have 

𝑦1 ≡ 𝑥1 + 4(𝑚𝑜𝑑5) 

Substituting for 𝑦1 in �̃� = (2 + 𝑝𝑥1, 1 + 𝑝𝑦1)(𝑚𝑜𝑑5
2), we have 

�̃� = (2 + 5𝑥1, 21 + 5𝑥1)(𝑚𝑜𝑑5
2) 

Each point in 𝐸(𝐹5) may be lifted in a similar manner to obtain points in 𝐸(𝑄5)(𝑚𝑜𝑑5
2) as given in 

the table below. 



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Points in 𝑬(𝑭𝟓) Lifting point in 𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓
𝟐) 

𝑃1 = (0,1) (5𝑥1, 1 + 3𝑥1. 5)(𝑚𝑜𝑑5
2) 

𝑃2 = (0,4) (5𝑥1, 4 + (2𝑥1 + 4). 5)(𝑚𝑜𝑑5
2) 

𝑃3 = (2,1) (2 + 5𝑥1, 1 + (4𝑥1 + 1). 5)(𝑚𝑜𝑑5
2) 

𝑃4 = (2,4) (2 + 5𝑥1, 4 + (𝑥1 + 3). 5)(𝑚𝑜𝑑5
2) 

𝑃5 = (3,1) (3 + 5𝑥1, 1 + (3 + 4𝑥1). 5)(𝑚𝑜𝑑5
2) 

𝑃6 = (3,4) (3 + 5𝑥1, 4 + (𝑥1 + 1). 5)(𝑚𝑜𝑑5
2) 

𝑃7 = (4,2) (4 + 5𝑥1, 2 + 5(2 + 𝑥1))(𝑚𝑜𝑑5
2) 

𝑃8 = (4,3) (4 + 5𝑥1, 3 + (2 + 4𝑥1). 5)(𝑚𝑜𝑑5
2) 

𝒪 �̃� 

 

Figure 3: Lifting of Points in 𝐸(𝐹5) to 𝐸(𝑄5)(𝑚𝑜𝑑 52) 

Assigning 𝑥1 = 0,1,2,3,4 in the above table, we have all lifting points in 𝐸(𝑄5)(𝑚𝑜𝑑5
2) in the 

following table. 

Lifting point in 

𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓
𝟐) 𝒙𝟏 = 𝟎 𝒙𝟏 = 𝟏 𝒙𝟏 = 𝟐 𝒙𝟏 = 𝟑 𝒙𝟏 = 𝟒 

(5𝑥1, 1 + 3𝑥1. 5)(𝑚𝑜𝑑5
2) (0,1) (1.5,1

+ 3.5) 

(2.5,1

+ 1.5) 

(3.5,1

+ 4.5) 

(4.5,1

+ 2.5) 

(5𝑥1, 4

+ (2𝑥1 + 4). 5)(𝑚𝑜𝑑5
2) 

(0,4

+ 4.5) 

(1.5,4

+ 1.5) 

(2.5,4

+ 3.5) 

(3.5,4) (4.5,4

+ 2.5) 

(2 + 5𝑥1, 1

+ (4𝑥1 + 1). 5)(𝑚𝑜𝑑5
2) 

(2,1

+ 1.5) 

(2 + 1.5,1) (2 + 2.5,1

+ 4.5) 

(2 + 3.5,1

+ 3.5) 

(2 + 4.5,1

+ 2.5) 

(2 + 5𝑥1, 4

+ (𝑥1 + 3). 5)(𝑚𝑜𝑑5
2) 

(2,4

+ 3.5) 

(2 + 1.5,4

+ 4.5) 

(2 + 2.5,4) (2 + 3.5,4

+ 1.5) 

(2 + 4.5,4

+ 2.5) 



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Lifting point in 

𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓
𝟐) 𝒙𝟏 = 𝟎 𝒙𝟏 = 𝟏 𝒙𝟏 = 𝟐 𝒙𝟏 = 𝟑 𝒙𝟏 = 𝟒 

(3 + 5𝑥1, 1

+ (3 + 4𝑥1). 5)(𝑚𝑜𝑑5
2) 

(3,1

+ 3.5) 

(3 + 1.5,1

+ 2.5) 

(3 + 2.5,1

+ 1.5) 

(3 + 3.5,1) (3 + 4.5,1

+ 4.5) 

(3 + 5𝑥1, 4

+ (𝑥1 + 1). 5)(𝑚𝑜𝑑5
2) 

(3,4

+ 1.5) 

(3 + 1.5,4

+ 2.5) 

(3 + 2.5,4

+ 3.5) 

(3 + 3.5,4

+ 4.5) 

(3 + 4.5,4) 

(4 + 5𝑥1, 2

+ 5(2 + 𝑥1))(𝑚𝑜𝑑5
2) 

(4,2

+ 2.5) 

(4 + 1.5,2

+ 3.5) 

(4 + 2.5,2

+ 4.5) 

(4 + 3.5,2) (4 + 4.5,2

+ 1.5) 

(4 + 5𝑥1, 3

+ (2 + 4𝑥1). 5)(𝑚𝑜𝑑5
2) 

(4,3

+ 2.5) 

(4 + 1.5,3

+ 1.5) 

(4 + 2.5,3) (4 + 3.5,3

+ 4.5) 

(4 + 4.5,3

+ 3.5) 

�̃� �̃� �̃� �̃� �̃� �̃� 

 

In order to obtain the lift points E(Q5)(mod5
3), we repeat the above process for considered 

point in E(Q5)(mod5
2) and obtain all lifts in E(Q5)(mod5

3) . On repeating in such a manner, we can 

obtain lift points in E(Q5)(mod5
n) for any positive integer n. The lifting of points from E(F5) to 

E(Q5)(mod5
3) are represented pictorially in fig 4. 

 

Figure 4: Lifting of Points in E(F5) to E(Q5)(mod 52) 

The table below shows the difference in number of points from E(F5) to E(Q5)(mod 52) which 

provides large key space for cryptographic purpose. 

Points in 

𝐸(𝐹5) 
Points in 𝐸(𝑄5)(mod 52) 

(0,1) (0,1) (1.5,1 + 3.5) (2.5,1 + 1.5) (3.5,1 + 4.5) (4.5,1 + 2.5) 

(0,4) 
(0,4

+ 4.5) 
(1.5,4 + 1.5) (2.5,4 + 3.5) (3.5,4) (4.5,4 + 2.5) 

(2,1) 
(2,1

+ 1.5) 
(2 + 1.5,1) 

(2 + 2.5, 1 +

4.5)  

(2 + 3.5,1

+ 3.5) 

(2 + 4.5,1

+ 2.5) 



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Points in 

𝐸(𝐹5) 
Points in 𝐸(𝑄5)(mod 52) 

(2,4) 
(2,4

+ 3.5) 

(2 + 1.5,4

+ 4.5) 
(2 + 2.5,4) 

(2 + 3.5,4

+ 1.5) 

(2 + 4.5,4

+ 2.5) 

(3,1) 
(3,1

+ 3.5) 

(3 + 1.5,1

+ 2.5) 

(3 + 2.5,1

+ 1.5) 
(3 + 3.5,1) 

(3 + 4.5,1

+ 4.5) 

(3,4) 
(3,4

+ 1.5) 

(3 + 1.5,4

+ 2.5) 

(3 + 2.5,4

+ 3.5) 

(3 + 3.5,4

+ 4.5) 
(3 + 4.5,4) 

(4,2) 
(4,2

+ 2.5) 

(4 + 1.5,2

+ 3.5) 

(4 + 2.5,2

+ 4.5) 
(4 + 3.5,2) 

(4 + 4.5,2

+ 1.5) 

(4,3) 
(4,3

+ 2.5) 

(4 + 1.5,3

+ 1.5) 
(4 + 2.5,3) 

(4 + 3.5,3

+ 4.5) 

(4 + 4.5,3

+ 3.5) 

𝒪 �̃� �̃� �̃� �̃� �̃� 

 

4. Implementing Arithmetic of points on elliptic curve 𝑬(𝑸𝒑) over 𝒑-adic field 𝑸𝒑 to 

𝑬(𝑸𝒑)(𝒎𝒐𝒅 𝒑𝟐)  

In the context of improvising the efficiency of cryptosystems with elliptic curves, the purpose of 

studying arithmetic of points in elliptic curves 𝐸(𝑄𝑝) over 𝑝-adic field 𝑄𝑝 is necessary. To study 

arithmetic of points in 𝐸(𝑄𝑝), we first study arithmetic of points in 𝐸(𝑄𝑝) (𝑚𝑜𝑑 𝑝2) by extending 

arithmetic of points in 𝐸(𝐹𝑝) and then the arithmetic of points in 𝐸(𝑄𝑝) (𝑚𝑜𝑑 𝑝3) by extending 

arithmetic of points in 𝐸(𝑄𝑝) (𝑚𝑜𝑑 𝑝2) and so on. In this section, we have derived formula for 

arithmetic of points in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and had given an algorithm along with code in Python 

language. 

Now, to implement the arithmetic of points on elliptic curve 𝐸(𝑄𝑝) over 𝑄𝑝 to 𝐸(𝑄𝑝) (𝑚𝑜𝑑 𝑝2), 

if𝐸(𝜅) is an elliptic curve defined over a field 𝜅 with 𝐶ℎ𝑎𝑟𝜅 ≠ 2,3, given as 

𝐸 :  𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 

then for 𝑃1 =(𝛼1, 𝛽1) and 𝑃2 =(𝛼2, 𝛽2) in 𝐸(𝜅), the point addition of 𝑃1 and 𝑃2 denoted as 𝑃1 + 𝑃2 and 

is given as 

𝑃1 + 𝑃2 = {
(𝑚2 − 𝛼1 − 𝛼2, 𝑚(𝛼1 − 𝛼3) − 𝛽1)    with   𝑚 =

𝛽2−𝛽1

𝛼2−𝛼1
    if   𝑃1 ≠ 𝑃2

(𝑚2 − 2𝛼1,  𝑚(𝛼1 − 𝛼3) − 𝛽1)    with   𝑚 =
3𝛼1

2+𝐴

2𝛽1
    if   𝑃1 = 𝑃2

   ------------- (1) 

 

Now, In particular for 𝐾 = 𝑄𝑝, the 𝑝-adic field For an elliptic curve 𝐸 over 𝑄𝑝 defined over 𝑍𝑝, the 

points �̃�1 and �̃�2 are given as 



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�̃�1 = (�̃�1, �̃�1) = (𝑥10 + 𝑥11𝑝 + 𝑥12𝑝
2+. . . , 𝑦10 + 𝑦11𝑝 + 𝑦12𝑝

2+. . . ) 

𝑎𝑛𝑑 

�̃�2 = (�̃�2, �̃�2) = (𝑥20 + 𝑥21𝑝 + 𝑥22𝑝
2+. . . , 𝑦20 + 𝑦21𝑝 + 𝑦22𝑝

2+. . . ) 

with each coordinate in its 𝑝-adic expansion and by the point addition in 𝐸(𝑄𝑝), we have �̃�1 + �̃�2 =

�̃�3 say with �̃�3 = (�̃�3, �̃�3) given as 

�̃�3 = (�̃�3, �̃�3) = (𝑥30 + 𝑥31𝑝 + 𝑥32𝑝
2+. . . , 𝑦30 + 𝑦31𝑝 + 𝑦32𝑝

2+. . . ) 

The point �̃�3 could be known with the evaluation of 𝑥3𝑖’s and 𝑦3𝑖’s for all 𝑖 = 0,1,2, . .. . Now to obtain 

𝑥3𝑖’s and 𝑦3𝑖’s for all 𝑖 = 0,1,2, . .. , we implement the arithmetic of points �̃�1 and �̃�2 in 𝐸(𝑄𝑝) to the 

points �̃�1𝑛 and �̃�2𝑛 in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝𝑛+1) and obtain 𝑥3𝑖’s and 𝑦3𝑖’s for all 𝑖 = 0,1,2, . . . 𝑛, where the 

points �̃�1𝑛 and �̃�2𝑛 are the points obtained by considering �̃�1 and �̃�2 modulo 𝑝𝑛+1. 

In particular, on considering �̃�1 , �̃�2 to 𝑚𝑜𝑑 𝑝2, we have 

�̃�11 = (�̃�11, �̃�11) = (𝑥10 + 𝑥11𝑝, 𝑦10 + 𝑦11𝑝)𝑎𝑛𝑑�̃�21 = (�̃�21, �̃�21) = (𝑥20 + 𝑥21𝑝, 𝑦20 + 𝑦21𝑝) 

Now we obtain the arithmetic of points �̃�11 and �̃�21 in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) by implementing the point 

addition in 𝐸(𝑄𝑝) and obtain 𝑥30, 𝑥31, 𝑦30 , 𝑦31 . 

In the following theorem we describe the implementation of the arithmetic of points on elliptic curve 

𝐸(𝑄𝑝) defined over 𝑍𝑝 to 𝐸(𝑄𝑝) (𝑚𝑜𝑑 𝑝2). 

Theorem 4.1.  Consider an elliptic curve 𝐸(𝑄𝑝) defined over 𝑍𝑝 given as 

𝐸(𝑄𝑝): 𝑦
2 = 𝑥3 + 𝐴𝑥 + 𝐵 

over 𝑄𝑝 defined over 𝑍𝑝. For any points �̃�1, �̃�2 in 𝐸(𝑄𝑝), the arithmetic of points �̃�11 = (�̃�11, �̃�11) =

(𝑥10 + 𝑥11𝑝, 𝑦10 + 𝑦11𝑝) and �̃�21 = (�̃�21, �̃�21) = (𝑥20 + 𝑥21𝑝, 𝑦20 + 𝑦21𝑝) in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) may 

be obtained by implementing the point addition of points �̃�1, �̃�2 in 𝐸(𝑄𝑝) to the points �̃�11, �̃�21 in 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and �̃�31is given as 

�̃�31 = �̃�11 + �̃�21 = (�̃�31, �̃�31) = (𝑥30 + 𝑥31𝑝, 𝑦30 + 𝑦31𝑝) 

such that 

{
𝑥30 + 𝑥31𝑝 is the 𝑝-adic expansion of 𝑥30′ + 𝑥31′𝑝 modulo 𝑝2

𝑦30 + 𝑦31𝑝 is the 𝑝-adic expansion of 𝑦30′ + 𝑦31′𝑝 modulo 𝑝2
 

where 𝑥30′, 𝑥31′, 𝑦30′, 𝑦31′ given as follows 

 

for �̃�11 ≠ �̃�21 

{
 

 
𝑥30′ = 𝑚0

2 + (𝑝 − 1)(𝑥10 + 𝑥20)

𝑥31′ = 2𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11 + 𝑥20 + 𝑥21)

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10)

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11)

 



Communications on Applied Nonlinear Analysis 

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where �̃�1 =
�̃�21−�̃�11

�̃�21−�̃�11
= 𝑚0 +𝑚1𝑝. 

for �̃�11 = �̃�21 

 

{
 
 

 
 𝑥30′ = 𝑚0

2 + (𝑝 − 1)2𝑥10
𝑥31′ = 2(𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11))

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10)

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11)

 

where �̃�1 =
3�̃�11

2 +𝐴

2�̃�11
= 𝑚0 +𝑚1𝑝. 

Proof. Consider �̃�11 = (�̃�11, �̃�11) = (𝑥10 + 𝑥11𝑝, 𝑦10 + 𝑦11𝑝) and �̃�21 = (�̃�21, �̃�21) = (𝑥20 +

𝑎21𝑝, 𝑦20 + 𝑦21𝑝) 

For �̃�11 ≠ �̃�21 : 

By the implementation of arithmetic of points �̃�1, �̃�2 in 𝐸(𝑄𝑝) as in (1) to the points �̃�11, �̃�21 in 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2), we have 

�̃�31 = �̃�11 + �̃�21 = (�̃�1
2 − �̃�11

2 − �̃�21
2 , �̃�1(�̃�11 − �̃�31) − �̃�11) 

The 𝑥-coordinate of �̃�31 is given as 

 

𝑥(�̃�31) = �̃�1
2 − �̃�11

2 − �̃�21
2  

= (𝑚0
2 + 2𝑚0𝑚1𝑝) + ((𝑝 − 1) + (𝑝 − 1)𝑝)(𝑥10 + 𝑥11𝑝) + ((𝑝 − 1) + (𝑝 − 1)𝑝)(𝑥20 + 𝑥21𝑝)      

= 𝑚0
2 + 2𝑚0𝑚1𝑝 + (𝑝 − 1)𝑥10 + ((𝑝 − 1)(𝑥10 + 𝑥11))𝑝 + (𝑝 − 1)𝑥20 + ((𝑝 − 1)(𝑥20 + 𝑥21))𝑝 

= 𝑚0
2 + (𝑝 − 1)𝑥10 + (𝑝 − 1)𝑥20 + 2𝑚0𝑚1𝑝 + ((𝑝 − 1)(𝑥10 + 𝑥11))𝑝 + ((𝑝 − 1)(𝑥20 + 𝑥21))𝑝 

= 𝑚0
2 + (𝑝 − 1)(𝑥10 + 𝑥20) + (2𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11) + (𝑝 − 1)(𝑥20 + 𝑥21))𝑝             

Therefore, 

𝑥30′ = 𝑚0
2 + (𝑝 − 1)(𝑥10 + 𝑥20) 

𝑥31′ = 2𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11 + 𝑥20 + 𝑥21) 

Now, by considering 𝑝-adic expansion of 𝑥30′ + 𝑥31′𝑝 modulo 𝑝2, we have 

𝑥(�̃�31) = 𝑥30 + 𝑥31𝑝 

The 𝑦-coordinate of �̃�31 is given as 

𝑦(�̃�31) = �̃�1(�̃�11 − �̃�31) − �̃�11 

= (𝑚0 +𝑚1𝑝) (𝑥10 + 𝑥11𝑝 + ((𝑝 − 1) + (𝑝 − 1)𝑝)(𝑥30 + 𝑥31𝑝)) + (((𝑝 − 1) + (𝑝 −

1)𝑝)(𝑦10 + 𝑦11𝑝))  



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= (𝑚0 +𝑚1𝑝)(𝑥10 + (𝑝 − 1)𝑥30 + (𝑥11 + (𝑝 − 1)(𝑥30 + 𝑥31))𝑝) + (𝑝 − 1)𝑦10 + (𝑝 − 1)(𝑦10

+ 𝑦11))𝑝 

= 𝑚0𝑥10 + (𝑝 − 1)𝑚0𝑥30 + (𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31)𝑝)

+ (𝑝 − 1)𝑦10 + (𝑝 − 1)(𝑦10 + 𝑦11) 

= 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10)

+ (𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11))𝑝 

Therefore, 

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10) 

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11) 

Now, by considering 𝑝-adic expansion of 𝑦30′ + 𝑦31′𝑝 modulo 𝑝2, we have 

𝑦(�̃�31) = 𝑦30 + 𝑦31𝑝 

For �̃�11 = �̃�21 : 

By the implementation of arithmetic of points �̃�1, �̃�2 in 𝐸(𝑄𝑝) as in (1) to the points �̃�11, �̃�21 in 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2), we have 

�̃�31 = �̃�11 + �̃�21 = (�̃�1
2 − 2�̃�11

2 − �̃�21
2 , �̃�1(�̃�11 − �̃�31) − �̃�11) 

The 𝑥-coordinate of �̃�31 is given as 

𝑥(�̃�31) = �̃�1
2 − 2�̃�11

2  

= (𝑚0
2 + 2𝑚1𝑝) + 2((𝑝 − 1) + (𝑝 − 1)𝑝)(𝑥10 + 𝑥11𝑝)   

   = 𝑚0
2 + 2𝑚0𝑚1𝑝 + 2(𝑝 − 1)𝑥10 + 2((𝑝 − 1)(𝑥10 + 𝑥11))𝑝 

  = 𝑚0
2 + 2(𝑝 − 1)𝑥10 + 2𝑚0𝑚1𝑝 + 2((𝑝 − 1)(𝑥10 + 𝑥11))𝑝 

= 𝑚0
2 + 2(𝑝 − 1)𝑥10 + (2𝑚0𝑚1 + 2(𝑝 − 1)(𝑥10 + 𝑥11))𝑝 

Therefore, 

𝑥30′ = 𝑚0
2 + 2(𝑝 − 1)𝑥10 

𝑥31′ = 2(𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11)) 

Now, by considering 𝑝-adic expansion of 𝑥30′ + 𝑥31′𝑝 modulo 𝑝2, we have 

𝑥(�̃�31) = 𝑥30 + 𝑥31𝑝 

The 𝑦-coordinate of �̃�31 is given as 

 

𝑦(�̃�31) = �̃�1(�̃�11 − �̃�31) − �̃�11 

which on repeating above process, we have 



Communications on Applied Nonlinear Analysis 

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𝑦30 = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10) 

𝑦31 = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11) 

Now, by considering 𝑝-adic expansion of 𝑦30′ + 𝑦31′𝑝 modulo 𝑝2, we have 

𝑦(�̃�31) = 𝑦30 + 𝑦31𝑝 

Fi

el

d  

κ 

Elli

pti

c 

cur

ve 

Slope 𝒎                                            𝑷𝟏𝟏 + 𝑷𝟐𝟏 = 𝑷𝟑𝟏 

 P11 

≠ 

P21 

𝑃11 

=

 𝑃21 

 P11 ≠ P21 with �̃�11 = �̃�21  

P11  

≠ 

P21  

wi

th 

�̃�11 ≠

�̃�21 

𝑃11 = 𝑃21 

C

ha

r  

κ  

≠ 

2,

3  

𝛽2

= ∝3

+ 𝐴

∝ +𝐵 

�̃�21 − �̃�11
�̃�21 − �̃�11

 
3�̃�11

2 + 𝐴

2�̃�11
 

{
 
 

 
 
𝑥30′ = 𝑚0

2 + (𝑝 − 1)(𝑥10 + 𝑥20)

𝑥31′ = 2𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11 + 𝑥20 + 𝑥21)

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10)

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1) (
𝑚1𝑥30 +𝑚0𝑥30 +
𝑚0𝑥31 + 𝑦10 + 𝑦11

)

 

 

𝒪 

{
 
 

 
 𝑥30′ = 𝑚0

2 + (𝑝 − 1)2𝑥10
𝑥31′ = 2(𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11))

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10)

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11)

 

 

Table 2: Arithmetic of Points in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2 

Example 4.1.  For �̃�11 = (4 + 2.5,2 + 4.5) and �̃�21 = (2 + 1.5,4 + 4.5) then by above formulas, 

we have 

 �̃�11 + �̃�21 = (3.5 + (2 + 2.5)5,4 + 4.5 + (3.5 + 1.5
2)5)           

                 = (3.5 + 2.5 + 2.52, 4 + 4.5 + 3.52 + 1.53)  

                 = (3.53, 4 + 4.5 + 3.52 + 1.53)      

                 = (0,4 + 4.5)(𝑚𝑜𝑑52)              

 

The step-by-step procedure for point addition on elliptic curve 𝐸(𝑄𝑝)𝑚𝑜𝑑 𝑝2 over p-adic field 𝑄𝑝 

defined over 𝑍𝑝 using addition and multiplication process as in 𝑄𝑝 was discussed below. The code 

for arithmetic of points in 𝐸(𝑄𝑝)𝑚𝑜𝑑 𝑝2 and arithmetic operations of numbers in 𝑄𝑝 are included 

below. 

Algorithm: The step-by-step procedure is termed as Algorithm. 



Communications on Applied Nonlinear Analysis 

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Algorithm for Point Addition in (𝑸𝒑)(𝒎𝒐𝒅 𝒑𝟐) : Consider an Elliptic curve𝛽2 = 𝛼3 + 𝐴𝛼 + 𝐵 

over 𝑄𝑝. Let �̃�1, �̃�2 be two points in 𝐸(𝑄𝑝) then for �̃�1 and �̃�2 considered in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) given 

as �̃�11 = (�̃�11, �̃�11) = (𝑥10 + 𝑥11𝑝, 𝑦10 + 𝑦11𝑝) and �̃�21 = (𝑥20 + 𝑥21𝑝, 𝑦20 + 𝑦21𝑝) respectively. 

The point addition �̃�11 + �̃�21 = �̃�31 = (𝑥30 + 𝑥31𝑝, 𝑦30 + 𝑦31𝑝) in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) is obtained by 

the steps in the following algorithm. 

Step-I: Compute the slope �̃�1 = 𝑚0 +𝑚1𝑝 of �̃�11 and �̃�21 given as 

�̃�1 =

{
 
 

 
 �̃�21 − �̃�11
�̃�21 − �̃�11

 for �̃�11 ≠ �̃�21

3�̃�11
2 + 𝐴

2�̃�11
 for �̃�11 = �̃�21

 

Step-II: Compute 𝑥30′, 𝑥31′, 𝑦30′, 𝑦31′ using formulas 

For �̃�11 ≠ �̃�21: 

{
 

 
𝑥30′ = 𝑚0

2 + (𝑝 − 1)(𝑥10 + 𝑥20),

𝑥31′ = 2𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11 + 𝑥20 + 𝑥21),

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10),

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11)

 

For  �̃�11 = �̃�21: 

{
 
 

 
 𝑥30′ = 𝑚0

2 + (𝑝 − 1)2𝑥10,

𝑥31′ = 2(𝑚0𝑚1 + (𝑝 − 1)(𝑥10 + 𝑥11)),

𝑦30′ = 𝑚0𝑥10 + (𝑝 − 1)(𝑚0𝑥30 + 𝑦10),

𝑦31′ = 𝑚1𝑥10 +𝑚0𝑥11 + (𝑝 − 1)(𝑚1𝑥30 +𝑚0𝑥30 +𝑚0𝑥31 + 𝑦10 + 𝑦11)

 

Step-III: To evaluate 𝑥30  and 𝑥31, consider the value 𝑁 = 𝑥30′ + 𝑥31′𝑝 and write the 𝑝-adic 

expansion of 𝑁 and consider 𝑁(𝑚𝑜𝑑 𝑝2) to obtain 𝑥30 and 𝑥31. 

Step-IV: To evaluate 𝑦30  and 𝑦31 , consider the value 𝑀 = 𝑦30′ + 𝑦31′𝑝 and write the 𝑝-adic 

expansion of 𝑀 and consider 𝑀(𝑚𝑜𝑑 𝑝2) to obtain 𝑦30 and 𝑦31. 

Step-V: From the values of 𝑥30 and 𝑥31 in Step-III and 𝑦30  and 𝑦31 in Step-IV, the point addition 

�̃�11 + �̃�21 is given as: 

�̃�31 = (𝑥30 + 𝑥31𝑝, 𝑦30 + 𝑦31𝑝) 

For an Elliptic curve 𝑦2 = 𝑥3 + 𝑥 + 1 over 𝑄999983(𝑚𝑜𝑑 9999832), Consider two points �̃�11 =

(371181 + 9738 × 999983 ,  209555 + 151202 × 999983) and �̃�21 = (540108+ 4976 ×

999983 ,  254286 + 183355 × 999983) with slope of 𝑃1 and 𝑃2 given as �̃�′ = 383473 +

214267 × 999983 and �̃�11 + �̃�21 = �̃�31 is given as 

�̃�31 = (130341 + 144599× 999983 ,  997817 + 451277 × 999983) 

which is a lift of a point (130341,997817) ∈ 𝐸(𝐹𝑝). 

The code for addition, subtraction, multiplication and division of 𝑝-adic numbers and finding slope 

points in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and Arithmetic of Points in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) were given below in Python. 



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# to initialize run this cell 

from sage.all import * 

# make same length 

def pad_with_zeros(expansion, length): 

    return expansion + [0] * (length - len(expansion)) 

# function to add 

def add_p_adic(expansion1, expansion2, p): 

    #ensure both expansions are of same lengh 

    max_length = max(len(expansion1), len(expansion2)) 

    expansion1 = pad_with_zeros(expansion1, max_length) 

    expansion2 = pad_with_zeros(expansion2, max_length) 

    result = [] 

    carry = 0 

    for digit1, digit2 in zip(expansion1, expansion2): 

        total = digit1 + digit2 + carry 

        result.append(total % p) 

        carry = total // p 

    if carry > 0: 

        result.append(carry) 

    return result 

# function to subtract 

def subtract_p_adic(expansion1, expansion2, p): 

    # Ensure both expansions are of same length 

    max_length = max(len(expansion1), len(expansion2)) 

    expansion1 = pad_with_zeros(expansion1, max_length) 

    expansion2 = pad_with_zeros(expansion2, max_length) 

    result = [] 

    borrow = 0 

    for digit1, digit2 in zip(expansion1, expansion2): 

        total = digit1 - digit2 - borrow 

        if total < 0: 

            total += p 



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            borrow = 1 

        else: 

            borrow = 0 

        result.append(total) 

    # Remove trailing zeros 

    while result and result[-1] == 0: 

        result.pop() 

    return result 

# Function to multiply 

def multiply_p_adic(expansion1, expansion2, p): 

    #ensure both expansions are of same length 

    max_length = max(len(expansion1), len(expansion2)) 

    expansion1 = pad_with_zeros(expansion1, max_length) 

    expansion2 = pad_with_zeros(expansion2, max_length) 

    result = [0] * (2 * max_length) 

    for i in range(max_length): 

        carry = 0 

        for j in range(max_length): 

            total = expansion1[i] * expansion2[j] + result[i+j] + 

carry 

            result[i + j] = total % p 

            carry = total // p 

        result[i + max_length] += carry 

 

    # Removing trailing zeros 

    while len(result) > 1 and result[-1] == 0: 

        result.pop() 

    return result 

def divide_p_adic(expansion1, expansion2, p): 

    #ensure both expansions are of same length 

    max_length = max(len(expansion1), len(expansion2)) 

    expansion1 = pad_with_zeros(expansion1, max_length) 

    expansion2 = pad_with_zeros(expansion2, max_length) 



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    K = pAdicField(p,max_length) 

    padic_number_a = sum(c * K(p**i) for i, c in 

enumerate(expansion1)) 

    padic_number_b = sum(c * K(p**i) for i, c in 

enumerate(expansion2)) 

    result = padic_number_a/padic_number_b 

    expansion = result.expansion() 

    coefficients = [int(coef) for coef in expansion] 

    return coefficients 

def to_p_adic(n, p, precision): 

    if p<=1: 

        raise ValueError("Base p must be a prime number greater 

then 1.") 

    if n == 0: 

        return [0] 

    digits = [] 

    while n != 0: 

        digits.append(n % p) 

        n //= p 

    return digits[0:precision] 

def print_expansion(value, base, coefficients, precision): 

    print(f"The {base}-adic expansion of {value} is: ", end=' ') 

    for i in range(min(precision, len(coefficients))): 

        print(f"{coefficients[i]}*{base}^{i}", end=' ') 

        if (i < min(precision, len(coefficients)) - 1): 

            print(f"+", end=' ') 

        else: 

            print(" ") 

def calculate_p3(x1, x2, y1, y2, p, A, B): 

    max_length = max(len(x1), len(x2), len(y1), len(y2)) 

    x1 = pad_with_zeros(x1, max_length) 

    x2 = pad_with_zeros(x2, max_length) 

    y1 = pad_with_zeros(y1, max_length) 



Communications on Applied Nonlinear Analysis 

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    y2 = pad_with_zeros(y2, max_length) 

    precision = max_length 

    x10 = x1[0] 

    y10 = y1[0] 

    x11 = x1[1] 

    y11 = y1[1] 

    x20 = x2[0] 

    y20 = y2[0] 

    x21 = x2[1] 

    y21 = y2[1] 

    # Cheking P1=P2 

pointEquality = x10 == x20 and x11 == x21 and y10 == y20 and y11 == 

y21 

    if pointEquality: 

        print("P1 == P2") 

        Kx = pAdicField(p,max_length) 

        x_for_solpe = sum(c * Kx(p**i) for i, c in enumerate(x1)) 

        y_for_solpe = sum(c * Kx(p**i) for i, c in  

enumerate(y1)) 

        m_out = ((3 * x_for_solpe^2 ) + A) / (2 * y_for_solpe) 

        m_exp = m_out.expansion() 

        m = [int(coef) for coef in m_exp] 

    else: 

        print("P1 != P2") 

    m = divide_p_adic(subtract_p_adic(y2, y1, p),  

    subtract_p_adic(x2, x1, p), p) 

    m0 = m[0] 

    m1 = m[1] 

    print(f"m0: {m0}") 

    print(f"m1: {m1}") 

    # P1 + P2 = P3 

    if not pointEquality: 

x3 = (m0 ** 2) + (p - 1) * (x10 + x20) + 2 * m0 * m1 *  



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    p + (p * (p - 1) * (x10 + x11 + x20 + x21)) 

        print(f"x3 = {x3}") 

        try: 

            p_adic_expansion_x = to_p_adic(x3, p, precision) 

            print_expansion(x3, p, p_adic_expansion_x, precision) 

        except ValueError as e: 

            print(e) 

y3 = m0 * x10 + (p - 1) * (m0 * p_adic_expansion_x[0] + y10) +  

p * (m1 * x10 + m0 * x11) + p * (p - 1) * (m1 * 

p_adic_expansion_x[0] + 

m0 * p_adic_expansion_x[0] + m0 * p_adic_expansion_x[1] + y10 + 

y11) 

        print(f"y3 = {y3}") 

        try: 

            p_adic_expansion_y = to_p_adic(y3, p, precision) 

            print_expansion(y3, p, p_adic_expansion_y, precision) 

        except ValueError as e: 

            print(e) 

        return p_adic_expansion_x, p_adic_expansion_y 

     

    else: 

x3 = m0 ** 2 + (p - 1) * 2 * x10 + 2 * p * (m1 + (p - 1) * (x10 + 

x11)) 

        print(f"x3 = {x3}") 

        try: 

            p_adic_expansion_x = to_p_adic(x3, p, precision) 

            print_expansion(x3, p, p_adic_expansion_x, precision) 

        except ValueError as e: 

            print(e) 

y3 = m0 * x10 + (p - 1) * (m0 * p_adic_expansion_x[0] + y10) +  

p * (m1 * x10 + m0 * x11) + p * (p - 1) * ( m1 * 

p_adic_expansion_x[0] +  

m0 * p_adic_expansion_x[0] + m0 * p_adic_expansion_x[1] + y10 + 

y11) 



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        print(f"y3 = {y3}") 

        try: 

            p_adic_expansion_y = to_p_adic(y3, p, precision) 

            print_expansion(y3, p, p_adic_expansion_y, precision) 

        except ValueError as e: 

            print(e) 

        return p_adic_expansion_x, p_adic_expansion_y 

def take_input(): 

    expansion1 = [int(x) for x in input("Enter the first p-adic 

expansion (comma-separated): ").split(',')] 

    expansion2 = [int(x) for x in input("Enter the second p-adic 

expansion (comma-separated): ").split(',')] 

    p = int(input("Enter the base (prime number): ")) 

    return expansion1, expansion2, p 

#%%# For addition run this cell 

exp1, exp2, p = take_input() 

result = add_p_adic(exp1, exp2, p) 

print(f"The sum of the p-adic expansions is: {result}") 

#%%# For subtraction run this cell 

exp1, exp2, p = take_input() 

result = subtract_p_adic(exp1, exp2, p) 

print(f"The sum of the p-adic expansions is: {result}") 

#%%# For multiplication run this cell 

exp1, exp2, p = take_input() 

result = multiply_p_adic(exp1, exp2, p) 

print(f"The sum of the p-adic expansions is: {result}") 

#%%# For division run this cell 

exp1, exp2, p = take_input() 

result = divide_p_adic(exp1, exp2, p) 

print(f"The sum of the p-adic expansions is: {result}") 

#%%# For calculation of slope run this cell 

x1 = [int(x) for x in input("Enter the x1 p-adic expansion  

(comma-separated): ").split(',')] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

881 
https://internationalpubls.com 

x2 = [int(x) for x in input("Enter the x2 p-adic expansion  

(comma-separated): ").split(',')] 

y1 = [int(x) for x in input("Enter the y1 p-adic expansion  

(comma-separated): ").split(',')] 

y2 = [int(x) for x in input("Enter the y2 p-adic expansion  

(comma-separated): ").split(',')] 

p = int(input("Enter the base (prime number): ")) 

max_length = max(len(x1), len(x2), len(y1), len(y2)) 

x1 = pad_with_zeros(x1, max_length) 

x2 = pad_with_zeros(x2, max_length) 

y1 = pad_with_zeros(y1, max_length) 

y2 = pad_with_zeros(y2, max_length) 

slope_result = divide_p_adic(subtract_p_adic(y2, y1,p),  

subtract_p_adic(x2, x1, p), p) 

print(slope_result) 

#%%# To calculate P3 run this cell 

print("Calculation of point on curve y^2=x^3 + Ax + B") 

x1 = [int(x) for x in input("Enter the x1 p-adic expansion  

(comma-separated): ").split(',')] 

x2 = [int(x) for x in input("Enter the x2 p-adic expansion  

(comma-separated): ").split(',')] 

y1 = [int(x) for x in input("Enter the y1 p-adic expansion  

(comma-separated): ").split(',')] 

y2 = [int(x) for x in input("Enter the y2 p-adic expansion  

(comma-separated): ").split(',')] 

p = int(input("Enter the base (prime number): ")) 

A = int(input("Enter the value of A: ")) 

B = int(input("Enter the value of B: ")) 

p3_x, p3_y = calculate_p3(x1, x2, y1, y2, p, A, B) 

print(f"P3: (x: {p3_x}, y:{p3_y}) ") 

#%% 

5. Conclusions 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

882 
https://internationalpubls.com 

In this paper, the focus is on the points on elliptic curve 𝐸(𝑄𝑝) over 𝑝-adic field 𝑄𝑝 and its arithmetic. 

All the points in elliptic curve 𝐸(𝑄𝑝) are obtained by starting with points in 𝐸(𝐹𝑝) and lifting to 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) and then lifting each point in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) to 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝3) and so on. The 

arithmetic of points on 𝐸(𝑄𝑝) may be evaluated by implementing arithmetic of points in 𝐸(𝑄𝑝) to 

𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2). This study of arithmetic in 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) provides a key space for cryptosystems 

with 𝐸(𝑄𝑝)(𝑚𝑜𝑑 𝑝2) bigger than the key space of cryptosystems with 𝐸(𝐹𝑝) with an increase in level 

of security. 

References 

[1] J. H. Silverman and J. Tate. “Rational points on Elliptic Curves” Undergraduate Texts 

in Mathematics. Springer-Verlag, New York, 1992. 

[2] J. H. Silverman “The Arithmetic of Elliptic Curves” volume 106 of Graduate texts in Mathematics. Springer-

Verlag, New York, 1996. 

[3] Fernando Gouvea p-adic Numbers: An Introduction (Second Editon). Springer, NewYork, 1997. 

[4] Neal Koblitz “A course in number theory and cryptography” ISBN 3-5780718 SPIN 10893308. 

[5] J. Buchmann “Introduction to cryptography” , Springer-Verlag 2001. 

[6] Lawrence C. Washington.“Elliptic curves number theory and Cryptography” CRC Press, Second Edition 

[7] “𝑝-adic numbers applied on elliptic curve cryptography” Maherindrainibelahasa, Ravaliminoarimalalason, 

Randimbindrainibe. Vol-5 Issue-2 2019, IJARIIE-ISSN(O)-2395-4396. 

[8] Rosa Winter “ Elliptic curves over 𝑄𝑝” 

[9] S. Katok. "𝑝-adic analysis compared with real" volume 37 of Student Mathematical Library. American 

Mathematical Society, Providence, RI, 2007 

[10] L. Praveen Kumar "Arithmetic of Elliptic curves with Affine and Projective Coordinates and some cryptographic 

aspects" 2015. 

[11] N. Koblitz. "𝑝-adic numbers, 𝑝-adic analysis and zeta-functions" volume 58 of Graduate texts in Mathematics. 

Springer-Verlag, New York, second edition, 1984. 

[12] Alain M. Robert "A Course in 𝑝-adic Analysis" Volume 198 of Graduate Texts in Mathematics Springer-Verlag 

2000.