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Bachet’s Equation
∆GSN
Clay McGowen
Scott Navert
Evan Smith
Boise State University
28 April, 2015
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Outline
Background and History
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications and Beyond
References
Background and History
Bachet’s equation is a famous Diophantine equation of the form:
y 2 = x 3 + k for k ∈ Z
It is named after famous French mathematician Claude Gaspard
Bachet de Mèziriac, who proposed the first strong integer
solution
Background and History
Bachet’s equation is a famous Diophantine equation of the form:
y 2 = x 3 + k for k ∈ Z
It is named after famous French mathematician Claude Gaspard
Bachet de Mèziriac, who proposed the first strong integer
solution
Background and History
Bachet’s equation is a famous Diophantine equation of the form:
y 2 = x 3 + k for k ∈ Z
It is named after famous French mathematician Claude Gaspard
Bachet de Mèziriac, who proposed the first strong integer
solution
Background and History
Bachet’s equation is a famous Diophantine equation of the form:
y 2 = x 3 + k for k ∈ Z
It is named after famous French mathematician Claude Gaspard
Bachet de Mèziriac, who proposed the first strong integer
solution
Background and History
Also known as Mordell Curves, Bachet’s Equation is an
elliptic curve that has been studied since the 17th century
(1621)
There exists more than one solution, as opposed to when it
was first proposed by Bachet
Louis Mordell refocused attention to the curve and its
properties in 1913
There exists a means to calculate general solutions for
certain values of k
Background and History
Also known as Mordell Curves, Bachet’s Equation is an
elliptic curve that has been studied since the 17th century
(1621)
There exists more than one solution, as opposed to when it
was first proposed by Bachet
Louis Mordell refocused attention to the curve and its
properties in 1913
There exists a means to calculate general solutions for
certain values of k
Background and History
Also known as Mordell Curves, Bachet’s Equation is an
elliptic curve that has been studied since the 17th century
(1621)
There exists more than one solution, as opposed to when it
was first proposed by Bachet
Louis Mordell refocused attention to the curve and its
properties in 1913
There exists a means to calculate general solutions for
certain values of k
Background and History
Also known as Mordell Curves, Bachet’s Equation is an
elliptic curve that has been studied since the 17th century
(1621)
There exists more than one solution, as opposed to when it
was first proposed by Bachet
Louis Mordell refocused attention to the curve and its
properties in 1913
There exists a means to calculate general solutions for
certain values of k
Background and History
Also known as Mordell Curves, Bachet’s Equation is an
elliptic curve that has been studied since the 17th century
(1621)
There exists more than one solution, as opposed to when it
was first proposed by Bachet
Louis Mordell refocused attention to the curve and its
properties in 1913
There exists a means to calculate general solutions for
certain values of k
Background and History
For some equation y 2 = x 3 + k, the following is a generalized
solution for any rational x, y, k is:
4 −8kx −x 6 −20kx+8k 2
,
)
( x 4y
2
8y 3
Background and History
For some equation y 2 = x 3 + k, the following is a generalized
solution for any rational x, y, k is:
4 −8kx −x 6 −20kx+8k 2
,
)
( x 4y
2
8y 3
Background and History
For some equation y 2 = x 3 + k, the following is a generalized
solution for any rational x, y, k is:
4 −8kx −x 6 −20kx+8k 2
,
)
( x 4y
2
8y 3
Important Definitions
Definition 1
Supersingular Primes A prime number with a certain relationship to an elliptic curve
Definition 2
E(Zn ) The elliptic curve defined over the group Zn such that all positive
integers 0 ≤ a ≤ n are quantified
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Our spreadsheet was organized into 4 major columns:
Zp , k , |S|, and |S| ∪ IDg
Where Zp is the range to which we confine our search
k is the specific constant we are looking at
|S| is the solution size
And |S| ∪ IDg is the solution size including the identity element.
Data Background
This looks like:
Figure: Data
Data Background
This looks like:
Figure: Data
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data Background
Questions began to arise as patterns became more evident:
What happens when |S| ∪ IDg = m where m is prime?
What happens when |S| > p?
What happens when |S| ∪ IDg = p?
What happens when |S| = p?
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 02
|S| > p
This was our smallest emphasis of focus.
We found solutions occurred for the values:
{7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139,
151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283,
307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433,
439, 457, 463, 487, 499, 523, 541}
Which turns out to be the primes that can be written as
6n + 1
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
|S| ∪ IDg = p
This was a major focus in the latter portion of research.
We found solutions occurred for the values:
{7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657,
1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447,
5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267,
11719, 12097, 13267, 13669, 16651, 19441, 19927,
22447, 23497, 24571, 25117, 26227}
Which are Cuban primes: primes which are the difference of
two consecutive cubes.
Data 03
We looked at the number of occurrences per prime value
This number seems to converge to around 0.1667
with the exception of the data point 2437 which had 381
occurrences and a ratio of 0.1563.
Figure: Converging dagta
Data 03
We looked at the number of occurrences per prime value
This number seems to converge to around 0.1667
with the exception of the data point 2437 which had 381
occurrences and a ratio of 0.1563.
Figure: Converging dagta
Data 03
We looked at the number of occurrences per prime value
This number seems to converge to around 0.1667
with the exception of the data point 2437 which had 381
occurrences and a ratio of 0.1563.
Figure: Converging dagta
Data 03
We looked at the number of occurrences per prime value
This number seems to converge to around 0.1667
with the exception of the data point 2437 which had 381
occurrences and a ratio of 0.1563.
Figure: Converging dagta
Data 03
We looked at the number of occurrences per prime value
This number seems to converge to around 0.1667
with the exception of the data point 2437 which had 381
occurrences and a ratio of 0.1563.
Figure: Converging dagta
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 04
|S| = p
This was the major and ongoing focus for research.
This is where we made multiple discoveries especially after
modular operations
Big sequences found using OEIS include:
#A040117 which are Primes congruent to 5 (mod 12)
#A061242 Primes of the form 9n − 1.
#A132236 Primes congruent to 29 (mod 30).
Data 05
Results of this investigation:
Conjecture: 1
If |S| = p ∀ k
Then p is a prime of the form 3n − 1 and where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is a supersingular prime.
Data 05
Results of this investigation:
Conjecture: 1
If |S| = p ∀ k
Then p is a prime of the form 3n − 1 and where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is a supersingular prime.
Data 05
Results of this investigation:
Conjecture: 1
If |S| = p ∀ k
Then p is a prime of the form 3n − 1 and where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is a supersingular prime.
Data 05
Results of this investigation:
Conjecture: 1
If |S| = p ∀ k
Then p is a prime of the form 3n − 1 and where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is a supersingular prime.
Data 05
Results of this investigation:
Conjecture: 3
The ratio
|k|
p
converges for the cuban primes
Conjecture: 4
In Bachet’s equaiton, all primes p of the form 6n + 1 have at
least 1 k value for which the solution size is greater than p itself.
Data 05
Results of this investigation:
Conjecture: 3
The ratio
|k|
p
converges for the cuban primes
Conjecture: 4
In Bachet’s equaiton, all primes p of the form 6n + 1 have at
least 1 k value for which the solution size is greater than p itself.
Data 05
Results of this investigation:
Conjecture: 3
The ratio
|k|
p
converges for the cuban primes
Conjecture: 4
In Bachet’s equaiton, all primes p of the form 6n + 1 have at
least 1 k value for which the solution size is greater than p itself.
Data 05
Results of this investigation:
Conjecture: 3
The ratio
|k|
p
converges for the cuban primes
Conjecture: 4
In Bachet’s equaiton, all primes p of the form 6n + 1 have at
least 1 k value for which the solution size is greater than p itself.
Elliptic Curve Factoring: Intro
This method was discovered in the 1980’s by Hendrik
Lenstra. [1]
Elliptic Curve Factorization is the fastest method for
numbers around 60 digits [1], and the third fastest factoring
method known. [3]
Limits our focus to the group (E(Zn ), +). [3]
Elliptic Curve Factoring: Intro
This method was discovered in the 1980’s by Hendrik
Lenstra. [1]
Elliptic Curve Factorization is the fastest method for
numbers around 60 digits [1], and the third fastest factoring
method known. [3]
Limits our focus to the group (E(Zn ), +). [3]
Elliptic Curve Factoring: Intro
This method was discovered in the 1980’s by Hendrik
Lenstra. [1]
Elliptic Curve Factorization is the fastest method for
numbers around 60 digits [1], and the third fastest factoring
method known. [3]
Limits our focus to the group (E(Zn ), +). [3]
Elliptic Curve Factoring: Intro
This method was discovered in the 1980’s by Hendrik
Lenstra. [1]
Elliptic Curve Factorization is the fastest method for
numbers around 60 digits [1], and the third fastest factoring
method known. [3]
Limits our focus to the group (E(Zn ), +). [3]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring: The Method
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Definitions
Let E be an elliptic curve of the form y 2 = x 3 + Ax + B.
3x 2 + A
is the slope at any points x and y .
Then, s =
2y
n·P = (x 0 , y 0 ), where x 0 = s2 − 2x and y 0 = s(x 0 − x) − y.
Definitions
Let E be an elliptic curve of the form y 2 = x 3 + Ax + B.
3x 2 + A
is the slope at any points x and y .
Then, s =
2y
n·P = (x 0 , y 0 ), where x 0 = s2 − 2x and y 0 = s(x 0 − x) − y.
Definitions
Let E be an elliptic curve of the form y 2 = x 3 + Ax + B.
3x 2 + A
is the slope at any points x and y .
Then, s =
2y
n·P = (x 0 , y 0 ), where x 0 = s2 − 2x and y 0 = s(x 0 − x) − y.
An Example
Let’s factor 455839
Choose E to be y 2 = x 3 + 5x − 5 mod 455839
Choose the first point to be P = (1, 1)
An Example
Let’s factor 455839
Choose E to be y 2 = x 3 + 5x − 5 mod 455839
Choose the first point to be P = (1, 1)
An Example
Let’s factor 455839
Choose E to be y 2 = x 3 + 5x − 5 mod 455839
Choose the first point to be P = (1, 1)
An Example
Let’s factor 455839
Choose E to be y 2 = x 3 + 5x − 5 mod 455839
Choose the first point to be P = (1, 1)
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Calculate the slope: s =
3(x)2 + 5
3(1)2 + 5
=
= 4.
2(y )
2(1)
Calculate 2P:
x 0 = 42 − 2·1 = 14.
y 0 = 4(1 − 14) − 1 = −53.
So, 2P = (14, −53).
An Example
Now we need to calculate 3!P = 3(2P).
Slope of 2P: s =
3(14)2 + 5
−593
=
.
2(−53)
106
Now we come to the magic step. Computing x, where
−593
≡ x mod 455839
106
An Example
Now we need to calculate 3!P = 3(2P).
Slope of 2P: s =
3(14)2 + 5
−593
=
.
2(−53)
106
Now we come to the magic step. Computing x, where
−593
≡ x mod 455839
106
An Example
Now we need to calculate 3!P = 3(2P).
Slope of 2P: s =
3(14)2 + 5
−593
=
.
2(−53)
106
Now we come to the magic step. Computing x, where
−593
≡ x mod 455839
106
An Example
Now we need to calculate 3!P = 3(2P).
Slope of 2P: s =
3(14)2 + 5
−593
=
.
2(−53)
106
Now we come to the magic step. Computing x, where
−593
≡ x mod 455839
106
An Example (Magic Step)
1. gcd(455839,106) = 1. (Note, this is where the factor will
come from if there is one (i.e. gcd(455839,k) 6= 1).
2. Compute x, where 106−1 ≡ x mod 455839. Use a
version of the extended Euclidean algorithm to get
106−1 = 81707 mod 455839.
3. Multiply by −593, so
−593
= −133317 mod 455839.
106
An Example (Magic Step)
1. gcd(455839,106) = 1. (Note, this is where the factor will
come from if there is one (i.e. gcd(455839,k) 6= 1).
2. Compute x, where 106−1 ≡ x mod 455839. Use a
version of the extended Euclidean algorithm to get
106−1 = 81707 mod 455839.
3. Multiply by −593, so
−593
= −133317 mod 455839.
106
An Example (Magic Step)
1. gcd(455839,106) = 1. (Note, this is where the factor will
come from if there is one (i.e. gcd(455839,k) 6= 1).
2. Compute x, where 106−1 ≡ x mod 455839. Use a
version of the extended Euclidean algorithm to get
106−1 = 81707 mod 455839.
3. Multiply by −593, so
−593
= −133317 mod 455839.
106
An Example (Magic Step)
1. gcd(455839,106) = 1. (Note, this is where the factor will
come from if there is one (i.e. gcd(455839,k) 6= 1).
2. Compute x, where 106−1 ≡ x mod 455839. Use a
version of the extended Euclidean algorithm to get
106−1 = 81707 mod 455839.
3. Multiply by −593, so
−593
= −133317 mod 455839.
106
An Example
Now compute 2(2P) = 4P.
x 0 = (−133317)2 − 2·14 = 259851 mod 455839.
y 0 = −133317(14 − 259851) − (−53) = 116255 mod
455839.
So, 4P = (259851, 116255) mod 455851.
An Example
Now compute 2(2P) = 4P.
x 0 = (−133317)2 − 2·14 = 259851 mod 455839.
y 0 = −133317(14 − 259851) − (−53) = 116255 mod
455839.
So, 4P = (259851, 116255) mod 455851.
An Example
Now compute 2(2P) = 4P.
x 0 = (−133317)2 − 2·14 = 259851 mod 455839.
y 0 = −133317(14 − 259851) − (−53) = 116255 mod
455839.
So, 4P = (259851, 116255) mod 455851.
An Example
Now compute 2(2P) = 4P.
x 0 = (−133317)2 − 2·14 = 259851 mod 455839.
y 0 = −133317(14 − 259851) − (−53) = 116255 mod
455839.
So, 4P = (259851, 116255) mod 455851.
An Example
Now compute 2(2P) = 4P.
x 0 = (−133317)2 − 2·14 = 259851 mod 455839.
y 0 = −133317(14 − 259851) − (−53) = 116255 mod
455839.
So, 4P = (259851, 116255) mod 455851.
An Example
Now 3!P can calculated. 3(2P) = 4P ⊕ 2P.
Now calculate 4!P, 5!P, 6!P, and so forth.
The factor come from 8!P in this example, which is 599.
Dividing 455839 by 599 yields 761 (the second factor of
455839).
An Example
Now 3!P can calculated. 3(2P) = 4P ⊕ 2P.
Now calculate 4!P, 5!P, 6!P, and so forth.
The factor come from 8!P in this example, which is 599.
Dividing 455839 by 599 yields 761 (the second factor of
455839).
An Example
Now 3!P can calculated. 3(2P) = 4P ⊕ 2P.
Now calculate 4!P, 5!P, 6!P, and so forth.
The factor come from 8!P in this example, which is 599.
Dividing 455839 by 599 yields 761 (the second factor of
455839).
An Example
Now 3!P can calculated. 3(2P) = 4P ⊕ 2P.
Now calculate 4!P, 5!P, 6!P, and so forth.
The factor come from 8!P in this example, which is 599.
Dividing 455839 by 599 yields 761 (the second factor of
455839).
An Example
Now 3!P can calculated. 3(2P) = 4P ⊕ 2P.
Now calculate 4!P, 5!P, 6!P, and so forth.
The factor come from 8!P in this example, which is 599.
Dividing 455839 by 599 yields 761 (the second factor of
455839).
Financial Applications
Mathematics and Finance are incredibly intertwined.
”Rocket Scientist” Mathematicians are fueling incredible
amounts of trades with data interpreted through the scope
of mathematics.
Currently, there exists no formal literature on the application
of Elliptic Curves and Supersingular Primes in the financial
world. This may be because the concept is relatively new, or
that financial institutions are reluctant to use unproven forms
of mathematics in monetary policy.
Financial Applications
Mathematics and Finance are incredibly intertwined.
”Rocket Scientist” Mathematicians are fueling incredible
amounts of trades with data interpreted through the scope
of mathematics.
Currently, there exists no formal literature on the application
of Elliptic Curves and Supersingular Primes in the financial
world. This may be because the concept is relatively new, or
that financial institutions are reluctant to use unproven forms
of mathematics in monetary policy.
Financial Applications
Mathematics and Finance are incredibly intertwined.
”Rocket Scientist” Mathematicians are fueling incredible
amounts of trades with data interpreted through the scope
of mathematics.
Currently, there exists no formal literature on the application
of Elliptic Curves and Supersingular Primes in the financial
world. This may be because the concept is relatively new, or
that financial institutions are reluctant to use unproven forms
of mathematics in monetary policy.
Financial Applications
A Quick Primer and Definitions:
Stock - A share of a company that is publicly traded on an
Exchange
Option - A contract between two parties usually on the
purchase price of a stock or commodity (ie, someone has
the option to purchase something at a previously agreed
upon value)
Price Level - The current market price for a stock or
commodity
Entry/Exit Price The price level for which an individual would
enter/exit the market
Efficient Market Hypothesis, EMH - The idea that markets
are self correcting and adjust themselves perfectly to the
availability of information
Financial Applications
A Quick Primer and Definitions:
Stock - A share of a company that is publicly traded on an
Exchange
Option - A contract between two parties usually on the
purchase price of a stock or commodity (ie, someone has
the option to purchase something at a previously agreed
upon value)
Price Level - The current market price for a stock or
commodity
Entry/Exit Price The price level for which an individual would
enter/exit the market
Efficient Market Hypothesis, EMH - The idea that markets
are self correcting and adjust themselves perfectly to the
availability of information
Financial Applications
Before we begin, we need to focus our attention to an important
concept of markets, which is the Stochastic process of Brownian
Motion: the idea that markets fluctuate due to information, and
take on a randomized, complex formation over time in some
space.
∆Bi = Bt+∆t − Bt
Financial Applications
Before we begin, we need to focus our attention to an important
concept of markets, which is the Stochastic process of Brownian
Motion: the idea that markets fluctuate due to information, and
take on a randomized, complex formation over time in some
space.
∆Bi = Bt+∆t − Bt
Financial Applications
Before we begin, we need to focus our attention to an important
concept of markets, which is the Stochastic process of Brownian
Motion: the idea that markets fluctuate due to information, and
take on a randomized, complex formation over time in some
space.
∆Bi = Bt+∆t − Bt
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
Financial Applications
The following ratios are calculated for our application:
F100 = 31.24
F61.8 = 29.85
F50 = 29.43
F38.2 = 29.00
F0 = 27.61
Financial Applications
And now, we superimpose the ratios on the graph:
Financial Applications
And now, we superimpose the ratios on the graph:
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the super singular primes
ratios like so:
En /En−1 = 13
7 − 1 ≈ 0.85
19
En+1 /En = 7 − 1 ≈ 0.46
19
En+2 /En+1 = 13
− 1 ≈ 0.63
Our ratios will be 0, 100, 46, 63 and 85
Financial Applications
And now, our new retracement:
Financial Applications (Group/Number Theory )
It is incredibly important to find a group that is reasonably
palpable to the Fibonacci Numbers in Finance because of
certain theories that relate natural occurrences in markets
(Stochastics, HMT, Kinetic Model Theory)
If we can show that the super singular primes and the
Fibonacci Numbers have some sort of connection, we can
then manipulate these financial processes over a new group
with significantly more complex data and
Financial Applications (Group/Number Theory )
It is incredibly important to find a group that is reasonably
palpable to the Fibonacci Numbers in Finance because of
certain theories that relate natural occurrences in markets
(Stochastics, HMT, Kinetic Model Theory)
If we can show that the super singular primes and the
Fibonacci Numbers have some sort of connection, we can
then manipulate these financial processes over a new group
with significantly more complex data and
Financial Applications (Group/Number Theory )
It is incredibly important to find a group that is reasonably
palpable to the Fibonacci Numbers in Finance because of
certain theories that relate natural occurrences in markets
(Stochastics, HMT, Kinetic Model Theory)
If we can show that the super singular primes and the
Fibonacci Numbers have some sort of connection, we can
then manipulate these financial processes over a new group
with significantly more complex data and
Financial Applications (Group/Number Theory)
Reviewing properties of the Fibonacci Sequence, recall that
[2] for any range from the Sequence g, where
g(n + 2) = g(n) + g(n + 1), g can be defined as a finite
vector space of abelian properties.
More generally, the range of g can be taken to be any
abelian group with kernel Fg , where Fg is the identity of the
group for that range.
If we can show some relation of super singular primes to the
Fibonacci Sequence, we can apply these properties to it
almost indefinitely.
Financial Applications (Group/Number Theory)
Reviewing properties of the Fibonacci Sequence, recall that
[2] for any range from the Sequence g, where
g(n + 2) = g(n) + g(n + 1), g can be defined as a finite
vector space of abelian properties.
More generally, the range of g can be taken to be any
abelian group with kernel Fg , where Fg is the identity of the
group for that range.
If we can show some relation of super singular primes to the
Fibonacci Sequence, we can apply these properties to it
almost indefinitely.
Financial Applications (Group/Number Theory)
Reviewing properties of the Fibonacci Sequence, recall that
[2] for any range from the Sequence g, where
g(n + 2) = g(n) + g(n + 1), g can be defined as a finite
vector space of abelian properties.
More generally, the range of g can be taken to be any
abelian group with kernel Fg , where Fg is the identity of the
group for that range.
If we can show some relation of super singular primes to the
Fibonacci Sequence, we can apply these properties to it
almost indefinitely.
Financial Applications (Group/Number Theory)
Let the following conjecture be made about the group of super
singular primes, Sn , and the Fibonacci Numbers, Fn .:
Conjecture: 5
Let some function Θ exist such that Θ : Sn → Fn . The order of
Sn = Fn , because both groups have one cycle of size n. Thus,
the groups are isomorphic to one another, and weakly
homomorphic.
Financial Applications (Group/Number Theory)
Let the following conjecture be made about the group of super
singular primes, Sn , and the Fibonacci Numbers, Fn .:
Conjecture: 5
Let some function Θ exist such that Θ : Sn → Fn . The order of
Sn = Fn , because both groups have one cycle of size n. Thus,
the groups are isomorphic to one another, and weakly
homomorphic.
Financial Applications (Group/Number Theory)
Let the following conjecture be made about the group of super
singular primes, Sn , and the Fibonacci Numbers, Fn .:
Conjecture: 5
Let some function Θ exist such that Θ : Sn → Fn . The order of
Sn = Fn , because both groups have one cycle of size n. Thus,
the groups are isomorphic to one another, and weakly
homomorphic.
Financial Applications (Group/Number Theory)
Because the groups are isomorphic, we have found a strong
relationship between the super singular primes and the
Fibonacci numbers over the function Θ.
Based on the properties of isomorphisms, we can show that
the group of super singular primes is both abelian and
locally finite, only because the Fibonacci numbers are so as
well. Group properties are preserved by isomorphisms. [2]
Financial Applications (Group/Number Theory)
Because the groups are isomorphic, we have found a strong
relationship between the super singular primes and the
Fibonacci numbers over the function Θ.
Based on the properties of isomorphisms, we can show that
the group of super singular primes is both abelian and
locally finite, only because the Fibonacci numbers are so as
well. Group properties are preserved by isomorphisms. [2]
Financial Applications (Group/Number Theory)
Because the groups are isomorphic, we have found a strong
relationship between the super singular primes and the
Fibonacci numbers over the function Θ.
Based on the properties of isomorphisms, we can show that
the group of super singular primes is both abelian and
locally finite, only because the Fibonacci numbers are so as
well. Group properties are preserved by isomorphisms. [2]
Finacial Applications (Group/Number Theory)
Now it is plainly obvious as to why the super singular primes
worked so well: They have a group relationship to the
Fibonacci Numbers.
However, we can say that with the super singular primes,
their ”elliptic-ness” is preserved over the isomorphism, the
same way Fibonacci Numbers keep their abelian properties.
That means that for any instance we use the super singular
primes, we can apply the elliptic properties to that function
This is a powerful tool for options trading specifically
Finacial Applications (Group/Number Theory)
Now it is plainly obvious as to why the super singular primes
worked so well: They have a group relationship to the
Fibonacci Numbers.
However, we can say that with the super singular primes,
their ”elliptic-ness” is preserved over the isomorphism, the
same way Fibonacci Numbers keep their abelian properties.
That means that for any instance we use the super singular
primes, we can apply the elliptic properties to that function
This is a powerful tool for options trading specifically
Finacial Applications (Group/Number Theory)
Now it is plainly obvious as to why the super singular primes
worked so well: They have a group relationship to the
Fibonacci Numbers.
However, we can say that with the super singular primes,
their ”elliptic-ness” is preserved over the isomorphism, the
same way Fibonacci Numbers keep their abelian properties.
That means that for any instance we use the super singular
primes, we can apply the elliptic properties to that function
This is a powerful tool for options trading specifically
Finacial Applications (Group/Number Theory)
Now it is plainly obvious as to why the super singular primes
worked so well: They have a group relationship to the
Fibonacci Numbers.
However, we can say that with the super singular primes,
their ”elliptic-ness” is preserved over the isomorphism, the
same way Fibonacci Numbers keep their abelian properties.
That means that for any instance we use the super singular
primes, we can apply the elliptic properties to that function
This is a powerful tool for options trading specifically
Finacial Applications (Group/Number Theory)
Now it is plainly obvious as to why the super singular primes
worked so well: They have a group relationship to the
Fibonacci Numbers.
However, we can say that with the super singular primes,
their ”elliptic-ness” is preserved over the isomorphism, the
same way Fibonacci Numbers keep their abelian properties.
That means that for any instance we use the super singular
primes, we can apply the elliptic properties to that function
This is a powerful tool for options trading specifically
Finacial Applications
Further fiscal applications of this elliptic group can be
defined over the variable space of Brownian Motion, which
we defined earlier:
∆Bi = Bt+∆t − Bt
Finacial Applications
Further fiscal applications of this elliptic group can be
defined over the variable space of Brownian Motion, which
we defined earlier:
∆Bi = Bt+∆t − Bt
Financial Applications
Black Scholes and PDE’s - An option pricing module based
on Monte Carlo methods and Partial Differential Equations.
Translating the distribution to an elliptic patter, and through
some analysis of PDE’s in an elliptic fashion, we can work
with a stronger retracement method.
Kinetic Model Theory - We can work over a more complex
space using super singular primes. This allows us to work
with a more marginal core group, which increases the
amount of derivative information we can model over and
move.
Financial Applications
Black Scholes and PDE’s - An option pricing module based
on Monte Carlo methods and Partial Differential Equations.
Translating the distribution to an elliptic patter, and through
some analysis of PDE’s in an elliptic fashion, we can work
with a stronger retracement method.
Kinetic Model Theory - We can work over a more complex
space using super singular primes. This allows us to work
with a more marginal core group, which increases the
amount of derivative information we can model over and
move.
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
Continuing the Research
In the future, we plan on continuing our research with elliptic
curves. Specifically, we plan on looking into:
Game Theory in Addition Games over Zp and Un
Elliptic Curve Cryptography
Data Analytics
Where groups come into play,
→ specifically in all other cases than when |S| ∪ IDg = m
Proofs for conjectures
References
Lawrence C. Washington.
Elliptic Curves: Number Theory and Cryptography.
Chapman & Hall/CRC, Boca Raton, Florida, 2008.
D. R. Morrison.
Ralf Zimmermann.
Optimized implementation of the elliptic curve factorization
method on a highly parallelized hardware cluster.
diploma thesis, Ruhr-University Bochum, November 2009.