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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.