Bachet s equation and groups formed from solutions in Z p

Transcription

1 Bachet s equation and groups formed from solutions in Z p Boise State University April 30, 2015

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3 Elliptic Curves and Bachet s Equation Elliptic curves are of the form y 2 = x 3 + ax + b Bachet equations are elliptic curves where a = 0. A.K.A Mordell s Equation Focus of our Project: Solving y 2 = x 3 + b where x, y, b Z p and p is prime.

4 Elliptic Curve Plot This is a specific type of elliptic curve. This one was generated from sage using the following command plot(ellipticcurve([0,10,0,4,4]))

5 Example y 2 = x For solution pairs to this equation in Z 7, we solve: y 2 mod 7 = (x 3 + 4) mod 7 Here, the only solutions are the ordered pairs (x, y) = (0, 2), (0, 5)

6 Overview of Current Applications of Elliptic Curves The basis for elliptic curve cryptography Used in an algorithm for proving primes Used in part of the Lenstra elliptic curve factorization method There are open problems associated with the equation that have yet to be solved

7 Connection to Class Elliptic curve solution pairs in Z p form a group with a i d element that is defined for each solution set. Using java we wrote a program that searched for solutions of y 2 = x 3 + b mod p where p was each of the first 100 primes and b Z p. Optimized our search using sage to find solutions and find solution sets. Generated and attempted to analyze over 24 thousand solution sets.

8 Elliptic Pairs and Cycles Through our initial data we focused on elliptic pairs and elliptic cycles. We collected basic elliptic pairs and analyzed them for patterns.

9 Elliptic Pairs We consider prime numbers p and q to be an elliptic pair if S p = {(x, y) : y 2 = x 3 + b, x, y, b Z p } {id g } has cardinality S p = q, and, S q = {(x, y) : y 2 = x 3 + b, x, y, b Z q } {id g } has cardinality S q = p.

10 Elliptic Pairs - Example Consider the following groups: S 13 = {(x, y) : y 2 = x 3 + 2, x, y Z 13 } {id g } S 19 = {(x, y) : y 2 = x 3 + 2, x, y Z 19 } {id g } Through our investigation, we discovered the following: S 13 = 19 S 19 = 13 Thus, (13, 19) is an elliptic pair. Note that we found this to be true for only a certain b value, as this is not an elliptic pair when b is a different value.

11 Observations of Elliptic Pairs There were certain properties that we observed about elliptic pairs. All elliptic pairs are primes of the form 6n + 1 If the prime was of the form 6n 1 then the number of solutions was equal to that prime plus the identity element for all values of b. There seemed to be a correlation between each pair having a b value that was prime. However this trend did not continue as we looked at larger pairs.

12 Consequences of Observations Even though there seemed to be a pattern for pairs being created there was no relation that followed a set amount of rules. This is one of the reasons why Elliptic Curve Cryptology is such a secure form of encryption. Unlike RSA, which uses the difficulty of factoring the product of large prime numbers, there are no proven shortcuts as of yet that give solutions to elliptic curves that are faster then going through the entire process.

13 Elliptic Cycle An elliptic Cycle is very similar to what an elliptic pair is. Using the same S p notation we have an elliptic cycle when the following is observed. p 1 = S p0 p 2 = S p1 p 3 = S p2 p 0 = S p3 Except the cycle can be any length long. In our beginning data gathering we have not found any elliptic cycles.

14 Elliptic List An elliptic list is similar to elliptic cycles except that the last number need not equal the first prime. p 1 = S p0 p 2 = S p1 p 3 = S p2 n = S p3

15 Example of elliptic list An example of an elliptic list that we have found is the following when b = 34. p 0 = 67 and S p0 = 79 = P p1 p 1 = 79 and S p1 = 97 = P p2 p 2 = 97 and S p2 = 112 Since the last prime and first prime are not the same we don t have a cycle yet, but we do have a list.

16 Weak Elliptic Structures Are exactly like Proper elliptic structures except the b can change between sets. Using a modification for our program above. Recursively searched for these weak elliptic structures. Example: 19 and 13 are proper elliptic pairs as well as weak elliptic pairs. 19 travels to 13 through b {2, 3, 14} 13 travels to 19 through b {2, 11}

17 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

18 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

19 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

20 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

21 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

22 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

23 Elliptic Graphs Here is the same graph generated from 19 and 13 not showing the b values

24 Elliptic Graphs con t Can visualise them by looking at a set of points

25 An Elliptic Chain A list that ends up connecting up and down we call an elliptic chain Each directed line is a b that connects.

26 Creating Up and Down Selecting a b between each number that connects them we can use the Chinese Remainder Theorem to make 2 unique integers that will connect up and down the list. Constructing the numbers from the previous graph we get. d = 13 (mod 73) d = 7 (mod 67) d = 3 (mod 79) d = 58 (mod 97) d = u = 5 (mod 103) u = 13 (mod 67) u = 6 (mod 79) u = 5 (mod 97) u =

27 What is next Next is going to be a summer long exhaustive search of primes and the cardinality of the groups that come from each prime. It will store the mod prime, then the number so solutions that have a prime cardinality and number that have a non prime cardinality. Since Elliptic cycles exist at 5 digit primes looking at the graphs of weak elliptic primes is a future goal of our project. Proving that each member of a weak elliptic prime s graph is isomorphic to each other.