Tag Archives: Elliptic Curve
NSA hacks our emails
How did the NSA hack our emails?
Using Math in Elliptic curve…
NSA Surveillance (an extra bit) – Numberphile:
Shimura-Taniyama-Weil Conjecture (Modularity Theorem)
Shimura and Taniyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.
The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.
It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:
Example:
— (I)
There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.
For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)
Note 1: the last solution when y=4,
Left-side = 16 + 4 = 20 = 4×5 = 0 (mod 5).
Right-side = 1-1= 0 (mod 5).
Note 2: We call the equation (I) a “Curve over a finite field” since {0, 1,2,.. p-1} is a Field with finite p elements.
Mathematicians for some time have known that if N is a prime number (p), there will be roughly p solutions.
However, the most interesting number is the difference between p and the actual number of solutions.
For N = p = 5, the above equation has actually 4 solutions,
Note: can be positive or negative.
There is a ‘general rule’ (generating function) to predict , and it is inspired from the ubiquitous Fibonacci numbers.
Recall:
Definition of the Fibonacci sequence as a recurrence relation:
Alternatively there is also a generating function for Fibonacci numbers:
Let’s expand it we get the infinite series:
The above coefficients coincide with
Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13…}
In 1954, the genius German mathematician Martin Eichler took the cue from the above, discovered another generating function:
— (II)
Let’s expand it, we get:
Let denotes the coefficient of the term
:
Eichler discovered that for any prime p,
Check:
The random numbers of solutions in the elliptic curve equation (I) lies on the generating function (II).
If we view q as a point inside a unit disc on the complex plane, there is a group of symmetries and the function (II) is invariant under this group. The function (II) is called a modular form.
The advanced generalisation of the Shimura-Taniyama-Weil Conjecture : we replace each cubic equation by a Representation of the Galois Group; and the modular form generalised by the generating function the “automorphic” function.:
Remarks:
1. The Shimura-Taniyama-Weil Conjecture is a special case of Langlands Program.
2. Weil’s “Rosetta stone”:
Number Theory -> Curves over Finite Fields -> Riemann Surfaces
References:
http://en.m.wikipedia.org/wiki/Modularity_theorem
Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY
Click below for more free loan at the Singapore National Library Branches: http://www.nlb.gov.sg/mobile/searches/view_availability/200154975
Elliptic Curve
Elliptic Curve (E):
1. Every E(Q) over Q is Modular
2. Δ = -16(4A³ + 27B²)
Elliptic Curve
Elliptic Curve (E):
1. Every E(Q) over Q is Modular
2. Δ = -16(4A³ + 27B²)
Modular Form
Modular Form (MF):
Is a function which takes Complex numbers from the upper half-plane as inputs and gives Complex numbers as outputs.
MF are notable for their high level of symmetry, determined not by a single number (2π for sine) but by 2×2 Matrices of Complex numbers.
Uses:
1. Proof of the FLT
2. Investigation of Monster Group.
3. Elliptic curve = MF
4. L-function provides dictionary for translating between Analysis and Number Theory.