How did the NSA hack our emails?

Using Math in Elliptic curve…

NSA Surveillance (an extra bit) – Numberphile:

How did the NSA hack our emails?

Using Math in Elliptic curve…

NSA Surveillance (an extra bit) – Numberphile:

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

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Elliptic Curve (E):

1. Every E(Q) over Q is Modular

2. Δ = -16(4A³ + 27B²)

Elliptic Curve (E):

1. Every E(Q) over Q is Modular

2. Δ = -16(4A³ + 27B²)

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.