By John Perry

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**Example text**

We claim that (M, ×) is isomorphic to (N, +). To see why, let f : M −→ N by f ( x a ) = a. First we show that f is a bijection. To see that it is one-to-one, let t , u ∈ M, and assume that f ( t ) = f ( u ). By definition of M, t = x a and u = x b for a, b ∈ N. By definition of f , f ( x a ) = f x b ; by substitution, a = b . In this case, x a = x b , so t = u. We assumed that f ( t ) = f ( u ) for arbitrary t , u ∈ M, and showed that t = u; that proves f is one-to-one. To see that it is onto, let a ∈ N.

It turns out that intersects E at another point S = ( s1 , s2 ) in R2 . Define P + Q = ( s1 , −s2 ). The last two statements require us to ensure that, given two distinct and finite points P , Q ∈ E, a line connecting them intersects E at a third point S. 4 shows the addition of P = 2, − 6 and Q = (0, 0); the line intersects E at S = −1/2, 6/4 , so P + Q = −1/2, − 6/4 . Exercises 5. 3. A plot of the elliptic curve y 2 = x 3 − x. 78. Let E be an arbitrary elliptic curve, defined as the roots of a function f ( x, y ) = ∂f ∂f y 2 − x 3 − a x − b .

2 + 1, 2 + 2 is a point on E. (b) Verify that (c) Compute the cyclic group generated by 2 + 1, 2 + 2 in E. 18 AA represents the field A of algebraic real numbers, which is a fancy way of referring to all real roots of all polynomials with rational coefficients. 18 Here 5. 4. Addition on an elliptic curve From then on, the symbol E represents the elliptic curve. You can refer to points on E using the command P = E(a , b , c ) where • if c = 0, then you must have both a = 0 and b = 1, in which case P represents P∞ ; but • if c = 1, then substituting x = a and y = b must satisfy equation 5.