Math and stuff
-
Modern Cryptography and Elliptic Curves: Congruence and Equality
Exercise from P.62 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show (by example) that the congruence ax \equiv ay \pmod n does not necessarily imply that x \equiv y \pmod n. 1 \not\equiv 0 \pmod 2 but 2 \cdot 1 \equiv 2 \cdot 0 \pmod 2. On...
-
Modern Cryptography and Elliptic Curves: Division Algorithm
Exercise from P.47 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Let a, b \in \mathbb{Z}, b = 0. Then there exist unique integers q and r with a = bq + r and 0 \leq r < \abs{b}, where \abs{b} is the absolute value of \abs{b}. Let...
-
Modern Cryptography and Elliptic Curves: Multiplicity at the Point of Tangency
Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Now consider y^2 = g(x) where g is a cubic, that is the zero set of f(x, y) = y^2 − g(x). We want to see that a nonvertical tangent has multiplicity at least 2 at the...
-
Modern Cryptography and Elliptic Curves: Orders of zeros and derivatives
Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that h has a zero of order k at x = a if and only if h(a) = h’(a) = \cdots = h^{(k−1)}(a) = 0 and h^{(k)}(a) = 0, where h^{(i)} is the ith derivative of...
-
Modern Cryptography and Elliptic Curves: Examples of zeros with multiplicity
Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that the curve y = (x − a)^k intersects the x-axis with multiplicity k at x = a and with multiplicity 0 at all other points x = b. The curve is the zero set of...