TABLE OF CONTENTS So far, we have been examining symmetric key algorithms that rely solely on secret keys for security. Now we are going to explore the realm of public key cryptography, but before we can do this, we have a significant piece of mathematics to cover.
Most standard public key algorithms are based on problems that are hard to solve in general. For example, the RSA algorithm is (loosely speaking) as secure as factoring is hard. That is, if factoring is hard, breaking RSA is, too (in practice). Similarly, elliptic curve algorithms are as hard to break as inverting point multiplication on the given curve.
In both cases, the problem becomes harder as you increase the size of the problem. In the case of RSA, as you increase the composite (public key), factoring becomes harder. Similarly, as you increase the order of the elliptic curve (do not worry if you do not know what that means at this point), the difficulty of inverting point multiplication increases.
To accommodate these larger parameters, we must deploy algorithms known collectively as BigNum algorithms.
As developers, you are most likely aware of the size limitations of your preferred data types. In C, for example, the int data type can represent only up to 32767 in a portable fashion. Even though on certain platforms it may be 32 or even 64 bits in length, you cannot always count on that.
Most programming languages at...
