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This page provides an overview of what ECC is. The primary advantage of using Elliptic Curve based cryptography is reduced key size and hence speed. Elliptic curve based algorithms use significantly smaller key sizes than their non elliptic curve equivalents.
The difference in equivalent key sizes increases dramatically as the key sizes increase. The approximate equivalence in security strength for symmetric algorithms compared to standard asymmetric algorithms and elliptic curve algorithms is shown in the table below. As can be seen, to get equivalent strength to a bit symmetric key, a standard asymmetric algorithm would have to use an enormous key of bits. Keys of this size are typically not practical due to the amount of processing power that would be required, and therefore the speed of the operations.
However, with elliptic curve algorithms, the equivalent key length is bits, which is entirely practical. First of all some terminology. We need to define what is meant by a field. In essence a field is a set of elements with operations defined for the elements of that set that equate to something like addition, substraction, multiplication and division.
The elements could be numbers, or they could be something else entirely. In order to be a field the following conditions also have to be met:. A finite field is simply a field where the set has a finite number of elements. So, for example, the set of all integers could not be used as the basis for a finite field because there are an infinite number of them.
However the set of integers from 0 to could form the basis of a finite field. So now we can define what an Elliptic Curve is. In general an Elliptic Curve is one of the form: In Elliptic Curve Cryptography we further restrict this such that x, y, a and b are elements of a finite field.
Ok, so far so good - but now it gets a bit more complicated! As well as the points on our curve we add an additional "special" point known as infinity. Using this set of points i. Points on a curve are given in terms of their x and y co-ordinates, x, y. Point Addition is essentially an operation which takes any two given points on a curve and yields a third point which is also on the curve. The maths behind this gets a bit complicated but think of it in these terms.
Plot two points on an elliptic curve. Now draw a straight line which goes through both points. That line will intersect the curve at some third point. That third point is the result of the addition operation. Point Doubling is similar and can be thought of as adding a point to itself. Imagine a point on the curve and draw a straight line which is a tangent to the curve at that point. The result of the Point Doubling operation is where that tangent line intersects the curve at some other point.
Point multiplication is the operation of taking a point on the curve and "multiplying" it by some number. In practice this is achieved through repeated addition and doubling operations.
So with our set of points on a curve plus the special point, infinity we can start doing something useful. First of all we pick a point on the curve called the generator we'll call it g. The set of points generated by repeatedly adding g to itself, along with the Point Addition operation together form a mathematical structure known as a group. If you are lucky then you may have chosen a curve and a g, such that continually adding g to itself will eventually visit all of the possible points on the curve - but often this is not the case.
The number n as defined above, is called the order of g. For various complicated mathematical reasons it also turns out that the total number of points that exist on the curve is divisble by n. Dividing the total number of points by n gives you another number known as the cofactor. The security of Elliptic Curve Cryptography comes from the fact that given some point on the curve kg, where k is a number and g is the known generator point , it is difficult to work out what the value of k is.
This is known as the discrete logarithm problem. In principle there are many different types of field that could be used for the values x and y of a point x, y. In practice however there are two primary ones used, and these are the two that are supported by the OpenSSL EC library.
The simplest is typically referred to as the prime field F p where p is a prime number. In cryptographic applications p must be a very large prime number. The elements of the set are simply the numbers 0 through to p-1, and addition and multiplication over the field have the normal meaning for modular or clock arithmetic. The next common type of field is referred to as the binary field F 2 m.
Elements of a binary field are typically represented as polynomials and not as numbers. So for example an element could be:. Addition of such polynomials is done as normal but with the result of each term reduced modulo 2. Multiplication in the binary field is done respective to an irreducible polynomial. Multiplication of polynomials is done in the normal way and the result is then divided by the irreducible polynomial.
The remainder is the result of the multiplication. Fortunately unless you are defining a new curve not recommended unless you know what you are doing , or you are using an unusual curve that OpenSSL does not have support for, you can usually utilise one of the named curves that are built-in to OpenSSL. These are a set of well known and widely used curves.
If a custom curve needs to be created, then it can be done as follows. This example code creates the same curve as the code above, but creates it "manually". In this example a prime field is being used, and the prime number is provided in the variable p. If a binary field was being created instead then a bit string representing the irreducible polynomial would have been provided in the p variable.
For further information on the low level EC functions being used refer to the EC manual pages:. There is no key data in it. To generate them using the low level API this can be done as follows:. Note that this operation generates a public and private key pair. Alternatively you may already know either the private key, the public key, or both.
If you set the private key then you must also set the public key. There have been occasional questions on the openssl-users email list from people who only have the private key but do not know the public key. Failure to do so will result in a SSL error of 0xa0c1 no shared cipher at the server. As an example, the following creates a elliptic curve key and saves it using a named curve rather than an expanded list of group paramters:.
If you want to detect the flags after reading a key or certificate from disk, then use the following code:. The certificates below were dumped with openssl x -in server-ecdsa-cert. The certificate on the left can be used with SSL server using ECDSA, but the certificate on the right cannot because it will result in 0xa0c1 at the server.
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