Discovery of Elliptic Curve Cryptographic Private Key in O(n)

An algorithm is presented that in context of public key use of Elliptic Curve Cryptography allows discovery of the private key in worst case O(n). Charles Sauerbier December 2009 Humans have for as long as recorded history sought means to keep secrets and to communicate in secret. In modern times we have turned to electronic computations as means to facilitate both. The later problem of communicating in secret has had many interesting solutions proposed; some succeeded for a time; others failed – sometimes miserably. Counted midst their numbers is the system of Elliptic Curve Cryptography; a subject on which we will never profess to be an expert, but one that does fail miserably in practice. Elliptic Curve Cryptography[1][2][3] The ECC system is based on having defined several factors, described by the tuple (p, a, b, G, n, h). Added to these common factors are the individual’s specific private key ‘d’ and public key ‘Q’. Of the many factors one is kept as a secret – the private key ‘d’ – while all others are shared. The factors shared include the tuple of common factors that are specific to definition of the Abelian group on which the keys are premised. Of the shared factor one is the public key ‘Q’. The security of ECC rests on how difficult it is to find ‘d’ – private key – given all the shared factors, and specifically the public key ‘Q’. The relationship between the private key and the public key is defined to be Q = d G, where G – a shared common factor – is a well known base point. The operation is defined as an arithmetic function that is fundamentally a difference equation; the consequence of which is to “walk” through a sequence of points. The specific points are described by an elliptic curve. The specifics of the curve are irrelevant to factoring the private key. Weakness in Scheme While the premise that factoring the Elliptic Curve is difficult may be valid, the problem is that to obtain a key pair (d, Q) one must perform some number of operations from some point of origin. One then, in the context of a fully observable algebraic structure provides half the key pair to others. However, the computation of the key pair is analogous to computing a Fibonacci number (by brute force without using the many tricks). Both the computation of the key pair and the pair (n, Fn) amount to the application of a difference equation on an algebraic group. The Fibonacci numbers have the [1] (Washington, 2008) [2] At time of writing subject related material could be found at: http://en.wikipedia.org/wiki/Elliptic_curve http://en.wikipedia.org/wiki/Elliptic_curve_cryptography, http://www.certicom.com/index.php/ecc tutorial. [3] The animation at time of writing at http://www.certicom.com/index.php/212 adding the points p and p can be visually assistive in understanding the nature of the problem, as well as providing geometric validity of the weakness. [4] (Goldberg, 1986) (Elaydi, 2005)