The study of Elliptic Curves has been closely connected with Machine computation almost since the invention of computers -- in 1952 Emil Artin had John von Neumann perform an extensive calculation relating to elliptic curves on the IAS MANIAC computer. The fundamental papers of Birch and Swinnerton-Dyer in 1965, which gave rise to the Birch-Swinnerton-Dyer conjecture were buttressed with extensive machine computation. There has been extensive interplay between theory and computation relating to ranks of elliptic curves, Heegner points, Galois representations, Sato-Tate distributions, and many other areas. The year 2010 marks the beginning of a 25 year period in which a number of influential papers initiated a fundamental connection between elliptic curves, cryptology and the theory of computation. . Rene Schoof about fast algorithms for counting points on elliptic curves over finite fields . Hendrik Lenstra about integer factorization using elliptic curves . Victor Miller and Neal Koblitz about the security of using elliptic Curves over finite fields in a Diffie-Hellman key exchange. . Shafi Goldwasser and Joe Kilian about primality proving using elliptic curves . Len Adleman and Ming-Deh Huang about primality proving using abelian varieties . Oliver Atkin and Francois Morain about primality proving using elliptic curves. Since 1997 there has been an annual workshop on Elliptic Curve Cryptography. To celebrate the 25th anniversary of the above papers we will hold a full week meeting intermixing talks which are concerned with the applications of elliptic curves in cryptography and other fundamental results concerning elliptic curves and computation. The meeting will be held from Oct 18-22, 2010 at Microsoft Research in Redmond, Washington, USA. The organizers are Victor Miller (Center for Communications Research), William Stein and Neal Koblitz (University of Washington), and Kristin Lauter (Microsoft Research)