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The history of cryptography is punctuated by the invention of clever systems to encipher messages and, sometime later, equally clever systems for cryptanalysing the enciphered messages to determine their meaning. Most enciphering schemes of any worth enjoy a relatively lengthy period of prominence before sufficiently determined cryptanalysts undermine their security by figuring out how to attack them. In response, cryptographers devise new and improved schemes and then the cycle repeats. Cryptographers have learned from history that it is dangerous to declare any enciphering scheme unbreakable; at best they are considered to be very secure. But there is one scheme, a scheme that has been around since 1917, that truly is unbreakable. It is the perfect cipher.

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Abstract.The object of this paper is to examine some radical properties of quadratic Jordan algebras and to show that under certain conditions, /?(23) = 23n/?(3) where 23 is an ideal of a quadratic Jordan algebra 3, R('&) is the radical of 23, and Ä(3) is the radical of 3-1. Preliminaries.We adopt the notation and terminology of an earlier paper [2] concerning quadratic Jordan algebras (defined by the quadratic operators Ux) as opposed to linear Jordan algebras (defined by the linear operators Lx). Thus we have a product Uxy linear in y and quadratic in x satisfying the following axioms as well as their linearizations:(UQJI) UX=I (1 the unit); (UQJII) Uu{x)y=UxU,JUx:Throughout this paper 3 will denote a quadratic Jordan algebra over an arbitrary ring of scalars O.Define a property R of a class of rings (e.g. associative rings or Jordan rings) to be a radical property it it satisfies the following three conditions