Proceedings of the American Mathematical Society, 1986
It is shown that the relation ⩽ \leqslant defined by x ⩽ y x \leqslant y if and only if x y = x 2... more It is shown that the relation ⩽ \leqslant defined by x ⩽ y x \leqslant y if and only if x y = x 2 xy = {x^2} , x 2 y = x y 2 = x 3 {x^2}y = x{y^2} = {x^3} is an order relation for a class of Jordan rings and we prove that a Jordan ring R R is isomorphic to a direct product of Jordan division rings if and only if ⩽ \leqslant is a partial order on R R such that R R is hyperatomic and orthogonally complete.
We classify indecomposable finite dimensional bimodules over Jordan superalgebras D ( t ) D(t) , ... more We classify indecomposable finite dimensional bimodules over Jordan superalgebras D ( t ) D(t) , t ≠ − 1 , 0 , 1 t \neq -1,0,1 .
Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of... more Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of modern cryptology. However, security practitioners are not always familiar with probability theory, and thus fail to foresee the impact of (seemingly small) deviations from the theoretical description of a scheme at the implementation level. On the other hand, many cryptographic scenarios involve mutually distrusting parties, which need however to cooperate towards a joint goal. In order to attain assurance of the good behavior of one party, interactive validation methods (also known as interactive proof systems) are employed. Randomness is at the core of such methods, which most often will only provide relative assurance, in the sense that they will establish correctness in a probabilistic way. In this paper we will briefly discuss the role of probability theory within modern cryptology, reviewing probabilistic proof systems as a powerful tool towards efficient protocol design, and pro...
It has been known some time ago that there are one-sided group codes that are not abelian codes, ... more It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring \(F_{5}S_{4}\). The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.
A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic... more A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.
Proceedings of the American Mathematical Society, 1986
It is shown that the relation ⩽ \leqslant defined by x ⩽ y x \leqslant y if and only if x y = x 2... more It is shown that the relation ⩽ \leqslant defined by x ⩽ y x \leqslant y if and only if x y = x 2 xy = {x^2} , x 2 y = x y 2 = x 3 {x^2}y = x{y^2} = {x^3} is an order relation for a class of Jordan rings and we prove that a Jordan ring R R is isomorphic to a direct product of Jordan division rings if and only if ⩽ \leqslant is a partial order on R R such that R R is hyperatomic and orthogonally complete.
We classify indecomposable finite dimensional bimodules over Jordan superalgebras D ( t ) D(t) , ... more We classify indecomposable finite dimensional bimodules over Jordan superalgebras D ( t ) D(t) , t ≠ − 1 , 0 , 1 t \neq -1,0,1 .
Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of... more Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of modern cryptology. However, security practitioners are not always familiar with probability theory, and thus fail to foresee the impact of (seemingly small) deviations from the theoretical description of a scheme at the implementation level. On the other hand, many cryptographic scenarios involve mutually distrusting parties, which need however to cooperate towards a joint goal. In order to attain assurance of the good behavior of one party, interactive validation methods (also known as interactive proof systems) are employed. Randomness is at the core of such methods, which most often will only provide relative assurance, in the sense that they will establish correctness in a probabilistic way. In this paper we will briefly discuss the role of probability theory within modern cryptology, reviewing probabilistic proof systems as a powerful tool towards efficient protocol design, and pro...
It has been known some time ago that there are one-sided group codes that are not abelian codes, ... more It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring \(F_{5}S_{4}\). The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.
A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic... more A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.
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