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Semirings and Semigroup Actions in PublicKey Cryptography
, 2002
"... In this dissertation, several generalizations of cryptographic protocols based on the Discrete Logarithm Problem (DLP) are examined. ..."
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In this dissertation, several generalizations of cryptographic protocols based on the Discrete Logarithm Problem (DLP) are examined.
On finite congruencesimple semirings
 J. Algebra
"... In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zeromultiplication rings of prime order, matrix rings over finite fields, and those that are additively idempotent. ..."
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In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zeromultiplication rings of prime order, matrix rings over finite fields, and those that are additively idempotent.
THE ENDOMORPHISM SEMIRING OF A SEMILATTICE
"... Abstract. We prove that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and describe its monolith. The endomorphism semiring is congruence simple if and only if the semilattice has both a least and a largest element. 1. ..."
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Abstract. We prove that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and describe its monolith. The endomorphism semiring is congruence simple if and only if the semilattice has both a least and a largest element. 1.
Public Key Cryptography Based on Simple Modules Over Simple Rings
 IN PROCEEDINGS OF MTNS 2002
, 2002
"... The Diffie Hellman key exchange and the ElGamal oneway trapdoor function are the basic ingredients of public key cryptography. Both these protocols are based on the hardness of the discrete logarithm problem in a finite ring. In this paper we show how the action of a ring on a module gives rise t ..."
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The Diffie Hellman key exchange and the ElGamal oneway trapdoor function are the basic ingredients of public key cryptography. Both these protocols are based on the hardness of the discrete logarithm problem in a finite ring. In this paper we show how the action of a ring on a module gives rise to a generalized DiffieHellman and ElGamal protocol. This leads naturally to a cryptographic protocol whose difficulty is based on the hardness of a particular control problem, namely the problem of steering the state of some dynamical system from an initial vector to some final location.
FINITELY GENERATED ALGEBRAIC STRUCTURES WITH VARIOUS DIVISIBILITY CONDITIONS
"... Abstract. Infinite fields are not finitely generated rings. Similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutat ..."
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Abstract. Infinite fields are not finitely generated rings. Similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutative semiring that is a group with respect to multiplication, can be twogenerated only if it belongs to the subclass of additively idempotent semirings; this class is equivalent to ℓgroups. 1.
THE SEMIRING OF 1PRESERVING ENDOMORPHISMS OF A SEMILATTICE
"... Abstract. We prove that the semirings of 1preserving and of 0,1preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way. 1. ..."
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Abstract. We prove that the semirings of 1preserving and of 0,1preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way. 1.
FINITELY GENERATED COMMUTATIVE DIVISION SEMIRINGS
"... Abstract. Onegenerated commutative division semirings are found. The aim of this (partially expository) note is to find all onegenerated (commutative) division semirings (see Theorem 8.5). In particular, all such semirings turn out to be finite. To achieve this goal, we have to correct some result ..."
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Abstract. Onegenerated commutative division semirings are found. The aim of this (partially expository) note is to find all onegenerated (commutative) division semirings (see Theorem 8.5). In particular, all such semirings turn out to be finite. To achieve this goal, we have to correct some results from [1] (especially Proposition 12.1 of [1]) and to complete some results from [2]. Anyway, all the presented results are fairly basic and (with two exceptions) we shall not attribute them to any particular source. 1.
ONE VERY PARTICULAR EXAMPLE OF A CONGRUENCESIMPLE SEMIRING
"... Abstract. The existence of congruencesimple semiring S with nonconstant multiplication such that 2s = 3t for all s, t ∈ S and S + S = S is proved and hence the most enigmatic class of congruencesimple semirings is not empty. 1. ..."
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Abstract. The existence of congruencesimple semiring S with nonconstant multiplication such that 2s = 3t for all s, t ∈ S and S + S = S is proved and hence the most enigmatic class of congruencesimple semirings is not empty. 1.