We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Di e-Hellman algorithm. We prove the existence of non-associative groupoids which are simultaneously power commuting and not power-associative. Note: Originally published in Diskretnaya Matematika (2014) 26, №3, 45-64 (in Russian).We study the possibility to generalize the well known Di e-Hellman algorithm [7] realizing an open key distribution protocol by means of cyclic group to the case when a non-associative groupoid is used instead of a group. Let us introduce necessary de nitions.For an element of a nite groupoid ( , * ) and given , ∈ ℕ we de ne the right -th and left -th powers respectively by equalitiesWe say that has commuting right powers, or that is a CRP-element ifIf this identity is valid for any element ∈ , then we say that ( , * ) is a CRP-groupoid. Similarly, by the identity ∀ , ∈ ℕ : [ ][ ] = [ ][ ] , we de ne elements and groupoids with commuting left powers, CLP-elements and CLP-groupoids, respectively. Algorithm 1 of the open key distribution. After choosing a (non-secret) CRP-element of a groupoid the users and independently choose arbitrary numbers , ∈ ℕ and exchange the elements [ ] and [ ] with each other. Then they form the common secret key [ ][ ] = [ ][ ] . The complexity of revealing the secret key by an observer which has access to the open information , [ ] , [ ] does not exceed the complexity of the right discrete logarithm in the groupoid computation, i.e. the complexity of solving the equation [ ] = ℎ.A natural generalization of this approach to the construction of open key distribution procedures is the combination of right and left powers.We say that a groupoid ( , * ) is a groupoid with commuting powers (CP-groupoid) if it is a CLP-and CRPgroupoid and, moreover, for any ∈ and any , ∈ ℕ the following equality holds:We say that an element of any groupoid ( , * ) is a CP-element if it generates a CP-groupoid.Let be a CP-element.

Описывается алгоритм, сводящий с полиномиальной сложностью задачу дискретного логарифмирования в конечномерной алгебре к задаче дискретного логарифмирования над конечным полем. Ключевые слова: открытое распределение ключей, неассоциативные группоиды, конечномерные алгебры, дискретное логарифмирование.

Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.