Using Vojvoda approach  we demonstrate that cryptographical primitives proposed in  are vulnerable relative to chosen ciphertext attack and chosen plaintext attack. We develop proposed in  modifications and add some new modifications of known quasigroup based stream ciphers [19, 26]. Systems of orthogonal n-ary groupoids are used.
Buchsteiner loops are those which satisfy the identity x\(xy · z) = (y · zx)/x. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent. A loop (Q, ·) is a set Q together with a binary operation · such that for each a, b ∈ Q, the equations a · x = b and y · a =… (More)