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Weak mutually unbiased bases with applications to quantum cryptography and tomography. Weak mutually unbiased bases.
Mutually unbiased bases is an important topic in the recent quantum system
researches. Although there is much work in this area, many problems
related to mutually unbiased bases are still open. For example, constructing
a complete set of mutually unbiased bases in the Hilbert spaces with composite
dimensions has not been achieved yet. This thesis defines a weaker
concept than mutually unbiased bases in the Hilbert spaces with composite
dimensions. We call this concept, weak mutually unbiased bases. There is
a duality between such bases and the geometry of the phase space Zd × Zd,
where d is the phase space dimension. To show this duality we study the
properties of lines through the origin in Zd × Zd, then we explain the correspondence
between the properties of these lines and the properties of the
weak mutually unbiased bases. We give an explicit construction of a complete
set of weak mutually unbiased bases in the Hilbert space Hd, where
d is odd and d = p1p2; p1, p2 are prime numbers. We apply the concept of
weak mutually unbiased bases in the context of quantum tomography and
quantum cryptography.Egyptian government
Система криптографического преобразования чисел линейными рекуррентными формами
Рассматривается двухступенчатая система кодирования чисел, основанная на представлении чисел в виде aPn + bQn, где Pn и Qn линейные рекуррентные последовательности. Последовательности Pn и Qn определяются разложением в цепные дроби квадратичных иррациональностей вида (a + √b)/c. В системах симметричной криптографии числа a, b и c является ключами.Розглядається двоступенева система кодування чисел, заснована на представленні чисел у вигляді aPn + bQn, де Pn та Qn лінійні рекурентні послідовності. Послідовності Pn і Qn визначаються розкладанням в ланцюгові дроби квадратичних іррациональностей виду (a + √b)/c. У системах симетричної криптографії числа a, b і c є таємними ключами.The purpose of the article is to develop and study a nondeterministic system of cryptographic integer encoding by means of linear recurrent sequences. Methods. We used methods of continued fractions, properties of linear forms, and bijective encoding of natural numbers. Results. We proved as a theorem that such a system of encoding is absolutely resistant to passive crypto-attacks. With some further additions it is also resistant to stronger types of attacks