Synchronous stream ciphers are classified up to the used keystream into two types: either to the synchronous ciphers that are used pseudorandom keystreams, or to the perfectly-secret ciphers that are used truly random keystreams; each of these classifications has its disadvantages, so the necessity to find a method that belongs to the synchronous stream ciphers and overcomes their disadvantages is required. In this article, the author proposes a method to satisfy this purpose; the proposed method is called symbolic integer splitting method and it is a mathematical method for representing an integer in form of a certain sequence of integers by using the modular arithmetic operation. Also this method can be classified under perfectly-secret ciphers because it satisfies asymptotic secrecy under a certain condition. The new scientific results are conducted as the ability of this method to success in dealing with the disadvantages of both the traditional synchronous stream ciphers and perfectly-secret ones, likes: insecure protection of information, hide the information about the length of original message from the cryptanalyst, no need to use each gamma only once, not expensive in terms of required resources to store gammas.

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The integer splitting method is a special method that is used to encrypt the text. In this method, each text character is represented as an integer in accordance with the selected code table after that this integer is replaced on the base of another number with a sequence of k integers (k-splitting level) and this is performed by the usage of modular arithmetic. This article studies the property of asymptotic secrecy of the simple integer splitting method when it is used in the field of cryptography; the research is conducted from the point view of the hacker. The purpose of using integer splitting method is to increase the level of security and provide the ability to control the degree of protection of information. So a lemma was proven in this article and as a conclusion we can notice that the more usage of the splitting level leads to the more level of safety and protection of information.

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