Encipherment of digital sequences by reversible transposition methods
Methods for transposing elements of a sequence according to a rule, wherein the rule is derived from pseudo-noise or pseudo-noise like binary and non-binary sequences are disclosed. Sequences of transposed symbols can be recovered by applying a reversing rule. Sets of orthogonal hopping and transposition rules are created by applying transposition rules upon themselves. Sets of orthogonal hopping and transposition rules are also created from binary and non-binary Gold sequences.
1. A method for transposing a first sequence of symbols into a second sequence of symbols, comprising:
inputting on a processor the first sequence of symbols;
applying by the processor on the first sequence of symbols to generate the second sequence of symbols a reversible transposition rule being expressed as a sequence of unique numbers derived from a third sequence of n-valued symbols that is pseudo-random with n>2 wherein the third sequence contains at least n k −1 different overlapping words, each overlapping word having k n-valued symbols with k an integer greater than 2; and
the processor outputting the second sequence of symbols, wherein a symbol is represented by a signal.
2. The method as claimed in claim 1 , wherein the third sequence contains n k different overlapping words, each overlapping word having k n-valued symbols with k an integer>2.
3. The method as claimed in claim 2 , wherein the reversible transposition rule comprises:
decomposing the third sequence into consecutive n k words of n-valued symbols; and
replacing each word by a unique number to form the sequence of unique numbers.
4. The method as claimed in claim 1 , wherein the reversible transposition rule comprises the steps:
decomposing the third sequence into consecutive n k −1 words of n-valued symbols; and
replacing each of a plurality of words of the consecutive n k −1 words by a unique number to form the sequence of unique numbers.
5. The method as claimed in claim 1 , further comprising:
creating a sequence of s unique numbers with s>n from the third sequence, wherein each unique number has a position in the sequence of s unique numbers;
identifying a unique number in a p th position in the sequence of s unique numbers with p<s; and
placing a symbol in a p th position in the second sequence that is identical to a symbol in the first sequence of which a position is determined by the unique number in the p th position in the sequence of s unique numbers.
6. The method as claimed in claim 1 , further comprising:
creating a sequence of s unique numbers with s>n from the third sequence, wherein each unique number has a position in the sequence of s unique numbers;
identifying a unique number in a p th position in the sequence of s unique numbers with p<s; and
placing a symbol that is identical to a symbol in a p th position in the first sequence in a position in the second sequence, the position in the second sequence being determined by the unique number in the p th position in the sequence of s unique numbers.
7. A method for allocating a plurality of transmission channels to at least a first and a second user by creating and applying a plurality of orthogonal hopping rules from a first transposition rule derived from a pseudorandom sequence of n-valued symbols with n an integer greater than 1, comprising the steps:
representing the first transposition rule as a sequence of unique numbers;
creating a second transposition rule orthogonal to the first transposition rule by applying the first transposition rule to the sequence of unique numbers;
implementing the first and the second transposition rules as a first and a second hopping rule on a processor that is part of a communication system; and
the processor allocating the plurality of transmission channels to the first user in accordance with the first hopping rule and the plurality of transmission channels to the second user in accordance with the second hopping rule.
8. A method for allocating a plurality of transmission channels to at least a first and a second user by creating and applying a plurality of orthogonal hopping rules from a set of p Gold sequences, each Gold sequence having p n-valued symbols and each Gold sequence can be formed from a first and second sequence, the first and second sequences being able to be decomposed into p different words of k n-valued symbols with n>2, p>2 and k>1, comprising:
selecting a Gold sequence from the set of p Gold sequences;
decomposing the selected Gold sequence from the set of p Gold sequences into a first sequence of up to p different words of 2*k n-valued symbols;
replacing each word of 2*k n-valued symbols in the first sequence by a unique number to form a sequence of unique numbers as a first hopping rule;
selecting a new Gold sequence from the set of p Gold sequences to form a second hopping rule;
implementing the first and the second hopping rule on a processor that is part of a communication system; and
the processor allocating the plurality of transmission channels to the first user in accordance with the first hopping rule and the plurality of transmission channels to the second user in accordance with the second hopping rule.
9. The method as claimed in claim 8 , further comprising normalizing the hopping rules to the total number of hops included in the hopping rules.
10. A system for transposing symbols in a first sequence of symbols into a second sequence of transposed symbols by applying a reversible transposition rule, comprising:
a memory;
a processor enabled to retrieve instructions from the memory to perform the steps:
receiving the first sequence of symbols on an input of the processor;
providing the second sequence of transposed symbols on an output of the processor, wherein a symbol is represented by a signal; and
wherein the processor applies the reversible transposition rule on the first sequence of symbols to generate the second sequence of transposed symbols, the reversible transposition rule being expressed as a sequence of unique numbers being derived from a pseudo-random third sequence of n-valued symbols with n>2 and wherein the third sequence contains at least n k −1 different overlapping words, each overlapping word having k n-valued symbols with k an integer greater than 2.
11. The system as claimed in claim 10 , wherein the pseudo-random sequence is created from n k different words of k n-valued symbols and has n k symbols with k>2.
12. The system as claimed in claim 11 , wherein the reversible transposition rule comprises:
decomposing the third sequence into consecutive n k words of n-valued symbols with k greater than 2; and
replacing each word by a unique number to form the sequence of unique numbers.
13. The system as claimed in claim 10 , wherein the reversible transposition rule is created by:
decomposing the third sequence into consecutive n k −1 words of n-valued symbols; and
replacing each word by a unique number to form the sequence of unique numbers.
14. The system as claimed in claim 13 , further comprising the processor performing the steps:
identifying a unique number in a p th position in the sequence of unique numbers with p>1; and
placing a symbol in a p th position in the second sequence of transposed symbols that is identical to a symbol in the first sequence of symbols of which a position is determined by the unique number in the p th position in the sequence of unique numbers.
15. The system as claimed in claim 13 , further comprising the processor performing the steps:
identifying a unique number in a p th position in the sequence of unique numbers with p>1; and
placing a symbol that is identical to a symbol in a p th position in the first sequence of symbols in a position in the second sequence of transposed symbols, the position in the second sequence of transposed symbols being determined by the unique number in the p th position in the sequence of unique numbers.