Symmetric Cryptosystem Using Permutation Cyclic Subgroups
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Solution Overview
Problem
Existing symmetric cryptography systems require the central controller to hold secret keys for authentication, which is impractical and insecure, especially in scenarios with a large number of users, and they lack efficient methods for publicly modifiable encryption keys.
Innovation Solution
The use of an algebraic group structure, specifically the cyclic subgroup of permutations, allows for a publicly modifiable and secure encryption key in symmetric cryptographic systems, enabling key modification for each message or authentication without the need for the central controller to hold secret keys, utilizing the structure of permutations and their cyclic subgroups.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a central controller holds secret keys for authentication in symmetric cryptographic systems, then authentication can be performed, but the system becomes impractical and insecure when the number of users is large
Solution Approach 1:
The secret key is segmented into multiple components: a master key held by the central controller and individual user keys derived from it. Each user has a unique key pair (public key and private key) generated from the master key, allowing the central controller to authenticate users without storing all individual secret keys centrally.
Solution Approach 2:
Public keys serve as intermediaries between users and the central controller. Instead of the central controller directly holding and managing all user secret keys, the public keys mediate the authentication process, allowing verification without exposing secret keys.
2Reliability
If encryption keys are kept secret in symmetric cryptography, then security is maintained, but key modification for each message becomes difficult
Solution Approach 1:
The system enables dynamic key modification where the effective encryption key can be changed for each message or session. The public key component can be modified while maintaining the underlying secret key structure, allowing flexible key rotation without compromising security or requiring central controller intervention for each change.
Data Source
AI summary
Symmetric cryptosystems with disposable keys are characterized in that, for each message encryption, a public piece of information is attached to the cryptogram, allowing the secret encryption key to be modified. This information is therefore a public key. The invention is characterized in that the secret key, called the mother key, is a permutation p belonging to the algebraic group structure (Se, o), where Se is the symmetric group of an ordered set E and o is the function composition operation. The permutation p is specially composed of numerous disjoints in order to generate a high-order cyclic subgroup G of Se defined by {p1, p2, ..., pj, ..., pa}, where pj is the bijection pop o.....op with the function composition operation o iterated j-1 times. G is then cyclic modulo a, the integer a being the least common multiple of the integers representing the lengths of the disjoints of p.The set of disposable keys is the cyclic subgroup G; the public keys consist of any integer j, or, according to a variant of the process, of an element w of the Cartesian product of the disjoints of p if the latter are chosen such that their lengths have no common divisor. Each integer j or each element w is associated with one and only one permutation belonging to G, which is computed by the algorithm {pa} → pj or {pa} → λ. According to another embodiment, authentication processes in which j or w is a public key and each permutation p' belonging to G is a private key, according to yet another embodiment, (j, p') or (w, p') are pairs of document authenticators.According to another use of the algebraic structure (Se, o) and its cyclic subgroups, the invention is a symmetric cryptosystem associated with a Latin square characterized by the fact that the secret key is no longer the Latin square in the form of a square matrix but a uniquely disjoint permutation p belonging to Se, said permutation enabling encryption by composition of the bijection p.