DESC:ENCRYPTION AND DECRYPTION OF MESSAGES ,CLAIMS:1. A method for encryption of messages transmitted over a computing network (702), the method comprising:
obtaining, by a processor (706), a pair of private keys (a0, d0), one or more predetermined numbers, a predetermined key (e), and a message;
generating, by the processor (706), one or more sequences of arithmetic progressions based on a first set of properties and the pair of private keys (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determining, by the processor (706), sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and one or more master arithmetic progressions in each of the sub-collections based on the one or more predetermined numbers;
selecting, by the processor (706), leading terms and common differences of the one or more master arithmetic progressions from each of the sub-collections;
determining, by the processor (706), master private keys based on at least the leading terms and the common differences of each master arithmetic progression; and
encrypting, by the processor (706), the message into a cipher-text using the master private keys.
2. The method as claimed in claim 1 further comprising transmitting the cipher-text to a computing device in the computing network.
3. The method as claimed in claim 1, wherein the message is encoded to a number (M) and split into smaller numbers (m¬1, m2, …, mi) before encryption.
4. A method for decryption of cipher-text received over a computing network (702), the method comprising:
obtaining, by a processor (706), a cipher-text, a pair of private keys (a0, d0), a predetermined key (e), and one or more predetermined numbers;
generating, by the processor (706), one or more sequences of arithmetic progressions based on a first set of properties and the pair of private keys (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determining, by the processor (706), sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and master arithmetic progressions in each of the sub-collections based on the one or more predetermined numbers;
selecting, by the processor (706), leading terms and common differences of the master arithmetic progressions from each of the sub-collections;
determining, by the processor (706), master private keys based on the leading terms and the common differences of each master arithmetic progression; and
decrypting, by the processor (706), the cipher-text into a message using the master private keys.
5. The method as claimed in claim 4, wherein the cipher text (C) is split into smaller numbers (c¬1, c2, …, ci) before decryption.
6. The method as claimed in claim 1 or 4, wherein the master private keys are determined based on at least the leading terms, the common differences, and a second common difference.
7. The method as claimed in claim 1 or 4, wherein the pair of private keys (a0, d0) is a pair of numbers, and wherein the numbers are the leading term and the common difference of the first arithmetic progression in the sequence.
8. The method as claimed in claim 7, wherein the pair of numbers are co-prime numbers.
9. The method as claimed in claim 1 or 4, wherein the first set of properties includes:
i) the product of jth terms of ith and (i+1)th arithmetic progressions is congruent to G modulo (j+1)th term of the ith arithmetic progression, where G is the greatest common divisor of the pair of private keys;
ii) the common difference of the arithmetic progressions are in non-increasing order;
iii) the product of common difference of (i+1)th arithmetic progression and the leading term of ith arithmetic progression is congruent to G modulo the common difference of the ith arithmetic progression; and
iv) the leading term of the (i+1)th arithmetic progression is equal to the addition of the common difference of the (i+1)th arithmetic progression and a predetermined function
10. The method as claimed in claim 1 or 4, wherein the second set of properties include:
i) the difference between the common differences of consecutive arithmetic progressions in the sub-collection is equal; and
ii) the sub-collection is maximal.
11. The method as claimed in claim 1 or 4, wherein the one or more predetermined numbers and the pair of private keys (a0, d0) are agreed between a first computing device (602) and a second computing device (604) of the computing network (702).
12. A method for encryption of messages transmitted over a computing network (702), the method comprising:
receiving, by the processor (706), one or more predetermined numbers, a predetermined key (e), and a public key (?), wherein the public key (?) is based on a secret key pair (p, q);
selecting, by the processor (706), a trap-door parameter (s), wherein the trap-door parameter (s) is co-prime to the public key (?);
determining, by the processor (706), a pair of private keys (a0, d0) and a trap-door information (v) based on one or more of the public key (?), the one or more predetermined numbers, and the trap-door parameter (s);
generating, by the processor (706), one or more sequences of arithmetic progressions based on a first set of properties and the private key (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determining, by the processor (706), sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and one or more master arithmetic progressions in each of the sub-collections based on the one or more predetermined numbers;
selecting, by the processor (706), leading terms and common differences of the one or more master arithmetic progressions from each of the sub-collections;
determining, by the processor (706), master private keys based on at least the leading terms and the common differences of each master arithmetic progression; and
encrypting, by the processor (706), the message into a cipher-text using the master private keys.
13. The method as claimed in claim 12, wherein the trap-door information (v) and the cipher-text are transmitted to a second computing device (604).
14. A method for decryption of cipher-text received over a computing network (702), the method comprising:
receiving, by the processor (706), one or more predetermined numbers, predetermined key (e), and a secret key pair (p, q);
receiving, by the processor (706), a trap-door information (v) from a computing device;
determining, by the processor (706), a pair of private keys (a0, d0) from the trap-door information (v) based on the secret key pair (p, q);
generating, by the processor (706), a sequence of arithmetic progressions based on a first set of properties and the pair of private keys (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determining, by the processor (706), sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and master arithmetic progressions in each of the sub-collections based on the one or more predetermined numbers;
selecting, by the processor (706), leading terms and common differences of the master arithmetic progressions from each of the sub-collections;
determining, by the processor (706), master private keys based on the leading terms and the common differences of each master arithmetic progression; and
decrypting, by the processor, the cipher-text into a message using the master private keys.
15. The method as claimed in claim 12 or 14, wherein the first set of properties includes:
i) the product of jth terms of ith and (i+1)th arithmetic progressions is congruent to G modulo (j+1)th term of the ith arithmetic progression, where G is the greatest common divisor of the pair of private keys;
ii) the common difference of the arithmetic progressions are in non-increasing order;
iii) the product of common difference of (i+1)th arithmetic progression and the leading term of ith arithmetic progression is congruent to G modulo the common difference of the ith arithmetic progression; and
iv) the leading term of the (i+1)th arithmetic progression is equal to the addition of the common difference of the (i+1)th arithmetic progression and a predetermined function.
16. The method as claimed in claim 12 or 14, wherein the second set of properties include:
i) the difference between the common differences of consecutive arithmetic progressions in the sub-collection is equal; and
ii) the sub-collection is maximal.
17. The method as claimed in claim 12 or 14, wherein the one or more predetermined numbers, the public key (?), predetermined key (e), and the secret key pair (p, q) are generated by a key management system.
18. The method as claimed in claim 12 or 14, wherein the secret key pair (p, q) is a pair of prime numbers, and wherein a product of the prime numbers is the public key (?).
19. The method as claimed in claim 12 or 14, wherein the trap-door parameter (s) is lesser than the public key (?) and greater than half of the public key (?).
20. The method as claimed in any of the preceding claims, wherein the pair of private keys (a0, d0) is refreshed to a new pair of numbers based on a key refreshing model.
21. A method for user authentication, the method comprising:
generating, by a processor of a computing device, a first sequence F(u, w) of arithmetic progressions based on a first set of properties and an authenticating key (u) and master key (w);
selecting and storing, by the processor, leading terms of predetermined arithmetic progressions in the first sequence F(u, w);
receiving, by the processor, an input key (x) provided by a user for authentication;
generating, by the processor, a second sequence F(x, w) of arithmetic progressions based on the first set of properties and the input key (x) and master key (w);
determining, by the processor, whether the leading terms of the predetermined arithmetic progressions of the second sequence F(x, w) are equal to the stored leading terms of the predetermined arithmetic progressions of the first sequence F(u, w); and
authenticating, by the processor, the user based on the determination.
22. The method as claimed in claim 21, wherein the first set of properties includes:
i) the product of jth terms of ith and (i+1)th arithmetic progressions is congruent to G modulo (j+1)th term of the ith arithmetic progression, where G is the greatest common divisor of the pair of private keys;
ii) the common difference of the arithmetic progressions are in non-increasing order;
iii) the product of common difference of (i+1)th arithmetic progression and the leading term of ith arithmetic progression is congruent to G modulo the common difference of the ith arithmetic progression; and
iv) the leading term of the (i+1)th arithmetic progression is equal to the addition of the common difference of the (i+1)th arithmetic progression and a predetermined function.
23. A computing device (704) for encrypting messages, the computing device (704) comprising:
a memory (708);
a processor (706) coupled to the memory (708);
an encryption module (608) executable by the processor, wherein the encryption module (608) is configured to:
receive a pair of private keys (a0, d0), a predetermined key (e), one or more predetermined numbers, and a message to be transmitted to a second computing device;
generate a sequence of arithmetic progressions based on a first set of properties and the pair of private keys (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determine sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and one or more master arithmetic progressions in each of the sub-collection based on the one or more predetermined numbers;
select leading terms and common differences of the master arithmetic progressions from each of the sub-collections;
determine master private keys based on the leading terms and the common differences of each master arithmetic progressions; and
encrypt the message into a cipher-text using the master private keys.
24. The computing device (704) as claimed in claim 23, wherein the encryption module (608) is further configured to:
receive the one or more predetermined numbers, and a public key (?), wherein the public key (?) is based on a secret key (p, q);
select a trap-door parameter (s), wherein the trap-door parameter (s) is co-prime to the public key (?); and
determine the pair of private keys (a0, d0) and a trap-door information (v) based on one or more of the public key (?), the one or more predetermined numbers, and the trap-door parameter (s).
25. A computing device (704) for decrypting messages, the computing device (704) comprising:
a memory (708);
a processor (706) coupled to the memory (708);
a decryption module (610) executable by the processor, wherein the decryption module (610) is configured to:
receive a cipher-text, a pair of private keys (a0, d0), a predetermined key (e), and one or more predetermined numbers;
generate a sequence of arithmetic progressions based on a first set of properties and the pair of private keys (a0, d0), wherein the number of sequences generated is the greatest common divisor of the pair of private keys (a0, d0);
determining, by the processor (706), a master sequence from the one or more sequences based on the predetermined key (e);
determine sub-collections of arithmetic progressions in the master sequence based on a second set of properties, and a master arithmetic progression in each of the sub-collection based on the one or more predetermined numbers;
select leading terms and common differences of the master arithmetic progression from each of the sub-collection;
determine master private keys based on at least the leading term and the common difference of each master arithmetic progression; and
decrypt the cipher-text into a message using the master private keys.
26. The computing device (704) as claimed in claim 25, wherein the decryption module (610) is further configured to:
receive the one or more predetermined numbers, and a secret key (p, q);
receive the cipher-text and a trap-door information (v) from a first computing device; and
determine the private key (a0, d0) from the trap-door information (v) based on the secret key (p, q).
27. The computing device (704) as claimed in claim 23 or 25, wherein the encryption and decryption module (610) comprises:
a key generation module (710-1) for generating a pair of private keys (a0, d0);
a parameter generation module (710-2) for generating leading terms and common differences based on the first set of properties; and

a key refreshing module (710-3) for refreshing the private keys based on a key refreshing model.