The 2nd International Conference on Software, Multimedia and Communication Engineering (SMCE2017), April 23-24, 2017,

Shanghai, ChinaA Proposal of Loose

Asymmetric Cryptography Algorithm

Loc NGUYEN1/4B Ton Duc Thang, Long Xuyen, An Giang, Vietnam

1. Introduction• Although traditional asymmetric algorithm along with its

implementations are successful in keeping important documents confidential, it uses only one private key.• This research proposes the loose asymmetric (LA) algorithm to satisfy

requirement of generating many access keys. Each access key is granted to only one user.• This demand is real because a group of members needs to retrieve

same documents but each member requires confidentiality in access.• It is “loose” because many generated private keys can be revealed but I

try my best to create a scheme of LA algorithm as secured as possible.

1. Introduction• Some keys in the loose asymmetric (LA) algorithm are column vectors.

Each vector Z is associated with linear mapping v(Z) called value of Z.

• Notations s, s’, and s’’ denote original document, firstly encrypted document, and secondly encrypted document, respectively.• In LA algorithm, Z is private key if it is known by only one person,

which is totally secret. Z is semi-public key if the number of persons who know it is more than 1 but limited. Z is public key if the number of persons who know it is unlimited, which is totally public.

2. Loose asymmetric (LA) algorithm• Let E be encryption key which is used to encrypt s into s’.

• Let D be decryption key to decrypt s’ encrypted by E, according to congruence with modulo η as same as RSA scheme [1, p. 286].

2. Loose asymmetric (LA) algorithm• Let M is called mask matrix, which is nxn singular matrix whose rank is

m (m < n). Let X be access key in LA algorithm, which is deliver to user.

• X is a solution of following equation:

2. Loose asymmetric (LA) algorithm• Let X be set of solutions X (s) which is called access key set. Let |X| be cardinality

of X which is number of trusted users. X has following form:

• F = {f1, f2,…, fm} is called retrieval function set and χ = {xm+1, xm+2,…, xn} is called independent set.

2. Loose asymmetric (LA) algorithm• Firstly, we generate two large primes p1 and p2 and create two positive keys e and d according to RSA

scheme [1, p. 286].

• Keys D and E are created such that v(E) = e and v(D) = d and we generate randomly n–1 integers a12, a13,…, a1n to create matrix M:

• We generate randomly n–1 integers λ2, λ3,…, λn. Access key X is created so that X is a solution of the equation MX = D.

3. Scheme of LA algorithm• The scheme uses 3 private keys (M, E, q) and 1 semi-public key

(encoded access key X’ = X/q).• Pool S, user U, and center C are responsible for keeping E, q, and M

totally secret, respectively.

3. Scheme of LA algorithm1. Secured pool S encrypts s into s’ by private key E.2. User U requests for s by providing semi-public key X’ to center C via

communication channel L1. Consequently, C receives X’.

3. C retrieves s’ from pool S via communication channel L2.4. C creates the key D’ = MX’ and encrypts s’ into s’’ by D’.

5. C sends s’’ to U via the channel L1. Consequently, U decrypts s’’ into s by private key q.

3. Scheme of LA algorithm (noisy technique)• We generate private key q = rt and create X’ with noisy key N:

• User U sends two semi-public keys N’ and X’ to center C at step 2. C creates both noisy data w’ and s’’ from s’ by MN’ and MX’ at step 4:

• C sends both w’ and s’’ to U and then, U decrypts s’’ into s by t and w’ at step 5:

3. Scheme of LA algorithm (group acceptance)• Let Yi be compound access key of user Ui with note that Yi is XOR sequence

of all encoded access keys except Xi’.

• In step 2, suppose user Uh request for s by providing Yh to center C. If C receives all Yi (s) that all users provide as their acceptance, C will make a full XOR sequence of Yh and all Yi (s) to recover the encoded access key Xh’.

• Center C will create the key D’ from the multiplication D’ = MXh’ in step 4 as usual.

Conclusion• The ideology of the loose asymmetric (LA) algorithm is simple, which

only applies some concepts in theory of linear algebra, but the scheme of implementation is complex. For each user, the scheme uses 4 keys whereas traditional asymmetric algorithms uses only 2 keys, which causes that the efficiency is decreased 2 times.• However, LA algorithm satisfies the requirement of many users who

want to access same documents in confidential manner and it is possible to keep track access log for restricting access attack.• Moreover, LA algorithm is more practical and flexible than traditional

asymmetric algorithm with regard to the aforementioned requirement.

Conclusion• In the future, I will build up a camera system for online monitoring via mobile

phone with a group of authenticated users.• Because video stream is dynamic data, we need to create one window W1 at

pool S, one windows W2 at center C and one window W3 for each user U• Following is possible flow of streaming data:• Firstly, streaming data is encrypted firstly by private key E after flowing through W1

and then it goes to W2.• It is encrypted secondly by key D’ = MX’ after flowing through W2 and then it goes to

W3.• Finally, it is decrypted by private key q after flowing through W3. User U now can view

it.

References1. Menezes, A. J., van Oorschot, P. C., & Vanstone, S. A. (1996).

Handbook of Applied Cryptography (5th ed.). CRC Press. Retrieved from http://cacr.uwaterloo.ca/hac

2. Nguyen, V. H. (1999). Linear Algebra. Hanoi, Vietnam: Hanoi National University Publishing House.

3. Rotman, J. J. (2002). Advanced Modern Algebra (1st ed.). Prentice Hall. Retrieved from https://books.google.com.vn/books/about/Advanced_Modern_Algebra.html?id=3B98QgAACAAJ

Thank you for attention