Cryptographic Protocols on the non-commutative Ring R AZIZ BOULBOT, ABDELHAKIM CHILLALI, ALI MOUHIB Sidi Mohamed Ben Abdellah University Department of Mathematics, Physical and Computer FP, LSI, Taza Morocco aziz.boulbot@usmba.ac.ma abdelhakim.chillali@usmba.ac.ma Abstract: In this paper we introduce one of the most famous problems in a non commutative ring R. In particular we are interested in cryptography is mainly encryption based on conjugal classical problem in R. We study the problem of conjugal over this non commutative ring. The problem as stated is generally impossible to solve. Next, we describe a new encryption scheme over this ring based on this problem. Key–Words: Finite field, Finite ring, Local ring, Cryptography. 1 Introduction Ever since the discovery of public-key cryptography by Diffie and Hellman in the year 1976 see, [5], the necessity for total privacy of digital data has become stronger and stronger, especially since the internet has become an indispensable part of both our pri- vate and work lives. Naturally, the question for more and more secure encryption schemes arose in the past few decades. One way to achieve confidentiality in applications, such as online banking, electronic vot- ing, virtual networks etc , are homomorphic and es- pecially fully homomorphic cryptographic schemes. Fully homomorphic cryptosystems or privacy homo- morphisms were introduced by Rivest, Adleman, and Dertouzous in 1978 see, [6] . In their paper they asked for a way to allow a third, untrusted party to carry out extensive computation on encrypted data, with- out having to decrypt first. Unfortunately, shortly af- ter its publication, major security as were found in the original proposed schemes of Rivest and al. The search for fully homomorphic cryptosystems began. The aim of homomorphic cryptography is to ensure privacy of data in communication and storage pro- cesses, such as the ability to delegate computations to untrusted parties. If a user could take a problem defined in one algebraic system and encode it into a problem in a different algebraic system in a way that decoding back to the original algebraic system is hard, then the user could encode expensive computations and send them to the untrusted party. This untrusted party then performs the corresponding computation in the second algebraic system, returning the result to the user. Upon receiving the result, the user can decode it into a solution in the original algebraic system, while the untrusted party learns nothing of which compu- tation was actually performed. Asked: ”Is there an encryption function Enc() such that both Enc(x + y) and Enc(x.y) are easy to compute from Enc(x) and Enc(y)?” Definition 1 A public-key encryption scheme E is a tuple, (K,E,D) of probabilistic polynomial-time al- gorithms (1) The key generation algorithm K takes the secu- rity parameter k as input and outputs a pair of keys (pk, sk). I refer to the first of these as the public key and the second as the private key or secret key. I as- sume that pk and sk each have length at least k, and that k can be determined from pk, sk. (2) The encryption algorithm E takes a public-key pk and a string m called the message from some under- lying message space M as input. It produces a cipher text c from an underlying cipher text space C , denoted as c = Encpk(m) or simple c = Enc(m), if it is ob- vious which public key is in use. (3) The decryption algorithm D takes a private-key sk and a cipher text c as input, and produces an output message m. Without loss of generality we assume that Dec is deterministic, and write this as m := Decsk(c). 2 Definitions and Notation Let d be a positive integer and q = p d be a power of a prime number p ≥ 5. Let F q a finite field of charac- teristic p and order q. We define the set F q [e],e 2 = e as: F q [e] := {α + βe|(α, β ) ∈ F q × F q and e 2 = e}. Aziz Boulbot et al. International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcm ISSN: 2367-895X 138 Volume 2, 2017