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A Study of Non-Abelian Public Key Cryptography

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A Study of Non-Abelian Public Key Cryptography International Journal of Network Security, Vol.20, No.2, PP.278-290, Mar. 2018 (DOI: 10.6633/IJNS.201803.20(2).09) 278 A Study of Non-Abelian Public Key Cryptography Tzu-Chun Lin Department of Applied Mathematics, Feng Chia University 100, Wenhwa Road, Taichung 40724, Taiwan, R.O.C. (Email: [email protected]) (Received Feb. 12, 2017; revised and accepted June 12, 2017) Abstract P.W. Shor pointed out that there are polynomial-time algorithms for solving the factorization and discrete log- Nonabelian group-based public key cryptography is a rel- arithmic problems based on abelian groups during the atively new and exciting research field. Rapidly increas- functions of a quantum computer. Research in new cryp- ing computing power and the futurity quantum comput- tographic methods is also imperative, as research on non- ers [52] that have since led to, the security of public key abelian group-based cryptosystems will be one of new re- cryptosystems in use today, will be questioned. Research search priorities. In fact, the pioneering work for non- in new cryptographic methods is also imperative. Re- abelian group-based public key cryptosystem was pro- search on nonabelian group-based cryptosystems will be- posed by N. R. Wagner and M. R. Magyarik [61] in 1985. come one of contemporary research priorities. Many inno- Their idea just is not suitable for practical applications. vative ideas for them have been presented for the past two For nearly two decades, numerous nonabelian groups have decades, and many corresponding problems remain to be been discussed to design efficient cryptographic systems. resolved. The purpose of this paper, is to present a survey The most frequently discussed nonabelian settings include of the nonabelian group-based public key cryptosystems matrix groups, braid groups, semidirect products, loga- with the corresponding problems of security. We hope rithmic signatures and algebraic erasers. that readers can grasp the trend that is examined in this In this paper, we give an overview of known pub- study. lic key cryptography designed by the above mentioned Keywords: Conjugacy Search Problem; Nonabelian nonabelian groups. These proposed nonabelian group- Groups; Public Key Cryptography based public key cryptosystems rely on either encryption- decryption or on key exchange agreement. A standard model for a public key cryptographic scheme is phrased 1 Introduction as two parties, which are referred to as Alice and Bob. Suppose that Alice wants to send a message M to Bob. The development of public key cryptography was a revo- A general model of encryption scheme is the following. lutionary concept that emerged during the twentieth cen- Alice uses the encryption map fk1 to encrypt the mes- tury. The first published study on public key cryptogra- sage C = fk1 (M), where fk1 is a one-way function and is phy was a key agreement scheme that was described by public. After receiving the cipher C, Bob uses the corre- W. Diffie and M.E. Hellman in 1976 [19]. The most com- sponding decryption map gk2 to decode gk2 (fk1 (M)) = M, mon public key cryptography presently in use, such as the where gk2 should be known only by Bob. Diffie-Hellman cryptosystem, the RSA cryptosystem, the Many non-abelian group-based key establishment pro- ElGamal cryptosystem and the elliptic curve cryptosys- tocols are related to the Diffie-Hellman (DH) protocol, tem are number theory based and hence depend on the and we therefore provide a brief description of the DH- structure of abelian groups. Their security depends on protocol. The Diffie-Hellman (DH) protocol functions as difficulties regarding resolving some hard problems of the follows: Let G be a cyclic group with a generator g. number theory. For instance, the RSA algorithm depends Suppose that Alice and Bob want to generate a shared on integer factorization problem. The Diffie-Hellman, El- secret key K. Alice then randomly selects an integer Gamal and ECC algorithms also depend on discrete log- 1 < a < o(g) and sends A := ga to Bob. Similarly, arithmic problems (DLP). Although there have not been Bob randomly selects an integer 1 < b < o(g) and sends any successful attacks on the above public key cryptosys- B := gb to Alice. Alice computes K = Ba, while Bob tems the security of public key cryptosystems in use today, computes K = Ab. The security of the DH-protocol relies will be questioned due to rapidly increasing computing on the Diffie-Hellman problem (or the Discrete Logarith- power and the futurity quantum computers. In 1997 [52], mic Problem). International Journal of Network Security, Vol.20, No.2, PP.278-290, Mar. 2018 (DOI: 10.6633/IJNS.201803.20(2).09) 279 Problem 1. (Diffie-Hellman Problem) Let G be a group. 3) Decryption: From the ciphertext C(X) and Bob's x y xy If g; g ; g 2 G are known, find the value of g . private key (M; a) the message b1 ··· bn can be re- covered by means of a procedure described in [64]. Problem 2. (Discrete Logarithmic Problem) Let G be a group. If h; g 2 G such that h = gx and h; g are known. 4) Security Analysis: Find the integer x. The protocol is based on conjugacy search problem Problem 3. (Conjugacy Search Problem) Let G be a and root problems. But, R. Steinwandt [55] in 1992) nonabelian group. Let g; h 2 G be known such that h = gx pointed out that the Yamamura's Encryption Scheme for some x 2 G. Find x. Here gx stands for x−1gx. is insecure. Suppose that an adversary Eve inter- cepted to the cipher C(X). She can compute Nonabelian group-based public key cryptography is a n relative new research field. In contrast to abelian groups Y D(X) := W (X)−1C(X) = (W (X)bi+1W (X)): the conjugacy search problem and its variant versions are 2 1 2 hard problems on some nonabelian groups. The conju- i=1 gacy search problem and its variant versions play an im- The entries of the matrix portant role for the security on nonabelian group-based ((W (X)bi+1W (X))−1D(X) public key cryptography. 1 2 In this paper, we give a survey of the representative should be polynomials over C. Beginning with nonabelian group-based public key cryptosystems so far. the first bit b1, if at least one of the entries of 2 −1 Their algorithms are very different. D1 := ((W1(X) W2(X)) D(X) involves a non- constant denominator then we can conclude b1 = 0; otherwise b1 = 1. Similarly, if the matrix 2 Matrix Groups 2 −1 D2 := ((W1(X) W2(X)) D1(X) contains a non- polynomial entry then we can conclude b = 0; oth- 2.1 Yamamura's Encryption Scheme 2 erwise b2 = 1. The process continues until all bits At PKC'98, A. Yamamura [64] proposed a public key en- bi; i = 1; ··· ; n are recovered. This means that the n cryption scheme based on the modular group SL(2; ZZ). plaintext b1 ··· bn 2 f0; 1g can be recovered effi- It is well known that SL(2; ZZ) is generated by two ma- ciently from ciphertext C(x) and the public data 0 −1 1 1 alone. trices S = and T = , where the orders 1 0 0 1 of both generators are o(S) = 4 and o(T ) = 1 and the 2.2 Two Rososhek-Matrix Cryptosys- 0 −1 matrix ST = is of order 6. Also, SL(2; ZZ) is tems 1 1 generated by the matrices S and ST subject to the re- In 2013, S. K. Rososhek [49] proposed a ElGamal-like en- lations S4 = (ST )4 = I and (ST )3 = S2. For a matrix cryption scheme -called BMMC ((Basic Matrix Modular 0 −1 Cryptosystem) - by using matrices over ZZ . N 2 SL(2; ZZ), the matrices A := N −1 N and n 1 1 1) BMMC: Let n be a large positive integer and let 0 −1 B := N −1 N satisfy the relations A6 = B4 = I G(α; β; γ) be a free subgroup of the general lin- 1 0 ear group GL(2; ZZ ) generated by three generators 3 2 n and A = B . Therefore, the matrices A and B generate A; B and C, where α; β; γ 2 ZZ with j α j; j β j SL(2; ZZ). 1 0 1 β ; j γ |≥ 3, A = , B = and C = 1) Key Generation: Bob α 0 0 1 1 − γ r i . Let q be the order of the group a. chooses two matrices V1 := (BA) and V2 := −γ γ + 1 (BA2)j 2 SL(2; ZZ) for some i; j 2 IN. GL(2; ZZn). All the data above is public. b. chooses matrices M 2 GL2(C) and a. Key Generation: Bob F1(X);F2(X) 2 Mat2(C[X]) and a 2 C such that F1(a) = V1 and F2(a) = V2. i. chooses two random matrices P1 and U in −1 G(α; β; γ) with P1U 6= UP1 and three inte- c. computes W1(X) := M F1(X)M and −1 gers k; s; l with −q ≤ k; s ≤ q and 2 ≤ q. W2(X) := M F2(X)M. −s k s l ii. computes P2 := U P1 U and P3 := U . d. Bob's public key: W1(X);W2(X). Bob's private key: M; a. iii. The public key: n; P1;P2;P3. The private key: U; k; s. n 2) Encryption: Let b1 ··· bn 2 f0; 1g be the message. b. Encryption: Let the message m 2 Mat(2; ZZ ) Alice computes the ciphertext n be a matrix.
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