A Cryptosystem Based on the Symmetric Group Sn

Summary This paper proposes a public key cryptosystem based on the symmetric group Sn, and validates its theoretical foundation. The proposed system benefits from the algebraic properties of Sn such as non commutative, high computational speed and high flexibility in selecting keys which make the Discrete Logarithm Problem (DLP) resistant to attacks by algorithms such as PohligHellman. Against these properties, the only disadvantage of the scheme is its relative large memory and bandwidth requirements. Due to the similarities in the algebraic structures, many other cryptosystems can be translated to their symmetric group analogs, and the proposed cryptosystem is in fact the Generalized ElGamal cryptosystem which is based on Sn instead of GF(p).

[1]  D. Shanks.  Class number, a theory of factorization, and genera , 1971 .

[2]  M. Newman,et al.  Topics in Algebra , 1978 .

[3]  Ueli Maurer,et al.  The Relationship Between Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms , 1999, SIAM J. Comput..

[4]  Taher El Gamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, IEEE Trans. Inf. Theory.

[5]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[6]  C. Cobeli,et al.  ON THE DISCRETE LOGARITHM PROBLEM , 2008, 0811.4182.

[7]  Leonard M. Adleman,et al.  A subexponential algorithm for the discrete logarithm problem with applications to cryptography , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[8]  Martin E. Hellman,et al.  A cryptanalytic time-memory trade-off , 1980, IEEE Trans. Inf. Theory.

[9]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[10]  Andreas Stein,et al.  Analysis of the Xedni Calculus Attack , 2000, Des. Codes Cryptogr..

[11]  Stephen C. Pohlig,et al.  An Improved Algorithm for Computing Logarithms over GF(p) and Its Cryptographic Significance , 2022, IEEE Trans. Inf. Theory.

[12]  S. Vanstone,et al.  Computing Logarithms in Finite Fields of Characteristic Two , 1984 .

[13]  M. Rabin.  DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION , 1979 .

[14]  Justin M. Reyneri,et al.  Fast Computation of Discrete Logarithms in GF(q) , 1982, CRYPTO.

[15]  J. Rotman.  Advanced Modern Algebra , 2002 .

[16]  Ronald L. Rivest,et al.  A Knapsack Type Public Key Cryptosystem Based On Arithmetic in Finite Fields , 1984, CRYPTO.

[17]  Don Coppersmith,et al.  Fast evaluation of logarithms in fields of characteristic two , 1984, IEEE Trans. Inf. Theory.

[18]  Adi Shamir,et al.  A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem , 1984, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[19]  Martin E. Hellman,et al.  Hiding information and signatures in trapdoor knapsacks , 1978, IEEE Trans. Inf. Theory.

[20]  Joseph H. Silverman,et al.  The Xedni Calculus and the Elliptic Curve Discrete Logarithm Problem , 2000, Des. Codes Cryptogr..

[21]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[22]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[23]  By J. M. Pollard.  Monte Carlo Methods for Index Computation (mod p) , 2010 .

[24]  T. Elgamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, CRYPTO 1984.

[25]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[26]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[27]  Roberto Mantaci,et al.  A permutations representation that knows what "Eulerian" means , 2001, Discret. Math. Theor. Comput. Sci..

[28]  Helmut Hasse.  The Class Number , 1980 .

[29]  Richard J. Lipton,et al.  Algorithms for Black-Box Fields and their Application to Cryptography (Extended Abstract) , 1996, CRYPTO.

[30]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[31]  D. H. Lehmer.  Teaching combinatorial tricks to a computer , 1960 .

[32]  A. L. Wells,et al.  A polynomial form for logarithms modulo a prime , 1984, IEEE Trans. Inf. Theory.

[33]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.