## Large Period Nearly deBruijn FCSR Sequences (Extended Abstract) (1995)

Venue: | In L.C. Guillou and J.J. Quisquater� editors� Advances in Cryptology � Eurocrypt �95 |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Klapper95largeperiod,

author = {Andrew Klapper and Mark Goresky},

title = {Large Period Nearly deBruijn FCSR Sequences (Extended Abstract)},

booktitle = {In L.C. Guillou and J.J. Quisquater� editors� Advances in Cryptology � Eurocrypt �95},

year = {1995},

pages = {263--273},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2-adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more difficult to find maximal period FCSR sequences. In this paper we exhibit a technique for easily finding FCSRs that generate nearly maximal period sequences. We further show that these sequence have excellent distributional properties. They are balanced, and nearly have the deBruijn property for distributions of subsequences.

### Citations

843 |
An Introduction to the Theory of Numbers
- Hardy, Wright
- 2005
(Show Context)
Citation Context ...mber c/q, the initial loading for an FCSR that gives output c/q can be found by the following procedure [a, 91. 131. Set m-1 = c. B2. For each i = 0, 1, . . . , r - 1 corriputc the following numbers:=-=(4)-=-, 268 If we use the initialloading (~'-1, a,.-z, . . . , u1, an) and initialmemory m,-l E R, then the resulting FCSR outputs the 2-adic expansion of c/q. If c is relatively prime to q, then the period... |

188 | p-adic Numbers, p-adic Analysis and Zeta-Functions - Koblitz - 1984 |

105 |
Analysis and Design of Stream Ciphers
- Rueppel
- 1986
(Show Context)
Citation Context ...er (Eds.): Advances in Cryptology - EUROCRYPT '95, LNCS 921, pp. 263-273, 1995. 0 Springer-Verlag Berlin Heidelberg 1995264 include nonlinear feedforward functions [6], nonlinear combining functions =-=[14]-=-, and clock controlled shift registers [a]. Recently a new class of binary sequence generators, feedback with carry shaft registers (or FCSRs) has been described by Klapper and Goresky [S, 91. They ha... |

64 |
On Artin’s conjecture
- Hooley
- 1967
(Show Context)
Citation Context ... modulo q. The search for primes q such that 2 is a primitive root is relatcd to a large body of corikmporary number theory. It is believed t,hat there are infinitely many primes q with this property =-=[5]-=-. However, finding such primes (and even finding large primes at all) is problematic. In this paper we consider two fundamental questions about FCSR sequences: 1. How can we guarantee the output seqne... |

60 |
Shift Register Sequences”, Aegean Park
- Golomb
- 1982
(Show Context)
Citation Context ...gister (LFSR) sequences. Maximal pcriod LFSR sequences (or m-sequences) are known to have large period and a balance of zeros and ones, and to become deRruijn sequences when a single zcro is inserted =-=[3]-=-. These properties, as well as the availability of algebraic tools for their analysis, have led to thcir use in a number of construct>ions of key stream generators. Examples L.C. Guillou and J.-J. Qui... |

40 |
An Analysis of the Structure and Complexity of Nonlinear Binary Sequence Generators
- Key
- 1976
(Show Context)
Citation Context ...les L.C. Guillou and J.-J. Quisquater (Eds.): Advances in Cryptology - EUROCRYPT '95, LNCS 921, pp. 263-273, 1995. 0 Springer-Verlag Berlin Heidelberg 1995264 include nonlinear feedforward functions =-=[6]-=-, nonlinear combining functions [14], and clock controlled shift registers [a]. Recently a new class of binary sequence generators, feedback with carry shaft registers (or FCSRs) has been described by... |

37 |
The fast Fourier transform in a finite field
- POLLARD
- 1971
(Show Context)
Citation Context ...nhage-Strassen algorithm [15], then S(r) = O(rlogrlog1ogr). This can be improved to S(r) - rlogr using Pollard’s nonasymptotic algorithm for r < 237 on a 32 bit machine or r < 270 on a 64 bit machine =-=[13]-=-. Finally, observe that gcd(2‘m - z, q) = 1 if and only if gcd(2‘m - zlp> = 1. This can be checked using the Euclidean algorithm in S(r log(p)’) bit operations. In summary, a desired initial loading c... |

25 |
Feedback Shift Registers, Combiners with Memory, and 2–adic
- Klapper, Goresky
- 1997
(Show Context)
Citation Context ...2, the best we can hope for in this regard is that the numbers of occurrences of any two sequences differ by at most one. This, in fact, is the case when y is prime and 2 is a primitive root modulo y =-=[8]-=-. When y is a power of a prime and 2 is a primitive root modulo q, we show that the numbers of occurrences of any two subsequences differ by at most two. This holds for subsequences of any length. Fin... |

14 |
2-adic Shift Register. Fast Software Encryption
- Klapper, Goresky
- 1994
(Show Context)
Citation Context ...output of a FCSR, or as being periodic, or as having any other property that should more properly be attributed to the sequence a. The following facts are known about FCSRs and their output sequences =-=[8,9]-=-. 1. A binary sequence a is eventually periodic if and only if its associated 2adic number cy is a rational number c/q. It is strictly periodic if and only if moreover -q < c 2 0. 2. If a is the outpu... |

14 |
Arithmetic codes with large distance
- Mandelbaum
- 1967
(Show Context)
Citation Context ...ations in the usual sense. There is, however, an arithmetic (or "with carry") analog of the cross-correlation. This has been studied previously in the case of autocorrelation funct,ions by Mandelbaum =-=[12]-=-. Definition6. Let a and 11 he tjwo evcntiially periodic sequences with period N, and let 0 5 r < N . Let b' be the sequence formed by shifting b by T positions, ba = bitT. Then the shifted udhmetic c... |

11 | Feedback with carry shift registers over finite fields - Klapper - 1995 |

5 |
personal communication
- Bombieri
(Show Context)
Citation Context ...1. Randomly choose bits au, . . , a,-l E {0,13 2. Compute r-1 r-d-I 3. Let 4. Check gcd(2'm - z, q) = 1. If so, use no, . . ', 0,-1 as the initial loading, and m as the initial memory. If not, repeat =-=(1)-=- (4). In some cases [(z+ y)/2'] = [z/2'1 + 1 and can also be tried as the the initial memory. To see that this gives a maximal period purely pcriodic seqiience for q, it suffices to check that 0 5 2'm... |

2 |
Pseudo Random Properties of Cascade Connections of ‘Clock COIItrolled Shift Registers
- Gollman
- 1985
(Show Context)
Citation Context ...oading (~'-1, a,.-z, . . . , u1, an) and initialmemory m,-l E R, then the resulting FCSR outputs the 2-adic expansion of c/q. If c is relatively prime to q, then the period of the sequence is T = ordq=-=(2)-=-. However if c and y have a common factor then the period may be smaller but at least it will divide ord,(2). Thus for q = pe with 2 primitive modulo p2 (and hence also modulo y), if we randomly choos... |

2 | Schnelle Multiplikation Grosset - SchSnhage, Strassen - 1971 |

1 | An Analysis of the structure and complexity of northnear binary sequence generators - Key - 1976 |

1 | Feedback Registers Bsed on Rmified Extensions of the 2-Adic Numbers, to appear - Klpper, Goresky - 1998 |

1 | Feedback Registers Rased on Ramified - Klapper, Goresky |