1 NAME OR TnU: 

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REMARKS eclassifi ed and approved for release by NSA 

n 01-26-2015 ursuantto E.O. 13526 

FROM NAME OR TITLE 

ORGAltlZATIOll AllD LDCA'llDll 

DD I nl'\o 9 4 REPUCl!S NME FORM ... I FEB 48, WHICH MAY .,.USED. 

INFORMATION 

NECESSARY 

ACTION 

llOTEAND 

RETURll 

SEE ME 

SIGNATURE 
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"ENCIPHERED K>TION11 Branching Structure 

Introduction a 

There has been considerable interest in observational da\a concerning the proporliiotl ot origins, single po~nt.s, and branch points 

ot a device ample>Jing wheels stepped b,v enciphe1"8d motion. To thia 

encl a program was written tor ATLAS II to conq:are the •pirical re-

...... 

aulta with the expectations under purel.1' ramom stepping. 

The probab1llt7 P(1) ot a branch point ot nml.t:lpllc:lt7 11111 tor the 

case where there are n random stepping wheels (each ot the "rand•" . wheels steps with proba.bll1t7 1/2, each wheel steps :lndepememl.7 of 

each other, am therefore each ot the i' motion asaumptiona la equal.11' 

like~) can be given 'bJ' ifha following t~nnula: 

1 2n-1 

P(i) •(fn)~) ~ -~) (1) 

As n increases, (1) approaches e(~I) as a limit. 

1 - I P(i)IV';m, (2) 

Note the expected number of ..origins and single points approaches the aame 

ftlue. 

In the ATLAS progralll, rotor aame wirings were sinulated and the 

motion determimd tor each setting ot the rotor, and attar stepping the 

rotors 1n accordance with the motion (the •ze encij.Sherment of the stepping 

SECRET Tba lntarrrat,on canlalned JD 

tbla dOllUIRDDl 111111 1101 Ila llllCIOHcl 

to loHl&n Htlo ala or tbelr real'G'• 

ee1tntlwe11 
~,~s was carried out, etc.) a listing ot the initial setting, the next 

t~ur aucpessive settipgs, and the lot.h setting wall recorded. Thia was 

ddne tor an·,.ettings composed or core settings A, B, or C in each ot 

the rotors step~ b7 the output ot the mze. Examination of these rune 

and recognition of confluences provided additional information in that 

tor some settings (those composed ot B or C on each of the rotors stepped 

by the output ot the maze) all possible predecessors and their immediate 

sucessors were present in the print out. ' For these settings actual 

enumeration would provide the number or them that were origins, single 

points, branch points ot order 2, branch points of order 3, etc. 

For example when n • 5 there are 25 settings which ire either origins, 

single point.a, or branch point.a of degree two or mre. We know tor a 

certaint7the character ot these 32 settings since all their possible 

predecessors have been examined among the 35 initial settings. 

A. The first dev.lce to be examined was a 10 wheel nachine with 

tour metric or c70le guarantee wheels and six wheels stepped 'b7 enciphered 

motion. Each of the latter six wheels baa probabilit7 1/2 or stepping. 

The toll.owing is a listing of tour samples nade on ATLAS. The samples 

dittertld in tl'Bt the metric rotors were set to a new position before each 

set ot runs was Ede. Shown also is the· expected number B(i) ot each kind 

ot point tor the aa.mple observed. 

. 

I' ~ -· ~ kto1 - = 
• 

• 

Or!gine 

Single point.11 

Double points 

Triple points 

Quadrupl• 

point a Quint~ple 

points 

SAMPLES 

B c D 1(1) 

24 22 25 Z1 23.4 

25 22 20 16 23.7 

11 16 14 15 11.8 

3 2 2 5 3.9 

l l ' 1 l.O 

- l - - .2 

61. 64 64 64 64.o 

' VII• l•IOrmGtJH IDIDOQ tg 

tt'lle ••ea•Ht wm 11et m aeooa 

to l•r•laa aatlo•olo 011 800111 r DGctlotllvoo 
• .. . .. I• .. 

.. - . ... 

~ 

B. The second device to be examined was a IO' wheel machine with 

five metric or cycle guarantee wheels ard five wheels stepped by 

enciphered motion. Ea.ch of these five latter wheels has probability 1/2 

of stepping. The following is a listing or nine samples made on ATLAS. 

E(i) is the expected number of branch points of multiplicity (i). 

Origins 

Single points 

Double points 

Triple points 

Quadruple 

points 

Quintuple 

points 

A B c 

9 14 13 

14 8 6 

7 7 11 

2 2 l 

- 1 1 

~ 

- - -

D 

11 

11 

9 

-

l 

- ~ 

SAMPLES 

E F 

12 12 

12 14 

4 2 

4 2 

- 2 

- r -

G H I E(i) 

15 13 11 11.6 

11 6 12 12.0 

4 11 5 6.o 

2 2 2 1.9 

·- - 1 .4 

- - 1 .1 . 

32 32 32 32 )2 32 32 32 32 32.0 

Robert Sengpienl 

NSA-314 

28 June 1955 

;;13 ieo~ormati~t" ~car.iocl i:i 

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