Historical Document · 1944
OP-20-G, "The Period of the CCM"
CCM period analysis (2 pp.) [writer A]
Historical Document · 1944
CCM period analysis (2 pp.) [writer A]
17 April 1944 TOR SECRET TOP SECRET THE PERIOD OF THE CCM . The CC1 has five Wheels of which the center steps every time, end the others irregularly. Designeting the wheels by 1,2.0,4,5, in order, the 3 wheel •steps continually, the 2 and 4 wheels stepping is under control of the 3 wheel, and the 1 and 5 wheels step under the the influence of the 2 and 4 wheels respectively. Since the stepping of the 4 and 5 wheels is similar (differing only in the notches) to the stepping of the 2 and 1 wheels, we need only discuss the motion of the 3-4-5 system* Let there be k notches on the 3 wheel which induce the motion of the 4 wheel. CASE I. k is reletively prime to 26. After the 3 wheel has revolved 26 times, each of the k notches has stepped the 4 wheel 26 times. Hence the 4 wheel has stepped 26k time-) or k complete revolutions. when the 4 wheel co s to the A position under the influence of (say) the 1st notch of the 3 wheel it staye until the 2nd notch of the 3 wheel moves it on to B position. The time wheel beelapsed in the A position is the interval on the tween the first and eecond notchee. During the cycle the A position is attained under the action of each of the k notches, and the total time elapeed in the A position is the sun of n the 3 wheel between the first and second the Interval notches, the second and third notchee, the (k-1)st end kth notches and the kth and let notches. This sum is cleerly once around the wheel or 26. By a similar argument each position of the 4 wheel is up 26 times during the cycle. Hence the 5 wheel has been kicked 26 tie e by each notch of the 4 wheel and hence has turned an integral number of times. . This shows that under the hypothesis k-relatively prime to 26, (where k is the number of notches on the center wheel) the period is 676. CASE II. k is an even number, 2m. After 13 revolutions of the 3 wheel, the 4 wheel has advanced 13 x 2m = 26m positions or m complete revolutions. -1ENCLOSURE (B) TOP SF,CliET TOP SECRET THE PERIOD OF THE CC! (Continued) As before, if the A position of .the 4 wheel is caused by the it noteh of the 3 wheel, the A position remains for en interval equal to the interval beteeeh 1st and 2nd notches of the 3 c.ee1. Durine the 13 revolutions of the 3 wheel, the e position of the 4 wh 1 coees up m times under the influence (in some order) of the 1st) 3rd, 5th, (2m-1)et notches of the 3 wheel end the total time elapeed in the A poeitloe ie the sum of the intervals on the 3 wheel betweeh the let and 2nd notchee, the 3rd an 4th notehes, the 5th and 6th notches, ...., end the (em-l)et end 2m th notchee. Let this sum be q. Th B position of tie 4 wheel comes ue fro:. the 2nd p 4th, 2m th notches of the 3 wheel, end the totel elapsed timee the sun of the interval between the 2nd end 3rd notches, the 4th end 5th, ;.,e1 and the 2= th ane 1st notches. This sue is 26-e. The C positioe heo1, ad the C=6-5 up fro; notches 1, 32 5, ... or the 3 Y totel elepeed time is q. Thee the positions e1C,L, have have elapsed time q, en6 the positions 1301),F, ..., elapsed time 26-q. The total .emount of kick received by the 5 wheel is qq (26-q)r2 where r1 is the number of notches of the 4 wheel in positions A te,E, Y, end r2 is the number of notches of the 4 wheel 2. in poeitions BID, The cycle of the whole 3-4-5 system depends on the factors common to grit (26-0 r2 and 26. The cycle is 338 if grit.(26-q) r2 o (mod ;()) q(r1-r2) = o (mod 26) i.e. Thus the cycle is Ahort if a) ri = r2 b) q = 13 and r1 r2 = o (mod 2) Sines all wheels have an even number of notches to prevent a short cycle occurring as in Case I, condition b) is simply q = 13. Several wheels In the present set have been examined and several satisfy- condition a). When such a wheel is in position 4 the cycle of the 3-4-5 system is 338. Similarly when such a wheel is in position 2 the cycle of the 3-2-1 eystee is 33c, and the cycle of the whole machine is short. If no wheels setisfy either a) or b) on either rie, then the minimum period weuld be 4394 letters.