Network Working Group S. Crocker Request for Comments #70 UCLA 15 October 70 A Note on Padding The padding on a message is a string of the form 10*. For Hosts with word lengths 16, 32, 48, etc., bits long, this string is necessarily in the last word received from the Imp. For Hosts with word lengths which are not a multiple of 16 (but which are at least 16 bits long), the 1 bit will be in either the last word or the next to last word. Of course if the 1 bit is in the next to last word, the last word is all zero. An unpleasant coding task is discovering the bit position of the 1 bit within its word. One obvious technique is to repeatedly test the low-order bit, shifting the word right one bit position if the low-order bit is zero. The following techniques are more pleasant. Isolating the Low-Order Bit Let W be a non-zero word, where the word length is n. Then W is of the form x....x10....0 \__ __/\__ __/ V V n-k-1 k where 0<=kp' => R(p) > R(p'), we obtain the following table of useful divisors for p < 100. [Page 4] Network Working Group A Note on Padding RFC 70 p R(p) p R(p) 1 1 29 28 3 2 37 36 5 4 53 52 9 6 59 58 11 10 61 60 13 12 67 66 19 18 83 82 25 20 Notice that 9 and 25 are useful divisors even though they generate only 6 and 20 remainders, respectively. Determination of R(p) If p is odd, the remainders 0 mod(2 ,p) 1 mod(2 ,p) . . . t will be between 1 and p-1 inclusive. At some power of 2, say 2 , there k t will be a repeated remainder, so that for some k < t, 2 = 2 mod p. t+1 k+1 Since 2 = 2 mod p t+2 k+2 and 2 = 2 mod p . . . etc. 0 t-1 all of the distinct remainders occur for 2 ...2 . Therefore, R(p)=t. [Page 5] Network Working Group A Note on Padding RFC 70 Next we show that R(p) 2 = 1 mod p R(p) k We already know that 2 = 2 mod p for some 0<=k =q, k q k-q mod(2 ,p) = 2 *mod(2 ,p'). [Page 6] Network Working Group A Note on Padding RFC 70 From this we can see that the sequence of remainders will have an q-1 initial segment of 1, 2, ...2 of length q, and repeating segments of length R(p'). Therefore, R(p) = q+R(p'). Since we normally expect R(p) ~ p, even p generally will not be useful. I don't know of a direct way of choosing a p for a given n, but the previous table was generated from the following Fortran program run under the SEX system at UCLA. 0 CALL IASSGN('OC ',56) 1 FORMAT(I3,I5) M=0 DO 100 K=1,100,2 K=1 L=0 20 L=L+1 N=MOD(2*N,K) IF(N.GT.1) GO TO 20 IF(L.LE.M) GO TO 100 M=L WRITE(56,1)K,L 100 CONTINUE STOP END Fortran program to computer useful divisors In the program, K takes on trial values of p, N takes on the values of the successive remainders, L counts up to R(p), and M remembers the previous largest R(p). Execution is quite speedy. [Page 7] Network Working Group A Note on Padding RFC 70 Results from Number Theory The quantity referred to above as R(p) is usually written Ord 2 and is p read "the order of 2 mod p". The maximum value of Ord 2 is given by p Euler's phi-function, sometimes called the totient. The totient of a positive integer p is the number of integers less than p which are relatively prime to p. The totient is easy to compute from a representation of p as a product of primes: n n n Let p = p 1 * p 2 ... p k 1 2 k where the p are distinct primes. Then i k -1 k -1 k -1 phi(p) = (p - 1) * p 1 * (p - 1) * p 2 ... (p - 1) * p k 1 1 2 2 k k If p is prime, the totient of p is simply phi(p) = p-1. If p is not prime, the totient is smaller. If a is relatively prime to p, then Euler's generalization of Fermat's theorem states phi(m) a = 1 mod p. It is this theorem which places an upper bound Ord 2, because Ord 2 is p p the smallest value such that Ord 2 2 p = 1 mod p Moreover it is always true that phi(p) is divisible by Ord 2. p [Page 8] Network Working Group A Note on Padding RFC 70 Acknowledgements Bob Kahn read an early draft and made many comments which improved the exposition. Alex Hurwitz assured me that a search technique is necessary to compute R(p), and supplied the names for the quantities and theorems I uncovered. [ This RFC was put into machine readable form for entry ] [ into the online RFC archives by Guillaume Lahaye and ] [ John Hewes 6/97 ] [Page 9]