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<=k p' =>
R(p) > R(p'), we obtain the following table of useful divisors for
p < 100.
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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.
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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').
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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.
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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
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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 ]
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RFC, FYI, BCP