1. Modulo Arithmetic & Text Coding ECE6612 - Slides-03a John Copeland
Georgia Tech

2. = 1001110000101111 Cryptographic Operations make use of :
XOR (faster than add - no “carry” operations) and
(A + B) + B = A if A + B = C then A = B + C
Modulo (%) 2n arithmetic:
% 216 =
Still to handle 256-bit numbers on a 32-bit
Computer requires representing the number as an array:
N = N[0] + N[1]*232 + N[2]*264 + … + N[7]*2224 2

3. To multiple two 256-bit numbers modulo(2256),
X = M * N % where int64 N[8], M[8], X[8]
N = N[0] + N[1]*232 + N[2]*264 + … + N[7]*2224
M = M[0] + M[1]*232 + M[2]*264 + … + M[7]*2224
X = N[0]*M[0] + (N[0]*M[1] + N[1]*M[0])* …
X[n] = ∑(i=0 to n) ∑(j=0 to [n-j]) N[i]*M[j] as array
Carries must then be processed:
X[i] = X[i] + (X[i-1] >> 32) ; X[i-1] &= 0xffffffff
Note that half the terms are not needed: those with i+j >7 because of the modulo(2256) 3

4. Compute the following: (A * B * C) % 10 A = 15897229742
Answer:
A * B * C % 10 = ( A %10 ) * ( B %10 ) * ( C %10 )
= (2 * 8 * 3) %10
= ( 48 ) %10
= 8
The modulo operation can be done on any intermediate values that exceed the modulus, for addition or multiplication. 4

5. Note that 9, 99, 999, 999…999 - all modulo 9 = 0, therefore:
Just for fun, ,without using a calculator, or pen and paper, compute the following:
(A * B * C) % 9
A =
B =
C =
Answer:
Note that 9, 99, 999, 999…999 - all modulo 9 = 0, therefore:
1, 10, 100, …000 - all modulo 9 = 1 (each is 1 + integer * 9)
A %9 = % add up the digits
= 11%9 = 1+1 = 2
etc.
So, (A * B * C) % 9 = (2 * 7 * 3) %9 = 14 %9 * 3 = 15 %9 = 6
Also (A + B + C) %9 = ( ) %9 = 3
The “method of nines” can check addition and multiplication. 5

6. If X %m = x then X = integerX * m + x (0<= x <m)
The modulo operation can be done on any intermediate values that exceed the modulus, for addition or multiplication.
Why does this work?
If X %m = x then X = integerX * m + x (0<= x <m)
(X + Y) %m = {(integerX + integerY) *m + (x + y)} %m
(X + Y) %m = (x + y) %m
(X * Y) %m = {(integerX * integerY) *m2 )} %m +
+ {(integerX *y *m)} %m + {(integerY *x *m)} %m
+ (x * y) %m
(X * Y) %m = (x * y) %m 6

7. End-Line(s) UTF-8 Since 2007 Teletype, MSDOS, Internet: LF and CR
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