The goal of this handout is to show you how to solve congruences of the form ax mod n = 1. You can use this to compute the private key during the initialization of an RSA cryptosystem.
Use Euclid's extended GCD algorithm to compute x. The algorithm, in Pascal, is:
begin (* compute x such that ax mod n = 1, where 0 < a < n *)
g[0] := n; g[1]:= a; u[0] := 1; v[0] := 0; u[1] := 0; v[1] := 1; i := 1;
while g[i] <> 0 do begin
(* loop invariant: g[i] = u[i] * n + v[i] * a *)
y[i] := g[i-1] div g[i];
g[i+1] := g[i-1] - y[i] * g[i];
u[i+1] := u[i-1] - y[i] * u[i];
v[i+1] := v[i-1] - y[i] * v[i];
i := i + 1;
end;
x := v[i-1];
if x < 0 then
x := x + n;
end.
For example, to solve 3x mod 7 = 1, calculate as follows:
| i | g[i] | u[i] | v[i] | y | u[i]*n+v[i]*a |
|---|---|---|---|---|---|
| 0 | 7 | 1 | 0 | 1*7+0*3=7 | |
| 1 | 3 | 0 | 1 | 2 | 0*7+1*3=3 |
| 2 | 1 | 1 | -2 | 3 | 1*7+(-2)*3=1 |
| 3 | 0 | (-3)*7+7*3=0 |
In the first example of an RSA cipher, we chose p = 5 and q = 7. This means n = pq = 35, and PHI(n) = (5-1)(7-1) = 24. We must find d and e such that de mod PHI(n) = 1. We chose d = 11 and need to determine e. As gcd(11, 24) = 1, we can use the above algorithm to solve 11e mod 24 = 1:
| i | g[i] | u[i] | v[i] | y | u[i]*n+v[i]*a |
|---|---|---|---|---|---|
| 0 | 24 | 1 | 0 | 1*24+0*11=24 | |
| 1 | 11 | 0 | 1 | 2 | 0*24+1*11=11 |
| 2 | 2 | 1 | -2 | 5 | 1*24+(-2)*11=2 |
| 3 | 1 | -5 | 11 | 2 | (-5)*24+11*11=1 |
| 4 | 0 |
In the second example, we chose p = 53 and q = 61. This means n = pq = 3233, and PHI(n) = (53-1)(61-1) = 3120. We must find d and e such that de mod PHI(n) = 1. We chose d = 791 and need to determine e. As gcd(791, 3120) = 1, we can use the above algorithm to solve 791e mod 3120 = 1:
| i | g[i] | u[i] | v[i] | y | u[i]*n+v[i]*a |
|---|---|---|---|---|---|
| 0 | 3120 | 1 | 0 | 1*3120+0*791=3120 | |
| 1 | 791 | 0 | 1 | 3 | 0*3120+1*791=791 |
| 2 | 747 | 1 | -3 | 1 | 1*3120+(-3)*791=747 |
| 3 | 44 | -1 | 4 | 16 | (-1)*3120+4*791=44 |
| 4 | 43 | 17 | -67 | 1 | 17*3120+(-67)*791=43 |
| 5 | 1 | -18 | 71 | 43 | (-18)*3120+71*791=1 |
| 6 | 0 |
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Department of Computer Science
University of California at Davis
Davis, CA 95616-8562