What is the smallest possible RSA encryption exponent for n = 19 * 31.? 1601 is

Question

What is the smallest possible RSA encryption exponent for n = 19 * 31.? 1601 is prime. Use Fermat's little theorem to find 24^4, 800,001 mod 1601. Use Euler's theorem to find integers x and y between 0 and 2120 such that 17^2, 400,002 x mod 2121 19^1200 y mod 2121. Solve for m m^17 42 mod 89, m^17 29 mod 88. TRUE or FALSE?: If p is prime, and x Z/p^x, then order (x^2) = order(x)^2. Let p be an odd prime number such that x^5 1 mod p for some x Z/p^x, x notequalto 1. Prove that the last decimal digit of p is 1. If x is invertible mod n, prove that the order of x mod n is the same as the order of the inverse of x mod n. Use n = 155150096071 and phi (n) = 155148967944 to factor n.

Explanation / Answer

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1. smallest possible RSA encryption for n = 19*31

= 589

now phi (n) = (19-1)*(31-1)

= 18 *30 = 540

now calculating the list of values 1 < e(exponent) < phi(n) and coprime with n 589

we get 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...... and so on

So the smallest exponent is 2 here.

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