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The most direct method of calculating a modular exponent is to calculate b e directly, then to take this number modulo m. Consider trying to compute c, given b = 4, e = 13, and m = 497: c ≡ 4 13 (mod 497) One could use a calculator to compute 4 13; this comes out to 67,108,864. Taking this value modulo 497, the answer c is determined to be 445.
Hence another name is the group of primitive residue classes modulo n. In the theory of rings , a branch of abstract algebra , it is described as the group of units of the ring of integers modulo n .
Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called "Montgomery's ladder" [2] addresses this concern.
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n , a modulo n (often abbreviated as a mod n ) is the remainder of the Euclidean division of a by n , where a is the dividend and n is the divisor .
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (). [ 26 ] This definition of exponentiation with negative exponents is the only one that allows extending the identity b m + n = b m ⋅ b n {\displaystyle b^{m+n}=b^{m}\cdot b^{n}} to negative exponents (consider the case m = − n ...
The RSA cryptosystem is based on this theorem: it implies that the inverse of the function a ↦ a e mod n, where e is the (public) encryption exponent, is the function b ↦ b d mod n, where d, the (private) decryption exponent, is the multiplicative inverse of e modulo φ(n).
This is also true for negative exponents. In particular, the reciprocal of an n th root of unity is its complex conjugate, and is also an n th root of unity: [8] = = = ¯. If z is an n th root of unity and a ≡ b (mod n) then z a = z b.