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240 CHAPTER 12. RSA
Theorem 5 Euler’s Theorem 8 (1760): If p and q are two distinct primes and
neither one of them divides a, then
a (p−1)(q−1) ≡ 1
(mod pq).
Leonard Euler (1707–1783) is history’s most prolific mathematician. Except
for 1741–1766 when he was at the Royal Academy in Berlin, from 1727 until his
death Euler lived in St. Petersburg and worked at the Imperial Academy there.
His contributed to most fields of mathematics, in particular geometry, calculus
and number theory, as well as to physics, especially acoustics, hydraulics and
the theory of light. As if more is needed, he became blind in 1766 at age
59 but, despite this, continued his work on optics, algebra and lunar motion,
producing almost half of his total works while blind. Euler wrote over 700 books
and papers, piling new papers atop older ones. The Imperial Academy, which
published the papers, published the top ones first, cause later, more advanced
results to appear before the ones they superseded or depended upon!
Examples: Of Euler’s Theorem.
(1) 12 60 ≡ 1 (mod 77), since 77 = 11 · 7 and (11 − 1)(7 − 1) = 60.
(2) 19 252 ≡ 1 (mod 301), since 301 = 43 · 7 and (43 − 1)(7 − 1) = 252.
(3) 23 24 ≡ 1 (mod 35) and 49 64 ≡ 1 (mod 85).
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Where will these theorems affect our work Only in one place. The only
times we’ve used “p − 1” so far is when doing modular arithmetic on the exponent.
So when the modulus is the produce of two prime pq, then we must
simply be careful to consider the exponent modulo (p − 1)(q − 1).
Examples:
(1) Compute 17 1803 %671.
671 factors as 61 · 11. Since (61 − 1, 11 − 1) = 600, we must reduce 1803
modulo 600. Since this is 3, we have 17 1803 ≡ 7 3 ≡ 35 (mod 77).
(2) Compute 25 448 %253.
253 = 11 · 23, and 10 · 22 = 220. 448%220 = 8, so 25 448 ≡ 25 8 ≡ 49
(mod 253).
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8 Euler actually proved his theorem for any modulus, rather than for the much simpler case
were are concerned with here. In general it takes the form a φ(n) ≡ 1 (mod n), where φ(n) is
an easily computed value. This φ is called “Euler’s phi function” and is not to be confused
with Friedman’s Φ.