Why do we need Euler's totient function $\varphi(N)$ in RSA?

Question

After we calculated $N = p * q$, we calculate $\varphi(N)$ and use it later to determine $e$ (PR) and $d$ (PU). But why?

For decryption and encryption, we only use $N$ and don't need $\varphi(N)$. So why can't we find $e$ and $d$ without Euler's totient function? I know that $\varphi(N)$ is giving me how many integers are coprime to $N$, and if $N$ is a prime then it would be $\varphi(N) = N - 1$. But why is this useful? Or rather why is it a must for determining $e$ and $d$ in RSA?

Edit: And why does $e$ need to be smaller than $\varphi(N)$?

Answer 1

What we really need is a number $\lambda$ satisfying $x^{\lambda+1} \equiv x \pmod n$ for all integers $x$ (which, by induction, then implies that $x^{k\lambda+1} \equiv x \pmod n$ for any $k$).

Given such a $\lambda$, and an arbitrary encryption exponent $e$ which is coprime to it, we can then find the multiplicative inverse of $e$ modulo $\lambda$, i.e. a number $d$ such that $ed \equiv 1 \pmod \lambda$, or in other words, $ed = k\lambda + 1$ for some integer $k$. Such $e$ and $d$ then satisfy $$(x^e)^d = x^{ed} = x^{k\lambda+1} \equiv x \pmod n,$$ meaning that, if we encrypt a number by raising it to the $e$-th power modulo $n$, we can recover the original number by raising the result to the $d$-th power and again reducing it modulo $n$. This is what we need for RSA encryption and decryption to work correctly.

The smallest such number $\lambda$ is given by the Carmichael totient function, which, for a product $n = pq$ of two primes, is $$\lambda(pq) = \operatorname{lcm}(p-1, q-1)$$ where $\operatorname{lcm}(p-1,q-1)$ denotes the least common multiple of $p-1$ and $q-1$. However, since we don't necessarily need the smallest such number, it's also possible to use the Euler totient function $$\varphi(pq) = (p-1)(q-1)$$ which is, by definition, always a multiple of $\lambda$.

I'm not aware of any particular reason for preferring $\varphi$ over $\lambda$, except that it's slightly easier to compute $\varphi$ and to explain why it has the necessary property $x^{\varphi+1} \equiv x \pmod n$, which may be why introductory texts tend to prefer it. I do believe, however, that actual practical RSA implementations (insofar as they explicitly compute the decryption exponent at all, rather than e.g. using the Chinese remainder theorem) generally use $\lambda$ rather than $\varphi$, since doing so yields the smallest possible decryption exponent $d$.

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Addendum: The reason why $\lambda$ (and $\varphi$) satisfies $x^{\lambda+1} \equiv x \pmod n$ is basically Fermat's little theorem, which says that, for any prime $p$ and any integer $x$, $$x^p \equiv x \pmod p.$$

This can be easily generalized to show that, for any multiple $k\lambda(p)$ of $\lambda(p) = p-1$, $$x^{k\lambda(p)+1} \equiv x^{\lambda(p)+1} = x^p \equiv x \pmod p.$$

In particular, since $\lambda(pq) = \operatorname{lcm}(p-1,q-1)$ is, by definition, a multiple (and in fact, the smallest common multiple) of both $\lambda(p) = p-1$ and $\lambda(q) = q-1$, it follows that $x^{\lambda(pq)+1} \equiv x$ modulo both $p$ and $q$, and therefore, also modulo $pq = n$.

Of course, since $\varphi(pq) = (p-1)(q-1)$ is also a multiple of both $p-1$ and $q-1$, it also has the same property, as does any other common multiple of those numbers.

The fact that $\varphi(n)$ also happens to be the order of the multiplicative group modulo $n = pq$ is basically a red herring; there's nothing special about $\varphi(n)$ among all the other multiples of $\lambda(n)$ as far as RSA is concerned. What's more relevant is that $\lambda(n)$ is the exponent of this group, which is essentially another way of stating the crucial property that $x^{\lambda(n)+1} \equiv x \pmod n$ for all $x$.

Answer 2

To complete Ilmari great answer, I would like to quote the Handbook of Applied Cryptography (p 286,291):

Proof that decryption works. Since $ed \equiv 1 \pmod \phi$, there exists and integer $k$ such as $ed = k\phi +1$. Now, if $gcd(m,p) = 1$ then my Fermat's little theorem

$m^{p-1} \equiv 1 \pmod p$

Raising both sides of this congruence to the power $k(q-1)$ and then multiplying both sides by $m$ yields

$m^{k(p-1)(q-1)+1} \equiv m \pmod p$

On the other hand, if $gcd(m,p) = p$, then this last congruence is again valid since each side is congruent to $0$ modulo $p$. Hence in all cases

$m^{ed} \equiv m \pmod p$

By the same argument,

$m^{ed} \equiv m \pmod q$

Finally, since $p$ and $q$ are distinct primes, it follows that

$m^{ed} \equiv m \pmod n$

This emphasises the reason for the use of $k\phi + 1 = k(p-1)(q-1)+1$ (or $k\lambda+1$ in Ilmari answer).

And on the use of $\phi$ over $\lambda$:

8.5 Note (universal exponent) The number $\lambda = lcm(p-1,q-1)$, sometimes called the universal exponent of $n$, may be used instead of $\phi = (p-1)(q-1)$ in the RSA key generation. Observe that $\lambda$ is a proper divisor of $\phi$. Using $\lambda$ can result in a smaller decryption $d$, which may result in faster decryption (cf. Note 8.9). However, if $p$ and $q$ are chosen at random, then $gcd(p-1,q-1)$ is expected to be smallm and consequently $\phi$ and $\lambda$ will be roughly the same size.

8.9 Note (small encryption exponents)
(i) If the encryption exponent $e$ is chosen at random, then RSA encryption using the repeated square-and-multiply algorithm takes $k$ modular multiplications and an expected $k/2$ (less with optimizations) modular multiplications, where $k$ is the bitlength of the modulus $n$. Encryption can be sped up by selecting $e$ to be small and/or by selecting $e$ with a small number of 1's in its binary representation. [$\ldots$] Another encryption exponent used in practice is $e = 2^{16}+1 = 65537$. This number has only two 1's in its binary representation, so encryption using the repeated square-and-multiply algorithm requires only 16 modular squaring and 1 modular multiplication.

Answer 3

The OP asks two questions. The first question is:

After we calculated $N = p * q$, we calculate $\varphi(N)$ and use it later to determine $e$ (PR) and $d$ (PU). But why?

This is exactly the prescription on page 6 of the original RSA paper, where $n=p\cdot q$ is the product of two (very large) prime numbers, and, hence the number of integers relatively prime to $n,$ or Euler's totient function, is the multiplication $\varphi(n)=(p-1)\cdot(q-1).$ From the RSA paper:

You then pick the integer $d$ to be a large, random integer which is relatively prime to $(p − 1) · (q − 1).$ That is, check that $d$ satisfies: $\gcd(d,(p − 1) · (q − 1)) = 1.$

There are two points to explain in the way the OP is formulated. Firstly, the introduction of Euler's totient function stems from Fermat-Euler's theorem. Again quoting the RSA original paper, page 7:

We demonstrate the correctness of the deciphering algorithm using an identity due to Euler and Fermat: for any integer (message) $M$ which is relatively prime to $n,$ $$M^{\varphi(n)}\equiv 1 \pmod n$$

Multiplying each side by $M,$ and rearranging:

$$\begin{align} M^{k\,\varphi(n)}& \equiv 1 \pmod n\\ M\cdot M^{k\,\varphi(n)}& \equiv M \cdot 1\pmod n\\ M^{k\,\varphi(n)+1}& \equiv M \pmod n \end{align}$$

we get to the last equation showing that the message to encrypt ($M$) is unchanged under modular exponentiation by multiples ($k$) of Euler's totient function of $n$ plus $1$, i.e. $\varphi(n) +1.$ This is great news, because we can figure out an encryption ($e$) and decryption ($d$) set of keys such that

$$e\cdot d = k\,\varphi(n) +1$$

as $$d = \frac{k\,\varphi(n)+1}{e}.$$

Of note, $k\,\varphi(n) +1,$ with $k$ being an integer, is the mathematical formulation of $1 \pmod{\varphi(n)},$ or, equivalently, $$e\cdot d\equiv 1 \pmod{\varphi(n)}\tag 1.$$

[See below the post for a toy example of a manual solution.]

Therefore modular exponentiation of the message $M$ will render the original message if both keys, $e$ and $d,$ are known - in this way, one key can be made public, while the other key is kept private:

$$\begin{align} M^{k\,\varphi(n)+1}& \equiv M \pmod n\\ M^{e\cdot d}&\equiv M \pmod n \end{align} $$

As on the first quote from the RSA article, $d$ needs to be coprime to $\varphi(n)$ precisely so that (from the RSA paper):

...it has a multiplicative inverse e in the ring of integers modulo $\varphi (n).$

Hence, allowing a solution to Eq. (1).

This introduces the abstract algebra concept of the finite ring of integers modulo $\varphi(n),$ which can represented as $\mathbb Z/\varphi(n)\mathbb Z.$ At first sight this is scary, but it is simply saying that the set of integers in modular arithmetic form a finite ring with the operations of addition and multiplication, whereby an element of the set will have a multiplicative inverse provided it is coprime to the modulus. From Wikipedia:

A modular multiplicative inverse of an integer $a$ with respect to the modulus $m$ is a solution of the linear congruence $ax\equiv 1\pmod {m}.$ [...] a solution exists if and only if $\gcd(a, m) = 1,$ that is, $a$ and $m$ must be relatively prime (i.e. coprime).

For example, in the ring of of integers modulus $10,$ i.e. $\mathbb Z/10\mathbb Z=\{0,1,2,3,\dots,9\},$ the element $9$ being coprime to $10$ secures a multiplicative inverse, i.e. $9\cdot 9 =81\equiv 1 \pmod {10}.$

The idea is that modular exponentiation of $M$ to $e\cdot d$ equals exponentiation to $1,$ returning the original message.

The second question in the OP was:

And why does e need to be smaller than $\varphi(N)$?

follows as $e$ and $d$ are elements of the ring of integers modulus $\varphi (n),$ that is $e,d\in \mathbb Z/\varphi(n)\mathbb Z.$

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Manual example:

Let's take $p=13$ and $q=23,$ yielding $n=299.$ The totient function is $\varphi(299)=12\times 22 = 264.$

To select the $e$ value we need a coprime to $\varphi(n)=264.$ Some of the coprime values of $264$ are $245, 247, 251, 257, 259,...$ If we select $e=245,$ the linear congruency to find a corresponding $d$ can be expressed as

$$245 d = 1 + 264k$$

This is equivalent to

$$245d + 264k =1\tag {*}$$

since is an arbitrary integer, $k\in \mathbb Z,$ and the rearrangement amounts to a change of sign, which wouldn't influence clock arithmetic.

Given that the values in equation $(*)$ are coprime, the expression amounts to Bézout's identity, $245x+264y=\gcd(245,264),$ and we can use the extended Euclidean theorem. This is explained in an example on this post.

Dividing the larger of the values ($\color{blue}{264}$) by the smaller value ($\color{magenta}{245}$) in the LHS of $(*),$ i.e. $\color{magenta}{245}d + \color{blue}{264}k =1,$ and keeping tally of the multiples in parenthesis, e.g. $\small\text{Dividend}=\text{Divisor}(\text{Quotient})+\text{Remainder}:$

$$\begin{align} \frac{\color{blue}{264}}{\color{magenta}{245}}=\color{tan}1{\small\text{, Rm }}\color{red}{19} \implies&\color{blue}{264}(1) = \color{magenta}{245}(\color{tan}1) + \color{red}{19} \\[2ex] \frac{\color{magenta}{245}}{\color{red}{19}}=\color{tan}{12} {\small\text{, Rm }}\color{purple}{17} \implies& \color{magenta}{245}(1) = \color{red}{19}(\color{tan}{12}) + \color{purple}{17} \\[2ex] \frac{\color{red}{19}}{\color{purple}{17}}=\color{tan}{1} {\small\text{, Rm }}\color{orange}{2} \implies& \color{red}{19}(1) = \color{purple}{17}(\color{tan}1) + \color{orange}2 \\[2ex] \frac{\color{purple}{17}}{\color{orange}{2}}=\color{tan}{8} {\small\text{, Rm }}\bf{1} \implies&\color{purple}{17}(1) = \color{orange}2(\color{tan}8) +\bf 1 \end{align}$$

Moving the remainders to the RHS...

$$\begin{align} \color{red}{19} &= 264(1) + 245(-1)\\ \color{purple}{17} &= 245(1) + 19(-12)\\ \color{orange}2 &= 19(1) + 17(-1) \\ \bf 1 &= 17(1) + 2(-8) \end{align}$$

Progressively linking these equations by substitution from the last one to the first, and distributing and rearranging terms...

$$\begin{align} \bf 1 &= 17(1) + \color{orange}2(-8)\\ &= 17(1) + \color{orange}{[19(1) + 17(-1)]}\bf{(-8)}\\ &= 17(1) + [19{\bf(-8)} + 17{\bf(8)}]\\ &= \color{purple}{17}(9) + 19(-8)\\ &= \color{purple}{[245(1) + 19(-12)]}{\bf(9)} + 19(-8)\\ &= [245{\bf(9)} + 19{\bf(-108)}] + 19(-8) \\ &= 245(9) + \color{red}{19}(-116) \\ &= 245(9) + \color{red}{[264(1) + 245(-1)]}{\bf(-116)}\\ &= 245(9) + [264{\bf(-116)} + 245{\bf(116)}]\\ &= \color{magenta}{245}(125) + \color{blue}{264}(-116) \end{align}$$

Comparing this last equation to $(*),$ the value of $d=125.$ And indeed, $245 \times 125 \pmod {264} = 1.$ The value $k=-116$ just spins the wheels on the clock face, and it is a single example of the general solution $1=245\times 125+264k.$

If we want to pass along the message "Hi", composed of the 8th and 9th letters of the alphabet, i.e. $89,$ we use the public key in the example, $(e,n)=(245, 299),$ and send the message $89^{245} \pmod {299}=111,$ which will be decrypted by the receiver by using the private key, $(d,n)= (125,299),$ exponentiating $111^{125}\pmod{299}=89,$ in other words, "Hi."

Answer 4

Why?

$\varphi(N)$, in the original RSA specification, works because it is a multiple of $\lambda(N)$.

Exponentiation of ring $R_N$ creates a period of length $\lambda(N)$. The cycle of this period starts as $m^0\equiv1 \pmod{N}, m^1\equiv m \pmod{N},...$

Using any multiple $k\lambda(N)$, including $\varphi(N)$, to compute the multiplicative inverse $d$ of $e$ can be viewed two ways. The first, $ed \equiv 1 \pmod{\lambda(N)}$ results in $m^{k\lambda(N)+1} \equiv m^{ed} \equiv m^1 \equiv m \pmod{N}$. And second, $ed=k\lambda(N)+1$ where $k\lambda(N)$ is a multiple of the period and $+1$ then becomes the second element of the cycle, aka $m^1 \equiv m \pmod{N}$.

Why do we need $\varphi(N)$ to compute $d$?

As noted above, any multiple $k\lambda(N)$, including $\varphi(N)$, can be used to compute a valid $d$, and each result may be unique. Because, $k\lambda(N)$ is a multiple of the period and $ed$ is the first element in the cycle of the period. Therefore, $m^{k\lambda(N)+1} \equiv m^{ed} \equiv m^1 \equiv m \pmod{N}$.

You can us any number (almost) to create a multiplicative inverse, however the resulting $ed$ will not align with the period. Therefore will not recover $m$.

Why $e$ needs to be smaller that $\varphi(N)$?

Numbers are typically smaller than the modulus, $\varphi(N)$ in this case. Though, technically, it doesn't need to be. It does help the computation cost by keeping these exponents smaller. It also makes sense that when smaller than $\varphi(N)$, computing $e$ as the multiplicative inverse of $d$ will result in the same $e$.