Computing the Square Roots of Unity to break RSA using Quantum Algorithms

Computing the Square Roots of Unity to
break RSA using Quantum Algo
Dr. Dharma Ganesan, Ph.D.,

Disclaimer
● The opinions expressed here are my own
○ But not the views of my employer
● The source code fragments and exploits shown here can be reused
○ But without any warranty nor accept any responsibility for failures
● Do not apply the exploit discussed here on other systems
○ Without obtaining authorization from owners
2

Goal
● Break RSA using Peter Shor’s quantum algorithm
○ But, we will not look into Quantum Computing
○ We will find the square roots of unity on a classical computer
● Investigate the square root of unity (i.e., sqrt(1)) in a Finite Group
○ Algorithm 1: Naïve/Bruteforce
○ Algorithm 2: Random Search
○ Algorithm 3: Peter Shor’s period calculation algorithm
○ Algorithm 4: Our little tweaks of Peter Shor’s period function (bounded search)
3

Prerequisite
Some familiarity with the following topics will help to follow the rest of the slides
● Group Theory
● Number Theory
● Algorithms and Complexity Theory
● If not, it should still be possible to obtain a high-level overview
4

Agenda
● Mathematical foundation/facts
● Brief overview of RSA
● Different algorithms to find the square roots of unity (in a finite group)
● Demos of finding the square roots of unity to factor composite numbers
● Conclusion
5

6
Notations and Facts
GCD(x, y): The greatest common divisor that divides integers x and y
Co-prime: If gcd(x, y) = 1, then x and y are co-primes
Zn
= { 0, 1, 2, …, n-1 }, n > 0; we may imagine Zn
as a circular wall clock
Z*
n
= { x ∈ Zn
| gcd(x, n) = 1 }; (additional info: Z*
n
is a multiplicative group)
φ(n): Euler’s Totient function denotes the number of elements in Z*
n
φ(nm) = φ(n).φ(m) (This property is called multiplicative)
φ(p) = p-1, if p is a prime number

Notations and Facts ...
● x ≡ y (mod n) denotes that n divides x-y; x is congruent to y mod n
● Euler’s Theorem: aφ(n)
≡ 1 (mod n), if gcd(a, n) = 1
● Fermat’s Little Theorem: ap
≡ a (mod p)
● Gauss’s Fundamental Theorem of Arithmetic: Any integer greater than 1 is
either a prime or can be written as a unique product of primes
○ Euclid’s work is the foundation for this theorem, see The Elements
● Euclid’s Lemma: if a prime p divides the product of two natural numbers a
and b, then p divides a or p divides b
● Euclid’s Infinitude of Primes (c. 300 BC): There are infinitely many primes
7

How can Bob send a message to Alice securely?
8
Public Key PuA
● Alice and Bob never met each other
● Bob will encrypt using Alice’s public key
○ Assume that public keys are known to the world
● Alice will decrypt using her private key
○ Private keys are secrets (never sent out)
● Bob can sign messages using his private key
○ Alice verifies message integrity using Bob’s public key
● Note: Alice and Bob need other evidence (e.g., passwords,
certificates) to prove their identity to each other
● Who are Alice, Bob, and Eve?
Private Key PrA
Public Key PuB
Private Key PrB

RSA Public Key Cryptography System
● Published in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman
● Rooted in elegant mathematics - Group Theory and Number Theory
● Core idea: Anyone can encrypt a message using recipient's public key but
○ (as far as we know) no one can efficiently decrypt unless they got the matching private key
● Encryption and Decryption are inverse operations (math details later)
○ Work of Euclid, Euler, and Fermat provide the mathematical foundation of RSA
● Eavesdropper Eve cannot easily derive the secret (math details later)
○ Unless she solves “hard” number theory problems that are computationally intractable
9

RSA - Key Generation Algo
1. Select an appropriate bitlength of the RSA modulus n (e.g., 2048 bits)
○ Value of the parameter n is not chosen until step 3; small n is dangerous (details later)
2. Pick two independent, large random primes, p and q, of half of n’s bitlength
○ In practice, q < p < 2q to avoid attacks (e.g., Fermat’s factorization)
3. Compute n = p.q (n is also called the RSA modulus)
4. Compute Euler’s Totient (phi) Function φ(n) = φ(p.q) = φ(p)φ(q) = (p-1)(q-1)
5. Select numbers e and d from Zn
such that e.d ≡ 1(mod φ(n))
○ e must be relatively prime to φ(n) otherwise d cannot exist (i.e., we cannot decrypt)
○ d is the multiplicative inverse of e in Zn
6. Public key is the pair <n, e> and private key is 4-tuple <φ(n), d, p, q>
10

RSA Trapdoor
● RSA: Zn
→ Zn
● Let x and y ∈ Zn
● y = RSA(x) = xe
mod n
○ We may view x as a plaintext, and y as the corresponding ciphertext
● x = RSA-1
(y) = yd
mod n
● e and d are also called encryption and decryption exponents, respectively
● In practice, some random padding is used (not relevant for this presentation)
11

RSA Trapdoor variables’ dependency graph
12
Private variable
Public variable
Note: If we can factor n and find p and q, then private keys are open to the world

What are the square roots of unity in Z?
● 1 and -1 are the two square roots of one because (±1)2
= 1
● This is true only if we work in the infinite set of integers Z
● In Cryptography, we often work with finite groups
● There are other non-trivial roots of unity other than ±1 in Z*
n
13

Square roots of unity in a finite group Z*
n
Consider this tiny multiplicative group Z*15
= {1, 2, 4, 7, 8, 11, 13, 14}
The square roots of unity are 1, 4, 11, and 14
Note that 14 ≡ -1 mod 15 (basically -1 is the same as 14)
But, we have two other roots 4 and 11 of unity (e.g., 4*4 = 1 mod 15)
1 and 14 are trivial roots of unity, whereas 4 and 11 are non-trivial roots of unity
Fact: Let n = pq, where p and q are odd primes, then there are four roots of unity
14

Algorithm: Factor n using the square roots of unity
Problem: Given a composite number n and sqrt(1), find the factors p and q of n
Let x be a square root of 1, then x2
= 1 (mod n)
Thus, x2
- 1 = 0 (mod n) ⇒ (x+1)(x-1) = 0 (mod n)
This means that n divides (x+1)(x-1)
If (x+1) and (x-1) are not multiples of n, then the factors of n are the following:
p = gcd(x+1, n) or gcd(x-1, n); q = n/p
15

Toy example: Let’s factor n = 15 using Z*15
Z*15
= {1, 2, 4, 7, 8, 11, 13, 14}
Since 4 and 11 are square roots of unity, we can use one of them to factor n
Let x = 4
x2
= 1 (mod 15)
(x+1)(x-1) = 0 (mod 15)
Factor p = gcd(x+1, 15) = gcd(5, 15) = 5; Factor q = n/p = 15/5 = 3
16

Factoring n using the square roots of unity ...
If we can find the non-trivial square roots on n, then we can factor n
Let’s investigate how easy it is to find the sqrt(1) in a finite multiplicative group
Four different algorithms to find the sqrt(1):
1. Bruteforce/Naive
2. Random Search
3. Peter Shor’s Quantum Algorithm
4. Our little tweaks of Shor’s algorithm
17

Algorithm 1: Naïve sqrt(1) in Zn
*
Step 1: x = 0
Step 2: if(x < n) goto step 3; otherwise goto step 5
Step 3: if(x2
= 1 mod n) then x is a square root of one.
Step 4: x = x + 1. goto step 2.
Step 5: End.
18

19
~/crypto/RSA$ java SqrtOneNaive 2086784737
Input bit length = 31
Sqrt(n) = 45681
sqrt(1) = 1 bit length = 1
sqrt(1) = 530886943 bit length = 29
sqrt(1) = 1555897794 bit length = 31
sqrt(1) = 2086784736 bit length = 31
Note: sqrt(n) is only 45681, but sqrt(1) can be the same bit length as n
(See the Appendix for the source code of Sqrt(1) naïve method)
Algorithm 1: Find the sqrt(1) in the group Z*2086784737

Algorithm 1: Analysis of the naïve algorithm
● The naïve algorithm is simple to understand and implement
● However, it is very slow because we enumerate all numbers of the group Zn
● If n is a 2048-bit composite number, this algorithm will perform 22048
iterations
● In other words, it will take several billion years to calculate sqrt(1)
20

Algorithm 2: Random search to find sqrt(1) in Zn
*
Step 1: Pick a random x from Zn
Step 2: if(x = ± 1 mod n) then goto step 1
Step 3: if(x2
= 1 mod n) then x is a sqrt(1) goto step 4; else step 1
Step 4: End.
21

22
dharmalingam_ganesan11@instance-1:~/crypto/RSA$ java RandomSqrtOne 45649
sqrt(1) = 43738
Number of random tries = 236
ddharmalingam_ganesan11@instance-1:~/crypto/RSA$ java RandomSqrtOne 45649
sqrt(1) = 43738
dharmalingam_ganesan11@instance-1:~/crypto/RSA$ java RandomSqrtOne 45649
sqrt(1) = 43738
Algorithm 2: Find the sqrt(1) in the group Z*45649

23
Algorithm 2: Analysis of Random Search Algo
● The random algorithm can find the square roots of one (for “small” n)
● But, it just needs a lots of tries to find a needle in the haystack
● In general, the number of tries is the size of the input composite number
● Random search is not able to find sqrt(1) for very large input n

25
Algorithm 3: Peter Shor’s Algorithm (core idea)
● To find an x such that x2
= 1 (mod n), we compute f(x) = ax
mod n, random a
● Fact: Every element of a finite group generates a cyclic subgroup
● In other words, the function f(x) is the same as the cyclic subgroup <a>
● Thus, f(r) = 1 mod n for some r; equivalently, ar
= 1 mod n for some r
● Once we find such an r, (ar/2
)2
= 1 mod n; (if r is even, the hacker is lucky)
● Thus, ar/2
is the sqrt(1); we know how to factor n once the sqrt(1) is exposed!

Algorithm 3: Finding the period r of the function f(x)
26
x f(x) = ax
mod n
0 a0
mod n
1 a1
mod n
2 a2
mod n
r ar
mod n = 1
● Peter Shor uses quantum computing
to find the period r of the function f(x)
● We implemented this function on my
classical computer (my laptop)
● Just to better understand the inner
beauty of this algorithm

27
~/crypto/RSA$ java -ea SqrtOneShor 45649
period = 1330
sqrt(1) = 1911
period = 4522
sqrt(1) = 43738
period = 22610
sqrt(1) = 1911
period = 22610
sqrt(1) = 1911
period = 4522
sqrt(1) = 1911
period = 22610
sqrt(1) = 43738

28
~/crypto/RSA$ time java -ea SqrtOneShor 1421727721
period = 236941974
Sqrt(1) = 521638680
real 0m47.429s
user 0m47.100s
sys 0m0.288s Period is very large

29
~/crypto/RSA$ java FactorShor 45649
period = 22610
p = 191
q = 239
period = 170
p = 239
q = 191
period = 3230
p = 239
q = 191
Algorithm 3: Factor a small number n = 45649

30
Algorithm 3: Analysis of the period of f(x) = ax
mod n
● The above set of runs shows that the period of f(x) can be very large
○ Thus, we cannot apply this algorithm to find sqrt(1) for a large group Zn
used in RSA
● Remember, we are running the algorithm on a classical computer
● It is good to play with this algorithm on a classical computer to understand it
● Algorithm 4 talks about some little tweaks of Algorithm 3

Algorithm 4: Let’s tweak Peter Shor’s algorithm
What if we put a bound b on the period r?
Step 1: Initialize bound b (e.g., b = n1/2
, n1/4
, etc.), r = 0, success = false
Step 2: Pick a random a from Zn
*
Step 3: if 0 ≤ r ≤ b goto step 4; otherwise goto step 7
Step 4: Compute f(r) = ar
mod n
Step 5: If f(r) != 1 then r = r+1 and goto step 3
Step 6: If r is even, then success = true; otherwise goto step 2
Step 7: If success = true, we found the period r; otherwise goto step 2
31

32
period = 33738
sqrt(1) = 521638680
real 0m45.786s
user 0m45.510s
sys 0m0.244s
Algorithm 4: bound = n1/2

33
period = 18
Square root of one = 521638680
real 6m14.713s
user 6m12.706s
sys 0m1.759s

34
period = 18
real 2m29.435s
user 2m28.400s
sys 0m0.936s

35
period = 6
real 2m31.106s
user 2m29.703s
sys 0m1.304s
Amazing, we found a cyclic subgroup of order 6

Algorithm 4: Analysis of our tweaks
● Z*
n
has even number of elements; there must be cyclic subgroups for each
divisor of |Zn
*|. (This is a theorem in Group Theory)
● The open question is how easy it is to find small cyclic subgroups of Z*n
● If we quickly find one cyclic subgroup, then sqrt(1) are easily revealed
● If we put a bound b on the period r, then sqrt(1) runs quick for a small n
● Our bounded random search variant of Shor’s algorithm is slow for a large n
36

Conclusion
● We studied the problem of finding the square roots of unity in a finite group
● We tried to factor composite numbers using sqrt(1)
● Experimental results showed that finding sqrt(1) in Z*n
where n = pq is “hard”
● We implemented Shor’s function f(x) = ax
mod n on a classical computer
● We learned that in general the period of f(x) is the same size as n
● We made minor tweaks by defining a bound on the period of f(x) = ax
mod n
● More experimentation of our tweaks is part of future work
37

39
import java.math.BigInteger;
import java.security.SecureRandom;
public class SqrtOneNaive implements IBigIntConstants
{
public static void main(String[] args)
{
/* Error handling is omitted */
BigInteger n = new BigInteger(args[0]);
BigInteger x = zero;
while(x.compareTo(n) <= 0) {
// is square(x) = 1 (mod n)?
if(x.multiply(x).mod(n).equals(one)) {
System.out.println(x + " is the square root of one ");
}
x = x.add(one);
}
}
}
Implementation of naïve Sqrt(1) algorithm

40
public static void main(String[] args)
{
BigInteger n = new BigInteger(args[0]); // Error handling code is removed
BigInteger x;
while(true) {
// Pick a random element from Zn
x = BigIntUtil.getRandomElt(n, new SecureRandom());
// We don't want non-trivial roots '1' or '-1'.
// If x = -1 (mod n) or x = 1 (mod n), then we have to restart.
if(x.add(one).mod(n).equals(zero) ||
x.subtract(one).mod(n).equals(zero)) { continue; }
if(x.multiply(x).mod(n).equals(one)) {
System.out.println(x + " is the square root of one ");
System.out.println("Number of random tries = " + counter);
break;
}
}
}
Implementation of random Sqrt(1) algorithm

41
Finding the Sqrt(1) using Peter Shor’s
algorithm (simplied)

References
● W. Diffie and M. E. Hellman, “New Directions in Cryptography,” IEEE
Transactions on Information Theory, vol. IT-22, no. 6, November, 1976
● R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital
signatures and public-key cryptosystems,” CACM 21, 2, February, 1978
● A. Menezes, P. van Oorschot, and S. Vanstone, “Handbook of Applied
Cryptography,” CRC Press, 1996
● P. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete
Logarithms on a Quantum Computer,” SIAM J.Sci.Statist.Comput. 26 (1997).
42

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