Updating headings on every page

content/posts/rsa-optimization.md

@@ -4,7 +4,7 @@ date: 2022-12-06
 draft: false
 ---
 
-# INTRODUCTION
+## INTRODUCTION
 
 RSA is a public key cryptosystem, which was named after the creators of
 the algorithm: Rivest, Shamir, and Adleman [@STALLINGS]. It is widely
@@ -55,7 +55,7 @@ decrypting messages. However the same instruction set architecture we
 propose in this paper can be used for signing and verifying messages
 with RSA.
 
-# CHARACTERISTICS OF RSA
+## CHARACTERISTICS OF RSA
 
 There are three areas that the RSA can be optimized: finding the
 encryption and decryption exponent, prime number generation, and
@@ -64,7 +64,7 @@ finding the encryption and decrypting exponent, prime number generation,
 and encrypting and decrypting the message is usually done, without
 specialized instructions.
 
-## ENCRYPTION AND DECRYPTION EXPONENT
+### ENCRYPTION AND DECRYPTION EXPONENT
 
 The current approach to verifying that the encryption exponent is
 coprime to the `φ(n)` is by using the Euclidean Algorithm. To find
@@ -150,7 +150,7 @@ we will use two instructions that will combine some of the instructions
 used in the Extended Euclidean Algorithm to reduce the number of stalls
 within the loop.
 
-## PRIME NUMBER GENERATION
+### PRIME NUMBER GENERATION
 
 The common approach to generating large primes in making encryption and
 decryption keys is to randomly select integers and test them for
@@ -199,7 +199,7 @@ shown below). The instructions to calculate `x^{2} (mod n)` would be
 executed `s` times. These two factors indicate a heavy reliance on the
 ability of a system to calculate exponentiation.
 
-## ENCRYPTION AND DECRYPTION
+### ENCRYPTION AND DECRYPTION
 
 One aspect of RSA to improve upon is performing large exponentiation.
 Currently the implementation of exponentiation is performed by the
@@ -222,13 +222,13 @@ to be available. We can lessen the number of stalls using a technique
 known as exponentiation by squaring. This technique is explained further
 in the design section.
 
-# DESIGN
+## DESIGN
 
 In this section, we will describe specialized instructions that will be
 used for prime number generation, computing the encryption and
 decryption exponent, and encrypting a decrypting a message.
 
-## ENCRYPTION AND DECRYPTION EXPONENT
+### ENCRYPTION AND DECRYPTION EXPONENT
 
 The issue with implementing the Euclidean Algorithm the traditional way
 is a divide, multiply, and subtract instructions are needed for each
@@ -320,7 +320,7 @@ an analysis to the speedup given to the Extended Euclidean Algorithm by
 using the modular instruction and the multiply-subtract instruction in
 the justification and analysis section.
 
-## PRIME NUMBER GENERATION, ENCRYPTION, AND DECRYPTION
+### PRIME NUMBER GENERATION, ENCRYPTION, AND DECRYPTION
 
 One issue already discussed in the previous section is that of stalls
 during large exponents. The way exponentiation is handled causes many
@@ -375,12 +375,12 @@ depending on the digit, may use the second accumulator. It will then run
 through one more multiplier to multiply the squares by the 1's
 multiplier.
 
-# JUSTIFICATION AND ANALYSIS
+## JUSTIFICATION AND ANALYSIS
 
 In this section, we will describe how our specialized instructions will
 improve the performance of the RSA encryption.
 
-## ENCRYPTION AND DECRYPTION EXPONENT
+### ENCRYPTION AND DECRYPTION EXPONENT
 
 Using the modular instruction in the Euclidean Algorithm, we can reduce
 the number of stalls needed. Instead of needing to stall for the result
@@ -441,7 +441,7 @@ speedup of 1.23. Also, an advantage to using the modular and
 multiply-subtract instructions is we reduce the number of temporary
 registers needed from five to three.
 
-## PRIME NUMBER GENERATION, ENCRYPTION, AND DECRYPTION
+### PRIME NUMBER GENERATION, ENCRYPTION, AND DECRYPTION
 
 Using the pow command we can reduce stalls of large exponents in half.
 Since the algorithm breaks the exponent into a binary representation of
@@ -472,7 +472,7 @@ the following equations to determin the overall speedup:
     Speedup = 1.25/1.125
     Speedup = 1.11
 
-# CONCLUSIONS
+## CONCLUSIONS
 
 In analyzing the typical algorithms used as a part of RSA, we have
 identified two primary bottlenecks in both encryption and decryption:
@@ -502,7 +502,7 @@ system with the sole task of encrypting and decrypting messages using
 RSA, we can create hardware that allows the specialized instructions to
 have a latency comparable to the traditional instructions.
 
-# Bibliography
+## Bibliography
 
 3 Beauchemin, Pierre, Brassard, Crepeau, Claude, Goutier, Claude, and
 Pomerance, Carl The Generation of Random Numbers That Are Probably Prime