Tiny Crypto – Exploring Cryptography with Small Prime Fields

A small-scale cryptography playground using tiny prime fields for easy manual verification.

Overview

TinyCrypto is an educational project that implements fundamental cryptographic algorithms using small prime fields. The goal is to simplify the math and make manual verification feasible, allowing learners to focus on concepts without getting lost in large numbers.

The project includes:

• Elliptic Curve Cryptography (ECC) – Weierstrass curves, point arithmetic, and scalar multiplication.
• Finite Field Arithmetic – Type-safe 𝔽ₚ implementation with modular ops and field-specific overloads.
• Shamir's Secret Sharing – Securely split and reconstruct secrets.
• Distributed Key Management – Collaborative cryptographic protocols.
• Other Cryptographic Routines – Prime testing, modular inverse, quadratic residue checks, and more.

Features

• Strong finite field typing via Fp{T,P}, ensuring safe and consistent modular arithmetic.
• Elliptic curve operations over tiny prime fields:
  • Curve generation over 𝔽ₚ for small primes
  • Point addition, doubling, scalar multiplication
  • Generator detection and validation
  • Point-at-infinity and ECPoint construction
• Clean ANSI-formatted output for symbolic math (e.g., 𝔽₃₁, 𝔾(6,2))
• Fully written in Julia, for clarity and performance.
• Educational focus – Ideal for teaching and cryptographic exploration.

Example Usage

using TinyCrypto
import TinyCrypto: is_identity

# Find a suitable Weierstrass curve over small field primes and coefficient ranges with iterative method
# general syntax: curve-name(prime-field-range, parameter-range₁, ..., parameter-rangeₙ, max_cofactor=8)
curve = Weierstrass(97:103, 10:15, 2:7)  # (prime range, a range, b range)
# Output: Weierstrass{𝔽₉₇}: y² = x³ + 10x + 3 | 𝔾(0,10), q = 101, h = 1, #E = 10

E = curve_points(curve) # Get all curve points
# → 101-element Vector{ECPoint₁₂₈}: (0𝔽₉₇,10𝔽₉₇), (0𝔽₉₇,87𝔽₉₇), ..., (96𝔽₉₇,63𝔽₉₇), (∞,∞)
S = subgroup_points(curve) # same as `curve_points` as #E(𝔽₉₇) ∈ primes

curve = Montgomery(30:40, 8:40, 3:40)
# Montgomery{𝔽₃₇}: 8y² = x³ + 3x² + x | 𝔾(15,2), q = 11, h = 4, #E = 44
S = subgroup_points(curve) # when cofactor greater than one, there are multiple sub groups
E = curve_points(curve)

curve = TwistedEdwards(50:100, 1:20, 1:20)
# TwistedEdwards{𝔽₉₇}: 3x² + y² = 1 + 10x²y² | 𝔾(48,93), q = 29, h = 4, #E = 116

curve = Edwards(50:100, 1:20, max_cofactor=8)
# Edwards{𝔽₇₉}: 1x² + y² = 1 + 6x²y² | 𝔾(7,58), q = 23, h = 4, #E = 92
is_singular(curve) # false

## Point arithmetic on curve
𝔾 = curve.G                          # (7𝔽₇₉,58𝔽₇₉) ∈ Edwards{Fp{UInt128, 79}}
𝔾 + 𝔾 + 𝔾 + 𝔾                        # (68𝔽₇₉,67𝔽₇₉)  point addition on a cyclic group
2𝔾                                   # (31𝔽₇₉,51𝔽₇₉) 
2𝔾 == 𝔾 + 𝔾                          # true

𝒪 = identity(curve)                   # identity point on Edwards curve variant: (0𝔽₇₉,1𝔽₇₉)
𝒪 + 𝔾 == 𝔾                            # true
𝔾 + 𝒪 == 𝔾                            # true  

# direct construction from paramters:
const 𝔽₃₁ = 𝔽ₚ{UInt8, 31}                   # define field prime 𝔽ₚ for the abelian group, or use non-unicode `Fp{base_type, prime_number}`
Weierstrass{𝔽₃₁}(6, 9, 37, 1, (0,3))        # Weierstrass{𝔽₃₁}: y² = x³ + 6x + 9 | 𝔾(0,3), q = 37, h = 1, #E = 37
# or pass the 𝔽ₚ field prime directly as in:
curve = Weierstrass(31, 6, 9, 37, 1, (0,3)) # Weierstrass{𝔽₃₁}: y² = x³ + 6x + 9 | 𝔾(0,3), q = 37, h = 1, #E = 37

𝒪 = identity(curve)                    # (∞,∞) Weierstrass curve identity is infinity
is_identity(𝒪)                         # true 
∞ = infinity(curve)                    # (∞,∞) as expected
is_infinity(∞)                         # true

E = curve_points(curve)                # 
is_point_on_curve(E[1], curve)         # true 
is_point_on_curve(curve.G, curve)      # true, G generator point defines the cyclic group, which in this case the entire abelien group 



## Tiny Hash (not for cryptographic use, for obvious reasons)
H("byte size hash of a string")        # hashes string to a byte, given the abelian group is byte size
H₁₆("16 bit hash")                     # in case you need more hash space
H₈("is same") == H("is same")          # true, it is just a alias, smae as H8 

## ECDSA Signature and Public Key Recovery Example
priv, pub = genkey(curve)                     # Key generation
msg = "hello ethereum"                        # Message to sign
signature = sign(curve, priv, msg)            # returns a NamedTuple (r = ..., s = ..., v = ...)
is_valid = verify(curve, pub, signature, msg) # Signature verification true
recovered = ecrecover(curve, msg, signature)  # Public key recovery from signature
@assert pub == recovered                      # Check if recovered public key matches original

Installation

using Pkg
Pkg.add(url="https://github.com/vargaconsulting/tiny-crypto.git")

Development

using Pkg
Pkg.develop(url="https://github.com/vargaconsulting/tiny-crypto.git")

Please follow these steps:

1. create new issue ie: gh issue create
2. branch on top of release tag: gh issue develop your-new-isse
3. for commits follow pattern: [#14]:svarga:feature, short description of your change
4. rebase to latest release tag and manage conflicts if any
5. upload changes: git push -u origin 14-svarga-feature-short-name
6. submit pull/merge request gh pr create --fill
7. We'll review your pull request and coordinate merging. Thanks for contributing!

Future Work

• Typed finite field system – 𝔽ₚ{T,P} representation for safe modular arithmetic
• Symbolic math output – pretty-printing with 𝔽ₚ, 𝔾(x,y) and Unicode formatting
• Robust EC point operations – scalar multiplication, inversion, curve validation
• Support for popular blockchain curves – secp256k1, ed25519, altbn128, ...
• Full signature schemes – ECDSA, EdDSA, Schnorr signatures
• Shamir’s Secret Sharing – secret splitting & reconstruction
• Proactive refresh & threshold MPC – secure key management with distributed trust
• Interactive playground – simulate distributed signing protocols, visually

Licence

MIT License. Copyright © 2025 Steven Varga, Varga Consulting, Toronto, ON, Canada 🍁 For questions, reach out at [email protected].